Mathematical Biosciences 242 (2013) 33–50

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Mathematical Biosciences

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Impact of enhanced malaria control on the competition between Plasmodiumfalciparum and Plasmodium vivax in India

Olivia Prosper ⇑, Maia MartchevaDepartment of Mathematics, University of Florida, 358 Little Hall, P.O. Box 118105, Gainesville, FL 32611–8105, United States

a r t i c l e i n f o

Article history:Received 13 September 2011Received in revised form 28 November 2012Accepted 30 November 2012Available online 19 December 2012

Keywords:MalariaVivaxFalciparumTwo-strain modelBasic reproduction numberInvasion numbers

0025-5564/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.mbs.2012.11.015

⇑ Corresponding author.E-mail address: olivia.f.prosper@dartmouth.edu (O

a b s t r a c t

The primary focus of malaria research and control has been on Plasmodium falciparum, the most severe ofthe four Plasmodium species causing human disease. However, the presence of both P. falciparum andPlasmodium vivax occurs in several countries, including India. We developed a mathematical modeldescribing the dynamics of P. vivax and P. falciparum in the human and mosquito populations and fit thismodel to Indian clinical case data to understand how enhanced control measures affect the competitionbetween the two Plasmodium species. Around 1997, funding for malaria control in India increaseddramatically. Our model predicts that if India had not improved its control strategy, the two species ofPlasmodium would continue to coexist. To determine which control measures contributed the most tothe decline in the number of cases after 1997, we compared the fit of seven models to the 1997–2010clinical case data. From this, we determined that increased use of bednets contributed the most to casereduction. During the enhanced control period, the best model predicts that P. vivax is out-competingP. falciparum. However, the reproduction numbers are extremely close to the invasion boundaries. Con-sequently, we cannot be confident that this outcome is the true future of malaria in India. We address thisuncertainty by performing a parametric bootstrapping procedure for each of the seven models. Thisprocedure, applied to the enhanced control period, revealed that the best model predicts that P. vivax out-competing P. falciparum is the most likely outcome, whereas the remaining candidate models predict theopposite. Moreover, the predictions of the top model are counter to what one expects based on the casedata alone. Although the proportion of cases due to falciparum has been increasing, the best fitting modelreveals that this observation is insufficient to draw conclusions about the longterm competitive outcomeof the two species.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Roughly 250 million people suffer from malaria infection eachyear, resulting in nearly one million deaths [41]. Malaria is the fifthleading killer among infectious diseases worldwide, and it is thesecond leading cause of death in Africa, behind HIV/AIDS [6]. De-spite many attempts at controlling malaria in India over the past60 years, India still produces roughly 70% of the malaria cases with-in Southeast Asia, resulting in about two million cases and 1000deaths each year. While mortality due to Plasmodium infection islow in India relative to the total morbidity, malaria still poses anenormous burden to the country. Several factors, including the biol-ogy and epidemiology of the disease, emerging drug-resistance ofparasites, insecticide resistance of mosquitoes, and socio-economicbarriers, have proven to be difficult obstacles to overcome in theongoing pursuit of malaria control.

ll rights reserved.

. Prosper).

1.1. Plasmodium vivax and Plasmodium falciparum parasites andobstacles they pose to malaria control

P. vivax and P. falciparum have very similar life cycles, with oneimportant exception. When a human is infected by a mosquitowith P. vivax, some of the parasites become hypnozoites, whichcan remain dormant in the human liver cells for some time, thenreactivate. Consequently, individuals infected with P. vivax areprone to relapses. In fact, P. vivax infections exhibit relapsesroughly 30% of the time after the initial clinical episode [1]. Fortu-nately, P. falciparum parasites do not have a hypnozoite stage, andthus relapses do not occur in falciparum infections.

Despite the absence of relapse in falciparum malaria infections,P. falciparum is associated with the highest risk of mortality for hu-mans among the malaria parasite species. Vivax infections are con-sidered to be benign, however the symptoms are still debilitatingand diminish both a person’s quality of life and their productivity.Moreover, some recent cases of P. vivax malaria have been far moresevere than is traditionally expected of this disease, sometimesresulting in death. The liver stages of P. vivax can also be extremely

34 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

long, up to three years, allowing P. vivax parasites to lay dormantand weather the low transmission seasons until conditions haveimproved, making vivax in some respects, a more formidable foethan P. falciparum in terms of malaria control.

Although symptoms due to malaria infection can be quitesevere and sometimes deadly, it is likely that, because multipleinfections can temporarily build a person’s immunity to the dis-ease, a large proportion of malaria cases in India are asymptomaticor display very mild symptoms, particularly in regions with meso-to hyper-endemicity [21]. A study of malaria infection in pregnantwomen in Jharkhand State, India, found that nearly half of the wo-men in the study carried asymptomatic malaria infection [18].Since asymptomatic human malaria infections are still infectiousto mosquitoes, unlikely to be treated, and consequently longer-lived, asymptomatic humans, in addition to liver stage vivaxinfected humans, create a reservoir for malaria parasites. Further-more, the long duration of untreated or unsuccessfully treatedinfections increases the likelihood of co-infection with P. vivaxand P. falciparum species. It can be very difficult to identify malariaco-infections because it is not yet very well understood how thetwo species interact. Co-infected individuals can also be very diffi-cult to treat because a drug that works for one infection may conferresistance in the other.

1.2. Objectives of modeling P. vivax–P. falciparum disease dynamics

In this paper, we develop a P. vivax–P. falciparum malaria modelwith co-infection to address questions regarding control measuresin the context of India. We want to find out what effect certain con-trol measures have on the competition between the two parasitespecies, how the presence of two circulating parasites affects whatcontrol measures should be implemented and how they can best beimplemented. The current literature on malaria in India dictatesthat there is a need to address not only P. falciparum malaria, whichis more commonly studied and modeled, but P. vivax as well. Chi-yaka et al. have published the first two-species malaria model,incorporating P. falciparum and P. malariae [7]. However, there isstill a need to model P. falciparum and P. vivax disease dynamics,particularly because the epidemiology of these two parasites is sodifferent. These differences, which are intrinsic to the parasite biol-ogy and are likely to greatly enhance the parasites’ ability to persistin a population in the face of numerous control efforts, need to beincluded in a mathematical model if we want to provide insight intoproblems regarding competition between parasite species in Indiaand how to develop an effective control policy for India. In Section2, we introduce the two-parasite ordinary differential equation ma-laria model. In Section 2.5 we present the disease-free equilibrium,the basic reproductive number for the model, and the control repro-ductive number. We present the isolated endemic equilibria of thesystem in Section 2.7 and a complete description and interpretationof the invasion numbers in Section 2.8.

Section 3 explains the parameters used in the model and the val-ues chosen for each parameter. In Section 3.3, for the 1987–1996period, we estimate transmission parameters by fitting the ODEmodel to Indian malaria case data. One of the assumptions of thisestimation procedure is that malaria transmission across India ishomogeneous. In actuality, the spatial distribution of malaria casesin India is heterogeneous. All states harbor malaria, but some mayhave more vivax than falciparum, and vice versa; the burden of ma-laria differs across the country as a consequence of many factors,including climate differences, differences in malaria vectors, and dif-ferences in drug resistance [9,10]. In addition to natural factors gen-erating heterogeneity in malaria burden across India, urbanizationand migration also contribute to the varied transmission landscape[9,10,30]. Furthermore, the Indian state, Orissa, contributes roughlya quarter of the cases to the national average [28,30]. Despite the

existing heterogeneity, for this initial study of parasite heterogene-ity, the underlying assumption of our analyses is that transmission ishomogeneous across India. Applying the methods developed in Sec-tion 3.3 to smaller regions of India would allow us to obtain moreaccurate parameter estimates by addressing the existing heteroge-neity in transmission, and is a subject of future research.

Lastly, a comparison of several models for the enhanced controlperiod (1997–2010) in Section 3.4 allowed us to determine whichcontrol measures contributed the most to the success of controlprograms. In the same section, we also present an uncertaintyanalysis to determine the most likely outcome of malaria in India.

2. P. falciparum and P. vivax malaria co-infection model

In the two-parasite malaria model below, it is assumed that themosquito population size, Nm, is constant, and that the size of thehuman population, N, exhibits logistic growth. In reality, mosquitopopulation size may vary, not only seasonally, but also inter-annu-ally as a consequence of changing climatic variables [32]. The strongdependence of suitable habitat for mosquito development on tem-perature, rainfall, and humidity, for example, can cause increasesin mosquito abundances during years with greater than averagerainfall, or warmer than average temperatures [16]. This depen-dence of mosquito dynamics on environmental factors has insti-gated questions about the effect of climate change on malariaprevalence. Although it is well-known that mosquito abundancefluctuates in response to deviations from average weather condi-tions, there are differing opinions on the extent to which climatechange will affect malaria dynamics, and which variables have thegreatest impact [19,43]. How changes in climate variables translateto changes in malaria dynamics depends on the region. For example,small deviations from the average may have the greatest impact inregions with low malaria endemicity, such as highland areas nor-mally protected by cooler temperatures [24,25]. For simplicity, weassume that yearly weather conditions remain close enough toaverage that it does not effect yearly mosquito abundance. Extend-ing our model to include variations in mosquito abundance is a po-tential area for future research provided sufficient temporal dataabout the vector become available. On the other hand, human pop-ulation data is readily available and indicates that India’s populationis steadily growing each year. Consequently, we felt it necessary toaddress the growing human population in our model.

The state variable M denotes the number of mosquitoes that arefully susceptible to both P. vivax and P. falciparum parasites. Simi-larly, S denotes the number of humans who are fully susceptible toboth malaria parasites. The number of infected mosquitoes at a gi-ven time is J, the sum of P. vivax infected mosquitoes (Jv) and P. fal-ciparum infected mosquitoes (Jf ).

Human deaths due to P. vivax infection are rare, and are thus con-sidered to be negligible. Although deaths due to P. falciparum do oc-cur, the associated mortality rate in India is very small comparedwith the total morbidity due to malaria. Once infectious individualsrecover fully from malaria, they again become susceptible to malar-ia infection and move to class S. As a result, in this two-parasitemodel, all humans recover from malaria infection. Once infectiousindividuals recover fully from malaria, they again become suscepti-ble to malaria infection and move to class S. Thus, when we refer toan individual ‘‘surviving’’ a particular stage, we mean that they didnot die due to natural mortality before the end of that stage.

2.1. Modeling the dynamics of P. vivax infection in the humanpopulation

First, we describe the dynamics of P. vivax malaria in the humanpopulation. When a P. vivax infected mosquito successfully

O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50 35

transmits a malaria parasite to a human, we assume that these hu-mans first go through a liver stage, denoted by L, in which the ma-laria parasites remain un-infectious. This liver stage acts as boththe initial incubation period for the P. vivax malaria parasites in ahuman and as the period between relapses in which malaria para-sites remain in a dormant liver stage as hypnozoites. Becauseasymptomatic individuals create a reservoir for malaria, posingsignificant challenges to malaria control, and because malaria in-fected individuals do not become infectious until the parasiteshave gone through the human liver stage [2], the model allowsfor a fraction of the individuals in the liver stage to bi-pass thesymptomatic stage and move directly to the P. vivax infectiousstage Iv . We refer to individuals who bi-pass the symptomaticstage as ‘‘asymptomatic’’, and those who do not are referred to as‘‘symptomatic’’.

A human presenting symptoms is considered a ‘‘clinical case’’and we let Cv denote the P. vivax clinical cases at any given time.Assuming an individual in Cv does not die of natural mortality,he or she will become infectious and move into the Iv stage. Oncein this infectious stage, individuals have the potential to fully re-cover, returning to the susceptible class S via either successfultreatment or natural recovery. Here, we define recovery as loss ofinfectiousness, that is, gametocyte clearance. We assume thatalthough individuals may begin treatment during the clinical stage,the treatment does not affect the person’s progression to the infec-tious stage. Consequently, we assume that even in treated individ-uals, gametocyte clearance occurs in the infectious stage.

As described previously, because some P. vivax parasites be-come hypnozoites during the liver stage, remain dormant in the li-ver for some period, and are reactivated at a later time, vivaxmalaria patients who are not successfully treated are prone to re-lapses. Thus, in our model, individuals in the vivax infectious classIv can return to the liver stage L and repeat the cycle of the vivaxinfection.

2.2. Modeling the dynamics of P. falciparum infection in the humanpopulation

P. falciparum infections, while typically more severe than vivaxinfections, exhibit simpler infection dynamics than vivax infections.In particular, P. falciparum parasites do not have a hypnozoite stage,and consquently, individuals infected with only P. falciparum do notexperience relapses. In light of this difference between P. falciparumand P. vivax infections, we omit the falciparum incubation periodwhich is typically shorter than that of P. vivax, meaning that oncea human is infected by a P. falciparum infectious mosquito, thatindividual moves directly either to the falciparum clinical stage Cf ,or moves to the falciparum infectious stage If . As noted in thedescription of vivax infection dynamics, we refer to the individualswho bi-pass the clinical stage as ‘‘asymptomatic’’ individuals. Thosewho pass through the clinical stage are referred to as ‘‘symptom-atic’’ individuals. Once in the P. falciparum infectious stage, as withP. vivax infection, individuals can fully recover via either successfultreatment or natural recovery.

The state variables Iv and If include both asymptomatic infec-tious individuals and infectious individuals who have shownsymptoms. It will be assumed that symptomatic individuals aretreated and asymptomatic individuals are not treated. Thus, therecovery rate from Iv and If will be a function of both the naturalrecovery rate and the treatment-recovery rate.

2.3. Modeling co-infection

Gupta et al. used 180 samples from six endemic regions in Indiato estimate the proportion of malaria cases that are mixed infec-tions. The samples showed that roughly 46% of the malaria

infections were P. falciparum–P. vivax co-infections [17]. Conse-quently, the ability for humans to obtain concurrent malaria infec-tions should play an important role in a P. falciparum–P. vivaxmalaria model for India. Mixed infection is incorporated into themodel by introducing two more ‘‘clinical case’’ state variables, Cvf

and Cfv . A P. vivax infected individual in either the liver stage orthe infectious stage who becomes co-infected with falciparum willmove to Cvf . At this stage we assume that individuals coming fromthe liver stage become infectious with vivax, those arriving from theinfectious stage remain infectious with vivax, and all individuals inCvf show symptoms of malaria infection, although it may not beclear which infection is causing the symptoms. According to Snou-nou et al., the assumption that co-infection with P. falciparum canreactivate hypnozoites in the dormant liver-stage, producing P. vi-vax blood-stage parasites, is plausible [39]. Similarly, a P. falciparuminfectious individual can become co-infected with P. vivax. Theseindividuals will move to the Cfv stage provided they have not suc-cumbed to natural mortality. Individuals in Cfv are still infectiouswith P. falciparum, but present symptoms associated with P. vivaxinfection. We will refer to individuals who have been in stages Cvf

and Cfv as ‘‘vivax co-infected’’ and ‘‘falciparum co-infected’’ individ-uals, respectively. If a co-infected individual survives the clinicalstage, they become infectious with both malaria parasites and moveto Ic.

In this two-parasite model, we assume that all co-infected indi-viduals are treated during the infectious co-infected stage Ic . Thisassumption is reasonable since most co-infected individuals showsymptoms [39]. The question is, what treatment do we give theseco-infected individuals? According to the 2009 malaria diagnosisand treatment guidelines for India, P. falciparum–P. vivax co-in-fected individuals should be given the same treatment that is givento P. falciparum infected patients [31]. However, malaria diagnostictests often only detect one of the two parasite species in the host,leading health-care providers to treat only the observed infection[26]. When only one of the two infections is treated, symptomsfor the other malaria infection emerge anywhere from 17 to63 days post-treatment [27]. The model incorporates this emer-gence of the hidden infection by allowing individuals in the infec-tious co-infected class Ic to move into either Iv or If after recoveryfrom the initial observed (and hence treated) infection. If P. falcipa-rum is treated first, then the P. vivax infection will emerge and indi-viduals move into the infectious class Iv . Likewise, those who aretreated for P. vivax first move to the infectious class If some timepost-vivax treatment. In our model, however, individuals treatedfor P. vivax first will not show symptoms following treatment ofthe co-infection since falciparum symptoms do not occur followingfalciparum infectiousness. Only co-infected individuals treated forP. falciparum first have the possibility of developing symptoms, inparticular vivax symptoms, since individuals in Iv can relapse. Con-sequently, our model does not capture the phenomenon describedabove where P. falciparum symptoms emerge following vivax treat-ment. This discrepancy can be resolved by adding a P. falciparumincubation period to the model, however for simplicity, and be-cause the majority of co-infections are treated for P. falciparumfirst, followed by the onset of P. vivax symptoms, we find thatincorporating only a vivax incubation/liver stage sufficient to cap-ture the most important features of the two-parasite species dis-ease dynamics.

2.4. Disease dynamics in the mosquito population

The human component of the two-parasite malaria model in-cludes five infectious classes: two classes are infectious with vivaxonly (Iv and Cvf ), two classes are infectious with falciparum only (If

and Cfv ), and one class is infectious with both vivax and falciparum(Ic). Thus, mosquitoes have five means by which they can become

36 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

infected. A susceptible mosquito infected by a human in class Iv orCvf will develop a P. vivax infection. A susceptible mosquito in-fected by a human in class If or Cfv will develop a P. falciparuminfection. In the event that a mosquito becomes infected by a co-infected infectious human (Ic), the model assumes that the mos-quito will contract only one of the two malaria parasite species.Which species it contracts will depend on the probability of ‘‘pick-ing up’’ that particular species. Since mosquitoes have a short life-span, we assume that all mosquitoes die natural deaths rather thandisease-induced deaths. Summaries of all human and mosquitostate variables are given in Tables 1 and 2, respectively. A descrip-tion of the model parameters, as well as the estimates used in latermodel simulations, is given in Tables 3 and 4.

The two-parasite malaria model diagram in Fig. 1 can bedescribed mathematically as follows:

Mosquito dynamics:

dJvdt¼ bv

Iv þ Cvf

N

� �ðNm � JÞ þ fbv

Ic

NðNm � JÞ � dJv ð1Þ

dJf

dt¼ bf

If þ Cf v

N

� �ðNm � JÞ þ ð1� fÞbf

Ic

NðNm � JÞ � dJf ð2Þ

where J ¼: Jv þ Jf ; Nm is constant, and M ¼ Nm � J.Since it is difficult to estimate how large the mosquito popula-

tion is, we modify the mosquito dynamics equations by considering

Table 1Description of human model state variables at time t.

State variables Description

Nm Mosquito population size – defined to be constantNðtÞ Human population size at time tLðtÞ Number of human P. vivax liver stage infections at time tCv ðtÞ Number of human P. vivax cases at time tCf ðtÞ Number of human P. falciparum cases at time tIv ðtÞ Number of P. vivax infectious humans at time tIf ðtÞ Number of P. falciparum infectious humans at time tCvf ðtÞ Number of symptomatic co-infected cases,

infectious with P. vivax only, at time tCfv ðtÞ Number of symptomatic co-infected cases,

infectious with P. falciparum only, at time tIcðtÞ Number of co-infected humans infectious

with both P. vivax and P. falciparum at time t

Table 2Description of mosquito model state variables at time t.

Statevariables

Description

MðtÞ Number of susceptible mosquitoes at time tSðtÞ Number of susceptible humans at time tmðtÞ Proportion of mosquitoes that are susceptible at time tJv ðtÞ Number of P. vivax infected mosquitoes at time tJf ðtÞ Number of P. falciparum infected mosquitoes at time t

jv ðtÞ Proportion of mosquitoes that are P. vivax infected at time tjf ðtÞ Proportion of mosquitoes that are P. falciparum infected at

time t

Table 3Description of model parameters pertaining to mosquito population dynamics and their e

Parameters Description

d Natural death ratef Probability that a susceptible mosquito that gets

1� f Probability that a susceptible mosquito that gets infectedby a co-infected human contracts P. falciparum

the proportion of mosquitoes infected rather than the number ofmosquitoes infected. Thus, dividing Eqs. (1) and (2) by the totalmosquito population size Nm, we arrive at the following set of equa-tions describing the mosquito infection dynamics:

djvdt¼ bv

Iv þ Cvf

N

� �ð1� jÞ þ fbv

Ic

Nð1� jÞ � djv

djf

dt¼ bf

If þ Cf v

N

� �ð1� jÞ þ ð1� fÞbf

Ic

Nð1� jÞ � djf

where now j ¼: jv þ jf ¼ 1NmðJv þ Jf Þ, denotes the fraction of the mos-

quito population that is infected with malaria parasites and hence,m ¼ 1� j represents the fraction of mosquitoes that are susceptibleto malaria infection. Note that

m0 ¼ �j0

¼ � bvIv þ Cv f

N

� �þ fbv

Ic

Nþ bf

If þ Cf v

N

� �þ ð1� fÞbf

Ic

N

� �m

� dð1�mÞ:

In the above system, bv and bf are human-to-mosquito transmissionrates, d is the mosquito natural mortality rate, and f is the probabil-ity that if a susceptible mosquito bites an Ic human, the mosquitowill contract vivax rather than falciparum.

Human Dynamics:

dSdt¼ dN

dt� dLv

dt� dC

dt� dI

dt

¼ rN 1� NK

� �þ qv Iv þ qf If � ðbv jv þ bf jf ÞS� lS

dLdt¼bvSjv þ dIv � avbf jf L� ðkþ lÞL

dCv

dt¼rvkL� ðmv þ lÞCv

dIvdt¼ð1� rvÞkLþ mvCv þ gf Ic � avbf Iv jf � ðdþ qv þ lÞIv

dCvf

dt¼avbf ðIv þ LÞjf � ðmvf þ lÞCvf

dCf

dt¼rf bf Sjf � ðmf þ lÞCf

dIf

dt¼ð1� rf Þbf Sjf þ mf Cf þ gv Ic � af bv If jv � ðqf þ lÞIf

dCf v

dt¼af bv If jv � ðmf v þ lÞCf v

dIc

dt¼ mvf Cv f þ mf vCf v � ðgv þ gf þ lÞIc

where C ¼: Cv þ Cf þ Cvf þ Cfv ; I ¼: Iv þ If þ Ic , and the total popula-

tion size is described by the logistic equation dNdt ¼ rN 1� N

K

� �� lN.

We chose to model human population growth with the logisticmodel to better approximate the situation in India. Although mos-quito population dynamics do fluctuate seasonally, the inter-annualchanges are less significant relative to the inter-annual changes inhuman population size. Because we are modeling disease-dynamicson a yearly time-scale, assuming a constant mosquito populationsize is an appropriate simplifying assumption.

stimates.

Value Reference

365=14 years�1 [3]670

670þ332 infected by a co-infected human contracts P. vivax See 3.1.4

Table 4Description of model parameters pertaining to human population dynamics and their estimates

Parameters Description Value Reference

1k

Duration of P. vivax liver stage 90/365 years See 3.1.31mv

Time until infectious after P. vivax symptom onset 4/365 years [14]1mf

Time until infectious after P. falciparum symptom onset 7/365 years [22]

1mvf

Duration of Cvf1mf

See 3.1.1

1mf v

Duration of Cfv 1mv

See 3.1.1

l Natural death rate 1=60:55 years�1 See 3.3cv P. vivax blood-stage parasite clearance rate with treatment 365=3 years�1 [33]cf P. falciparum treatment recovery rate 365=12 years�1 See 3.1.2rv P. vivax natural blood-stage parasite clearance rate 365/30 years�1 See 3.1.3

rf P. falciparum natural recovery rate 365=200 years�1 See 3.1.2qv Recovery rate from Iv to S See 3.1.2qf Recovery rate from If to S See 3.1.2pr Probability of post-treatment P. vivax relapse 0:23� 0:44 ð0:2904Þ [1]

d P. vivax relapse rate prrvcv þ ð1� rv Þrv See 3.1.3rv Probability that a P. vivax infected human becomes symptomatic 0.82 [38]rf Probability that a P. falciparum

infected human becomes symptomatic 0.90 assumedav Pv-induced cross-immunity to Pf 1af Pf-induced cross-immunity to Pv 1g Fraction of co-infected infectious individuals that recover first from P. falciparum 0.75 See 3.2gv Rate of progression from Ic to If due to P. vivax treatment gcvgf Rate of progression from Ic to Iv due to P. falciparum treatment ð1� gÞcf

Fig. 1. Mosquito and Human Population Dynamics Diagram under the influence oftwo circulating malaria parasites. Bold arrows indicate the acquisition of a newinfection, and dotted arrows indicate recovery from either P. vivax or P. falciparuminfection.

O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50 37

The mosquito-to-human transmission rates for vivax and falci-parum are denoted by bv and bf , respectively. The natural humanmortality rate is given by l. A proportion rv of vivax and a propor-tion rf of falciparum cases are symptomatic. We assume symptom-atic individuals get treated and clear blood-stage parasites at a rateci from infection i and asymptomatic individuals clear blood-stageparasites at a rate ri (i ¼ v ; f ). Thus, the rates of returning to thesusceptible class, denoted by qv and qf , are a function of bothtreatment and natural parasite-clearance rates: qi ¼ riciþð1� riÞri, for i ¼ v ; f . Vivax-infected individuals progress at a ratek from the liver stage to either Cv or Iv . Vivax-symptomatic and fal-ciparum-symptomatic individuals progress to the infectious stageat rate mv and mf , respectively. Similarly, vivax-co-infected and fal-ciparum-co-infected individuals enter Ic at rates mvf and mfv , respec-tively. av and af are cross-immunity coefficients. d is the rate atwhich vivax-infected individuals relapse. This parameter is given

by d ¼ prrvcf þ ð1� rv Þrv , where pr is the probability that atreated vivax patient relapses. Finally, gv and gf are the probabili-ties that an Ic individual is treated first for vivax and, respectively,for falciparum infection. A complete list of the model parametersand their descriptions is presented in Tables 3 and 4.

2.5. Derivation of the disease-free equilibrium, basic reproductivenumber R0 and control reproductive number RC

Note that dNdt can be rewritten in the form

dNdt ¼ ðr � lÞN 1� N

K 1�lrð Þ

� �so that the intrinsic growth rate of the

population r̂ is r � l, and the carrying capacity K̂ is K 1� lr

� �.

The disease-free equilibrium is ðN�;m�; j�v ; j�f ; S

�; L�;C�v ;C�f ;

I�v ; I�f ;C

�vf ;C

�fv ; I

�cÞDFE ¼ ðK̂;1;0; 0; K̂;0;0;0;0;0;0;0;0Þ. N� and S� are

easily determined by setting the right hand side ofdNdt ¼ r̂N 1� N

K̂

� �equal to zero and noting that when there is no dis-

ease, S ¼ N. Since m� ¼ 1� j�, we have that m� ¼ 1 when there isno disease.

The basic reproductive number, R0, of an epidemiological modelis the average number of secondary cases produced by one infec-tious individual in an otherwise fully susceptible population whereno control is being implemented. The control reproductive num-ber, RC , is defined similarly, with the exception that control mea-sures are assumed to be in place. If R0 < 1, the disease-freeequilibrium is locally asymptotically stable, implying that the dis-ease will eventually become extinct. On the other hand, if R0 > 1,the disease-free equilibrium is unstable [40]. Consequently, deter-mining an expression for the basic reproductive number from themodel and estimating its value is a key component to understand-ing how difficult it will be to control transmission of the diseaseand what control measures will be the most effective. An impor-tant goal of any infectious disease control program is to implementcontrol measures in such a way as to successfully bring the controlreproductive number below one. The isolation reproductive num-bers of a multi-parasite model, such as this two-parasite malariamodel, are the basic reproductive numbers for the model whenonly one parasite species is present at a time.

38 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

2.6. Expression of RC and R0 derived from the next generation approach

Using the next generation operator approach [40], we find thatthe control reproductive number (RC) for the malaria model is gi-ven by RC ¼ maxfRCv ;RCf g, where RCv is the isolation control repro-ductive number for P. vivax, and RCf is the isolation controlreproductive number for P. falicparum. For details of the derivationof RC , see appendix A.1. These isolation control reproductive num-ber for P. falciparum is described by the expression below:

RCf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibf

d� ð1� rf Þ þ rf

mf

mf þ l

bf

qf þ l

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRa

Cf þ RsCf

q;

where

RaCf ¼

bf

d� ð1� rf Þ �

bf

qf þ l

RsCf ¼

bf

d� rf �

mf

mf þ lbf

qf þ l

Observe that RaCf is the contribution of an asymptomatic infectious

individual to the basic reproductive number and RsCf is the contribu-

tion of a symptomatic infectious individual.The P. vivax isolation control reproductive number is given by

RCv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RaCv þ Rs

Cv

1� ddþqvþl

eRaCv þ eRs

Cv

� �vuutwhere

RaCv ¼

bv

dþ qv þ l� ð1� rvÞ �

kkþ l

� bv

d

RsCv ¼

bv

dþ qv þ l� rv �

kkþ l

� mv

mv þ l� bv

d; and

eRiCv ¼ Ri

Cv=bv bv

dðdþqvþlÞ

� �, for i ¼ a; s. Note that if d were zero, in other

words if P. vivax patients never relapsed, RCv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRa

Cv þ RsCv

qwhere

the interpretations of RaCv and Rs

Cv are analogous to that of RaCf and

RsCf , respectively. That is, Ra

Cv would be the contribution of an asymp-tomatic infectious individual to the basic reproductive number andRs

Cv the contribution of a symptomatic infectious individual. How-ever, the inclusion of the possibility of relapse in P. vivax infectedindividuals (d > 0) makes the expression for RCv more complicatedand its biological interpretation less straight-forward. The numera-tor squared of RCv is the number of new mosquito infections arisingfrom a single infected mosquito, without the intermediate humanhosts relapsing.

To interpret the denominator of RCv , first note thateRaCv þ eRs

Cv 2 ½0;1Þ since eRaCv þ eRs

Cv <k

kþl 1� rv þ rvð Þ ¼ kkþl < 1 and

clearly eRaCv þ eRs

Cv is positive. Let x ¼ ddþqvþl

eRaCv þ eRs

Cv

� �. Then

x < 1 implies that 11�x ¼

P1n¼0xn. Because d

dþqvþl is the probability

that an individual in Iv relapses when there are no falciparum-in-fected individuals in the population, x is the probability that a li-ver-stage human will relapse. Thus, xn is the probability that aliver-stage human will relapse n times. So,

R2Cv ¼

X1n¼0

RaCv þ Rs

Cv� �

� xn:

The ith term in the sum can be interpreted as the number of newmosquito infections generated by a single mosquito where theintermediate human hosts relapse exactly i times.

Since the only control measure explicitly implemented in themodel is treatment, the basic reproductive number for the modelis given by the control reproductive number evaluated with thetreatment recovery rates (cv and cf ) equal to the natural recoveryrates (rv and rf , respectively). Using our definition of qv ; qf , and dthis is equivalent to setting qv ¼ rv ; d ¼ rv , and qf ¼ rf . Thus,R0 ¼ maxfR0v ;R0f g, where

R0f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibf

d� ð1� rf Þ þ rf

mf

mf þ l

bf

rf þ l

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRa

0f þ Rs0f

qwhere

Ra0f ¼

bf

d� ð1� rf Þ �

bf

rf þ l

Rs0f ¼

bf

d� rf �

mf

mf þ lbf

rf þ l

Similarly,

R0v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ra0v þ Rs

0v

1� rv2rvþl

eRa0v þ eRs

0v

� �vuutwhere

Ra0v ¼

bv

d� ð1� rvÞ �

kkþ l

� bv

2rv þ l

Rs0v ¼

bv

d� rv �

kkþ l

� mv

mv þ l� bv

2rv þ l

and eRi0v ¼ Ri

0v=bv bv

dð2rvþlÞ

� �, for i ¼ a; s.

2.7. Isolated endemic equilibria and coexistence

Determining an analytic expression for the coexistence equilib-rium can be a difficult problem for more complicated models suchas this two-parasite malaria model. However, we can still gain in-sight into the conditions under which a coexistence equilibriumoccurs by studying the stability of the isolated endemic equilibria;that is, the equilibria where only one pathogen is present in a pop-ulation. Linearizing the system about these isolation equilibria pro-vides a condition under which the absent parasite species caninvade when introduced to the population. These threshold quan-tities are known as the invasion reproduction numbers.

First, we find the vivax-only equilibrium, Ev by assuming all fal-ciparum-infected variables are zero and setting each equation inthe resulting system equal to zero. Solving this system of equationsfor the non-trivial equilibrium, we find that

j�v ¼bv I�v

bv I�v þ dK̂

S� ¼ lK̂ þ qv I�vbv j�v þ l

L� ¼ bvS�j�v þ dI�vkþ l

C�v ¼rvk

mv þ lL�

where

I�v ¼1� R2

Cv

R2Cv

� lK̂

qv þ d 1� dþqvþld Ra

CvþRsCvð Þ

� �1þ l

bv

� � ð3Þ

It is simple to show that the denominator in Eq. (3) is always neg-ative. First recall that Ra

Cv þ RsCv < 1 so that dþqvþl

RaCvþRs

Cv> dþ qv þ l. So,

qvþd 1� dþqvþld Ra

CvþRsCvð Þ

� �1þ l

bv

� �<qv þ d 1� dþqvþl

d RaCvþRs

Cvð Þ

� �< �l < 0. The

numerator of Eq. (3) is negative if R2Cv > 1. Thus, I�v is positive only

O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50 39

if R2Cv > 1. In other words, the vivax-boundary equilibrium exists

only when RCv > 1.Now, we find the falciparum-only equilibrium Ef by setting all

vivax-infected variables equal to zero, and finding the non-trivialequilibrium of the resulting system.

j�f ¼bf I�f

bf I�f þ dK̂

S� ¼lK̂ þ qf I�fbf j�f þ l

C�f ¼rf bf

mf þ lS�j�f ;

where

I�f ¼1� R2

Cf

R2Cf

� lK̂

qf �bf bf

d RaCfþRs

Cfð Þ 1þ lbf

� � ð4Þ

Using the definition of RaCf þ Rs

Cf , we have that bf bf

d RaCfþRs

Cf

� � ¼qfþl

ð1�rf Þþrfmf

mf þl< qf þ l. Since qf �

qfþl

ð1�rf Þþrfmf

mf þlð1þ l=bf Þ < qf�

ðqf þ lÞ ¼ �l < 0, the denominator of Eq. (4) is always negative.Consequently, I�f is positive only when RCf > 1.

2.8. Invasion numbers Rfv and Rv

f

The basic reproduction number is a threshold that determineswhether a disease can invade the disease-free equilibrium or not.Likewise, invasion numbers are threshold quantities that determineif a disease can invade another disease’s endemic equilibrium. Thesequantities are very useful in understanding the competition be-tween pathogens in a multi-strain model. Here, we find analyticexpressions for Rf

v , the invasion number of P. vivax when the systemis at the P. falciparum-only equilibrium, and Rv

f , the invasion numberof P. falciparum at the P. vivax-only equilibrium. Typically, the fol-lowing result can be established: if RCf > 1 and Rf

v < 1, then the fal-ciparum-only equilibrium is locally asymptotically stable andunstable otherwise. Similarly, if RCv > 1 and Rv

f < 1, the vivax-onlyequilibrium is locally asymptotically stable and unstable otherwise.Both species coexist when Rf

v and Rvf are greater than one.

The invasion numbers were derived using the next generationapproach [13], the details of which are presented in appendicesA.2 and A.3. We find that

Rvf ¼

11� k5;3k6;5k3;6

� ðk2;1k3;2k1;3 þ k2;1k3;2k5;3k1;5

þ k2;1k3;2k5;3k6;5k1;6 þ k3;1k1;3 þ k3;1k5;3k1;5 þ k3;1k5;3k6;5k1;6

þk4;1k6;4k1;6 þ k4;1k6;4k3;6k1;3 þ k4;1k6;4k3;6k5;3k1;5Þ1=2

ð5Þ

where

k2;1 ¼rf bf S

�

d

k3;2 ¼mf

mf þ l

k1;3 ¼bf ð1� j�f ÞK̂

af bv j�v þ qf þ l

k5;3 ¼af bv j�v

af bv j�v þ qf þ l

k1;5 ¼bf ð1� j�vÞK̂

mf v þ l

k6;5 ¼mf v

mfv þ l

k1;6 ¼ð1� fÞbf ð1� j�f ÞK̂

gv þ gf þ l

k3;1 ¼ð1� rf Þbf S�

d

k4;1 ¼avbf ðI�v þ L�Þ

d

k6;4 ¼mvf

mvf þ l

k3;6 ¼gv

gv þ gf þ l

If we associate the subscripts 1 through 6 with a different compart-ment in the model, namely 1 ¼ jf ; 2 ¼ Cf ; 3 ¼ If ; 4 ¼ Cvf ;

5 ¼ Cf v ; 6 ¼ Ic , then each ki;j represents a progression of the diseasefrom compartment j to compartment i. These ki;j terms are carefullyderived in appendix A.3. The factor 1=ð1� k5;3k6;5k3;6Þ can be writ-ten as the geometric series

P1n¼0ðk5;3k6;5k3;6Þn, where

k5;3k6;5k3;6 ¼af bv j�v

af bv j�vþqfþl �mf v

mf vþl �gv

gvþgfþl is the probability that a falcipa-

rum-only infected human will loop through the disease states, orpath, If ! Cfv ! Ic ! If n times before infecting a mosquito. Thisloop arises when an If individual becomes co-infected, progressesto the Ic stage, and recovers from vivax malaria infection first,returning to the If stage. Note that an If individual can only transmitP. falciparum parasites by infecting a mosquito before leaving thatstage, or by becoming co-infected and recovering first from vivax

infection. Also note that k5;3 ¼af bv j�v

af bv j�vþqfþl is the transition probability

for If ! Cfv ; k6;5 ¼mfv

mf vþl is the transition probability Cf v ! Ic , and

finally k3;6 ¼ gvgvþgfþl represents the transition probability Ic ! If .

We can interpret the remaining terms in Rvf similarly. Instead of

a path representing a loop that a single individual takes, each pathbelow represents the path for how one falciparum infected mos-quito can lead to a new mosquito infection.

k2;1k3;2k1;3 ¼ jf ! Cf ! If ! jf

k2;1k3;2j5;3k1;5 ¼ jf ! Cf ! If ! Cf v ! jf

k2;1k3;2k5;3k6;5k1;6 ¼ jf ! Cf ! If ! Cf v ! Ic ! jf

k3;1k1;3 ¼ jf ! If ! jf

k3;1k5;3k1;5 ¼ jf ! If ! Cf v ! jf

k3;1k5;3k6;5k1;6 ¼ jf ! If ! Cfv ! Ic ! jf

k4;1k6;4k1;6 ¼ jf ! Cvf ! Ic ! jf

k4;1k6;4k3;6k1;3 ¼ jf ! Cvf ! Ic ! If ! jf

k4;1k6;4k3;6k5;3k1;5 ¼ jf ! Cv f ! Ic ! If ! Cf v ! jf

If we multiply any one of the terms above byP1

n¼0ðk5;3k6;5k3;6Þn,then the nth term in the resulting sum will have the same chainof events as above, with the exception that the If individual takesthe If ! Cfv ! Ic ! If loop n times before continuing to the nextstage in the chain. Thus, the next-generation approach leads to anexpression of the invasion numbers whose square has the biological

interpretation we desire: Rvf

� �2is the number of secondary falcipa-

rum mosquito infections caused by a single falciparum-infectedmosquito in a population at the vivax isolated endemic equilibrium.

Now we introduce the invasion number Rfv , whose expression is

more complicated than that of Rvf . Despite its more complicated

form, we can show that the square of this invasion number is thenumber of new vivax-infected mosquitoes arising from a single vi-vax-infected mosquito in a population at the falciparum isolatedendemic equilibrium. From the next generation approach, we findthat

40 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

Rfv ¼ f½k6;1ðk1;5k7;6k4;7ðk5;4 þ k2;4k5;2Þ þ k1;7k7;6ð1� k2;4ðk3;2k4;3

þ k4;2ÞÞÞ þ k2;1½ðk1;5 þ k1;7k7;5Þðk3;2k4;3k5;4 þ k4;2k5;4

þ k5;2Þ�g � ð1� k2;4ðk3;2k4;3 þ k4;2Þ � k4;7k7;5ðk5;4

þ k2;4k5;2ÞÞg1=2; where ð6Þ

k6;1 ¼af bv I�f

d

k1;5 ¼bvð1� j�f ÞK̂ðmvf þ lÞ

k7;6 ¼mvf

mvfþl

k4;7 ¼gf

gv þ gf þ l

k5;4 ¼avbf j�f

avbf j�f þ dþ qv þ l

k2;4 ¼d

avbf j�f þ dþ qv þ l

k5;2 ¼avbf j

�f

avbf j�f þ kþ l

k1;7 ¼ fk1;5

k7;6 ¼mf v

mf v þ l

k3;2 ¼rvk

avbf j�f þ kþ l

k4;3 ¼mv

mv þ l

k4;2 ¼ð1� rvÞk

avbf j�f þ kþ l

k2;1 ¼bvS�

d

k7;5 ¼mvf

mvf þ l

Here, we associate the subscripts 1 through 7 with a disease state inthe model as follows: 1 ¼ jv ; 2 ¼ L; 3 ¼ Cv ; 4 ¼ Iv ; 5 ¼ Cvf ;

6 ¼ Cf v ; 7 ¼ Ic . Again, ki;j (derived in appendix A.2) represents aprogression from compartment j to compartment i. To arrive atthe correct biological interpretation, we first observe that expand-

ing the expression for Rfv

� �2reveals that each term in the resulting

sum represents a path by which one vivax-infected mosquito resultsin another mosquito infection. For example, the first term (rear-ranged), k6;1k7;6k4;7k5;4k1;5, represents the number of If -humansinfected by a vivax-infected mosquito before dying, causing thosehumans to progress to the Cfv stage, times the fraction of peoplethat survive the Cfv stage and progress to the Ic stage, times thefraction of individuals that survive this stage and are treated for fal-ciparum prior to treatment for vivax, entering the Iv stage, times theprobability that these individuals are infected by a falciparum-infected mosquito and progress to the Cvf stage, and finally, timesthe number of susceptible mosquitoes a Cvf human infects priorto progressing to the co-infectious stage Ic . Each term in Rf

v repre-sents such a path from an infected mosquito to another mosquitoinfection. The negative terms, as we will demonstrate, account forinfections that arise because a human passes through the samestage more than once. In appendix A.2, we argue that the denomi-nator of Rf

v is positive. Thus, it must be that k2;4ðk3;2k4;3 þ k4;2Þþk4;7k7;5ðk5;4 þ k2;4k5;2Þ is less than one. Using the same reasoning aswe did for RCv and Rv

f , we can rewrite 1=ð1� k2;4ðk3;2k4;3 þ k4;2Þ�k4;7k7;5ðk5;4 þ k2;4k5;2ÞÞ as a geometric series, allowing us to fullyinterpret the invasion number.

Since 1=ð1� x� yÞ ¼P1

n¼0ðxþ yÞn when jxþ yj < 1, we can re-write Rf

v as

Rfv

� �2¼ k6;1k1;5k7;6k4;7ðk5;4 þ k2;4k5;2Þ þ k2;1ðk1;5 þ k1;7k7;5Þ½

k3;2k4;3k5;4 þ k4;2k5;4 þ k5;2Þð �

�X1n¼0

k2;4ðk3;2k4;3 þ k4;2Þ þ k4;7k7;5ðk5;4 þ k2;4k5;2Þ½ �n

þ k6;1k1;7k7;61� k2;4ðk3;2k4;3 þ k4;2Þ

1� k2;4ðk3;2k4;3 þ k4;2Þ � k4;7k7;5ðk5;4 þ k2;4k5;2Þð7Þ

Now, we can rewrite the fraction in the last term so that the expres-

sion for Rfv is fully interpretable. Note that this term is of the form

ð1� xÞ=ð1� x� yÞ, where x is precisely k2;4ðk3;2k4;3 þ k4;2Þ and y isk4;7k7;5ðk5;4 þ k2;4k5;2Þ. Using the fact that 1�x

1�x�y ¼1�x�yþy

1�x�y ¼ 1þ y1�x�y,

we have that

1� x1� x� y

¼ 1þ yX1n¼0

ðxþ yÞn ð8Þ

Hence, we arrive at a fully interpretable expression for the invasionnumber

Rfv

� �2¼ k6;1k1;5k7;6k4;7ðk5;4 þ k2;4k5;2Þ þ k2;1ðk1;5 þ k1;7k7;5Þ½

k3;2k4;3k5;4 þ k4;2k5;4 þ k5;2Þð �

�X1n¼0

ðxþ yÞn þ k6;1k7;6k1;7 1þ yX1n¼0

ðxþ yÞn !

: ð9Þ

The terms x ¼ k2;4ðk3;2k4;3 þ k4;2Þ and y ¼ k4;7k7;5ðk5;4 þ k2;4k5;2Þ rep-resent four different transmission paths. x represents two ways aperson can start in, and return to, stage Iv . One path travels throughthe symptomatic class while the other does not. Similarly, y repre-sents two ways in which a person in Ic can arrive at stage Cvf . One ofthese two paths travels through stage L, while the other path by-passes stage L.

The nth term in the summationP1

n¼0ðxþ yÞn represents theprobability of taking any combination of the four loops, resultingin a total of exactly n loops. The number one in parentheses repre-sents the contribution to secondary vivax cases by vivax-infectedindividuals who make no loops. Finally, the nth term in the expres-sion y

P1n¼0ðxþ yÞn represents the probability that an individual

first takes one of the loops in y, then makes a total of exactly nloops consisting of some combination of the four loops describedby x and y. The second summation in Eq. (9) arises because the onlyway in which an individual can enter path x (Iv ! LðL! Cv ! Iv þ L! IvÞ) from path k1;7k6;1k7;6 (Ic ! jv ! Cfv ! Ic)is by first entering path y (Ic ! Iv ðIv ! Cvf þ Iv ! L! Cvf Þ). Con-versely, paths x and y can be reached from all other paths repre-sented in Eq. (9).

By carefully rewriting the invasion numbers to consist of termsthat can be interpreted as either probabilities or fractions of a pop-ulation of individuals in a particular state, we have shown that it ispossible to link the mathematical expressions to a biological inter-pretation relevant to public health. In Section 3.4, we illustratehow these analytic expressions can be used to understand theinterplay between the use of malaria interventions and the compe-tition between falciparum and vivax.

3. Description of model parameters and choice of parametervalues for the years from 1987 to 1996

To answer questions about disease dynamics and the use ofcontrol measures in India, we must determine realistic estimatesto parameterize our mathematical model. To do this we found

O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50 41

reasonable estimates from the malaria literature for all parametersbut the transmission parameters (bv ; bf ; bv , and bf ) and humanpopulation growth parameters (r and K). Using these estimatesfrom the literature, we estimate the remaining parameters by fit-ting the model to malaria case data and population data for India.The case data, obtained from the National Vector Borne DiseaseControl Programme [31], consists of the yearly number of reportedmalaria cases in India, from 1987 to 2010. The population data wasobtained from the U.S. Census Bureau [37]. In the following sec-tions we first discuss the choice of estimates for parameters foundin the literature, then we describe the procedure for estimating thehuman population intrinsic growth rate, carrying capacity, andthe malaria transmission parameters. As discussed in Section 1,the underlying assumption of the parameter estimation procedureis that transmission is homogeneous across India. In future work,we plan to extend the procedure developed here to datasets forsmaller regions to address the existing heterogeneity.

3.1. Estimation of parameters from literature

3.1.1. Time to infectiousnessFollowing the onset of symptoms, it takes roughly 4 days for P.

vivax infections to become infectious in a human host [14], andapproximately 7 days for P. falciparum infection [22]. Thus, we takemv and mf , the rate of progression from symptomatic to infectiousfor P. vivax and P. falciparum, respectively, to be 365

4 years�1 for P. vi-vax, and 365

7 years�1 for P. falciparum. We assume that becoming co-infected does not alter the time it takes to become infectious. Thus,we let mfv ¼ mv and mvf ¼ mf .

3.1.2. Estimating recovery ratesThe rate of recovery from Iv and If to the susceptible class S is

estimated by qv ¼ ð1� rv Þrv þ rvcv and qf ¼ ð1� rf Þrf þ rf cf ,respectively. In other words, a fraction recover at the naturalrecovery rate, and a fraction recover at the treatment recovery rate.Since Chloroquine targets only the asexual blood stages of the par-asite, there may still be gametocytes remaining at the end of treat-ment. It takes roughly 8 days for gametocytes to mature, and thelifespan of a mature gametocyte is roughly between 3.5 and 4 days.From this, we estimate that individuals treated for falciparum withdrugs that do not kill the gametocytes can remain infectious for upto 12 (8 + 4) days after treatment is completed. Thus, treatment ofP. falciparum infections with Chloroquine reduces the infectiousperiod from roughly 200 days (ref) to 12 days, and we takerf ¼ 365

200 years�1, and cf ¼ 36512 years�1. A study of P. vivax gametocyt-

emia found that out of 516 patients treated with CQ, only four stillhad not cleared the gametocytes by the third day of treatment [29].Using this finding, we let cv ¼ 365

3 .

3.1.3. Parameterizing P. vivax relapseJoshi et al. [20] note that patterns of P. vivax relapse can be cat-

egorized into three groups. The first group is referred to as thetropical type which is characterized by an early primary attackwith frequent relapses. The time intervals between relapses ofthe tropical type are between one and three months. Group IIhas relapse intervals of intermediate length – approximately be-tween three and five months long. And finally, group III, alsoknown as the temperate type, is characterized by a long primarylatent period and relapses occurring every six to seven months.

In this malaria model, we assume that P. vivax infected individ-uals who relapse are those who either were never treated or wereunsuccessfully treated. Since P. vivax parasites inducing short-termrelapse patterns were found to be less susceptible to anti-relapsedrugs [20], we assume that individuals who were unsuccessfullytreated for P. vivax exhibit group I relapse patterns. Thus, theyshould relapse every one to three months. The rate at which a

relapsing individual progresses from Iv to L is the rate at which thatindividual loses infectiousness (i.e. the rate at which gametocytesare cleared from the blood). We assume that treated individualslose infectiousness at a rate cv , regardless of whether treatmentwas successful or not, and untreated (i.e. asymptomatic) individu-als lose infectiousness at a rate rv . Adak et al. determined that29.04% of P. vivax patients treated only with Chloroquine (CQ) re-lapsed following treatment [1]. Thus, the rate at which individualsprogress from Iv to the liver stage class L, is given byd ¼ :2904rvcv þ ð1� rv Þrv . If no one is treated, then d ¼ rv .

The time between P. vivax relapses is usually defined as the timebetween clinical episodes. However, in this model it is possible forindividuals who relapse, in the sense that the parasite repeats thecycle of infection within the human host, without passing throughthe symptomatic stage Cv . We will take the time between relapsesto be the time it takes to progress from Iv to L ( 1

rvfor an untreated

individual and 1cv

for an unsuccessfully treated individual), plus thetime it takes to progress from L to the next infected stage 1

k

� �. Thus,

if we take the average time between relapses to be three monthsfor an unsuccessfully treated individual, 1

k þ 1cv¼ 3 months �

90—93 days. Since 1cv

is approximately 3 days long, we take 1k to

be 90 days. In other words, k ¼ 36590 years�1. Asymptomatic individ-

uals could have relapse patterns associated with group I, II, or III –experiencing a relapse anywhere from every one to seven months.Thus, we take 1

rvþ 1

k to be the average of four months long. In otherwords, 1

k þ 1rv� 120—124 days. From this estimate and our estimate

for 1=k, we assume that it takes roughly 30 days for an untreated P.vivax infected individual to lose infectiousness.

3.1.4. Estimation of fA study conducted by Phimpraphi et al. [34] showed no signif-

icant difference in gametocyte production by P. vivax or P. falcipa-rum parasites in a co-infected human than in humans who wereonly infected with one of the two parasite species. Also, P. vivaxgametocyte densities were found to be higher than P. falciparumdensities in infected humans, with roughly 670 P. vivax gameto-cytes per ll of blood and 332 P. falciparum gametocytes per ll ofblood. Since gametocytes are the infectious stage of the malariaparasites in humans, we use these findings to determine a roughestimate of the parameter f, the proportion of mosquitoes infectedby a human in Ic that contract P. vivax. We assume that f is the den-sity of vivax gametocytes in the blood divided by the total gameto-cyte density. In other words, f ¼ 670

670þ332 � 0:67.

3.2. Estimation of g

P. vivax and P. falciparum are also endemic to Thailand withroughly half the cases resulting from P. vivax infection and halfdue to P. falciparum infection. Approximately 10% of cases in Thai-land initially diagnosed as P. vivax cases and 30% of cases initiallydiagnosed as P. falciparum cases turned out to be co-infections[42]. From this, we estimated that the proportion of co-infectedcases treated first for P. falciparum is g ¼ 0:75.

3.3. Estimation of population growth and transmission parametersusing population and malaria case data for India

From life expectancy data for India [23], we estimated that theaverage life expectancy between the years 1987 and 2009 isapproximately 60.55 years, giving us l ¼ 1=60:55 years�1. Usingthis estimate and a nonlinear least-squares fit of the logistic equa-tion to India’s population data, estimates are obtained for theparameters r and K (see Table 5). The best fit of the logistic curveis illustrated in Fig. 2.

Assuming that the use of control measures remained fairly sim-ilar during the period from 1987 to 1996, we can estimate the

1950 1960 1970 1980 1990 2000 20103

4

5

6

7

8

9

10

11

12x 108

Year

Popu

latio

n Si

ze

Fig. 2. Plot of time series data for India’s Population Size from 1950 to 2009 and thebest fit of the logistic curve to this data. Population data was obtained from [37].

Table 5Estimates of r and K.

Parameter Description Estimate CI

r Intrinsic growth rate 0.0398 years�1 0.0392–0.0404K Population Carrying capacity 7:5616� 109 humans 6:2919� 109–8:8313� 109

42 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

transmission rates bv ; bf ; bv ; bf by imputing the parameter values inTables 3–5, and fitting the model to the malaria case data. Moreprecisely, we used the ‘nlinfit’ function in MATLAB to minimizethe sum of squares of the difference between the data and the solu-tions curves by comparing solution curve Cv to the P. vivax dataand similarly comparing the solution curve Cf þ Cvf þ Cfv to theP. falciparum plus mixed-case data. From this fitting procedure,we obtain estimates for the transmission parameters, summarizedin Table 6; Fig. 4(a) illustrates the fit of the model to the 1987–1996 data with these parameter estimates. The controlled repro-duction numbers RCv and RCf can now be calculated using theexpressions in Section 2.5 (see Table 6).

The resulting reproduction number is larger than one, implyingmathematically that at least one of the two malaria parasites willpersist. Yet, in practice, the extremely close proximity of RCv andRCf to the persistence threshold makes it difficult to arrive at anydefinitive conclusion regarding the outcome of malaria in India.As a step towards addressing this concern, we use a parametricbootstrapping procedure to estimate confidence intervals for RCv

and RCf . The procedure, which we will re-iterate here with slightmodifications, is described in [8] by Chowell et al.

Let us denote the solution curves Cv and Cf þ Cfv þ Cvf that bestfit the data by Sv and Sf , respectively. In one iteration of the boot-strap procedure, we simulate new vivax case data by drawingpoints for each year (1987–1996) from a Poisson distribution with

Table 6Pre-1997 estimates of the transmission parameters.

Parameter Estimate CI

bv 14.54 14.2907–14.79bf 14.04 6.2573–21.83bv 191.33 188.35–194.31bf 51.73 23.26–80.20RCv 1.02RCf 1.01

mean equal to the value of Sv at the corresponding year. In thesame iteration we simulate new falciparum/mixed case data inthe same manner. New estimates for the transmission parametersare determined by fitting Cv and Cf þ Cfv þ Cvf to the simulateddata. This procedure is repeated 1000 times. Calculating the iso-lated controlled reproduction numbers for each of the 1000 runsallows us to produce histograms of the 1000 values of RCv andRCf . These figures (see Figs. 3) reveal that the values of thereproduction numbers generated by the bootstrapping procedureappear fairly symmetric. Consequently, it is simple to determineappropriate 95% confidence intervals for RCv and RCf (see Table 7)by determining the 0.025 and 0.975 quantiles of the 1000 esti-mates. Fig. 3 illustrates that the estimates of RCv ;RCf ;R

fv , and Rv

f

are consistently greater than one. Hence, we can conclude thatthe two Plasmodium species would likely continue to coexist after1997 had malaria intervention strategies not improved.

3.4. Estimation of parameters for the enhanced malaria control period

Around 1997, several programs arose that resulted in an up-surge in funding for malaria control in India. As a consequence ofenhanced malaria control, parameters related to different controlpolicies undoubtedly also changed around 1997. Here, we attemptto assess that change by again fitting our malaria model, this timeto case data for the period 1997–2010.

In general, an increase in the use of bednets decreases mosquitobiting rate, increased use of insecticide treated bednets (ITNs) bothdecreases biting rate and increases the mosquito mortality rate,improved treatment increases the recovery rate, and insecticidesincrease the mosquito mortality rate. A combination of these con-trol measures is often used. Our first goal here is to understandwhich of these control measures, or combination of control mea-sures, contributed the most to the decline in the number of malariacases after 1996. Secondly, we want to understand how the in-crease in funding for malaria has affected the competition betweenP. vivax and P. falciparum.

To address the first question – which parameters contributedthe most to the post-1996 decline in cases – we fit the model tothe 1997–2010 data several times, each time estimating a differentcombination of parameters relevant to malaria control while leav-ing the remaining parameters in the model fixed to their 1987–1996 estimates. We consider each of these parameterizations ofthe model to be a different model. For each model, we calculatethe corrected Akaike Information Criterion (AICc) – a measure ofthe goodness of fit of a model to the data, discounted by the num-ber of parameters estimated relative to the size of the dataset. TheAICc values allow us to order the models from best to worst: themodel with the smallest AICc is the best model, and the modelwith the largest AICc is the worst model. To make the distinctionbetween the models clearer, we calculate the 4AICc for each mod-el: the difference in AICc between the model and the model withthe smallest AICc. This means that the ‘‘best’’ model has a 4AICcof zero. The results of this model comparison are summarized inTable 8. The rule of thumb is that candidate models with 4AICc’sbetween 0 and 2 have strong support, models with4AICc between4 and 7 have considerably less support (but should still beconsidered), and models with4AICc greater than 10 should be dis-regarded as potential candidates [5].

Table 7Pre-1997 mean, median, standard deviations, and confidence intervals for RCv and RCf ,derived from parametric bootstrap.

Parameter Mean Median Stand. dev. 95% CI

RCv 1.0203 1.0204 0.0006 1.0192–1.0214RCf 1.0052 1.0052 0.0001 1.005–1.0055

1.004 1.006 1.008 1.024 1.026 1.028 1.03 1.004 1.0060

50

100

150

200

250

300Fr

eque

ncy

//

RCv

Rfv

1.006 1.008 1.07 1.072 1.074 1.076 1.0780

50

100

150

200

250

300

Freq

uenc

y

//

RCf

Rvf

Fig. 3. Histograms of (a) RCv and Rvf data and (b) RCf and Rf

v data generated by bootstrap for the period 1987–1996.

O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50 43

The results of this analysis yielded that modelA ¼ fcv ; cf ; av ; af g corresponding to estimating treatment recoveryrate and biting rate parameters best explains the observed data.Fig. 4(b) illustrates the fit of model A to the case data. The trans-mission rates bv ; bf ; bv ; bf are all dependent on mosquito bitingbehavior. Thus control measures that affect biting rate, which wedenote by av and af , will also modify the value of bv ; bf ; bv , andbf . We assume the transmission rates are directly proportional tobiting rates, and consequently, an x% reduction in biting rate ai,will result in an equal percent reduction in transmission rates bi

and bi. We allow av and af to be different because there is evidencethat control measures may affect vivax and falciparum transmission

Table 8Post-1996 models ordered by 4AICc value (difference from best AICc value �123.9).

Model Parameter Percent change from pre-1997 estimate

CI DAICc

A av �43.3 �67.3 – �19.3 0.0af �1.1 �1.2 – �1.1cv �68.1 �97.4 – �38.8cf �1.2 �1.5 – �0.9

B av �45.4 �67.9 – �22.9 2.8af 0.9 �7.9 – 9.8cv �73.3 �97.4 – �49.2cf -5.9 �22.6 – 10.8d 9.4 9.2 – 9.5

C av �3.1 �3.3 – �2.9 4.1af �0.5 �0.5 – �0.5

D cv 6.9 6.8 – 7.0 5.2cf 1.0 0.9 – 1.1

E av �0.6 �0.8 – �0.4 7.0af 2.0 2.0 – 2.1d 5.2 5.1 – 5.2

F cv 3.7 3.5 – 3.8 7.1cf �1.9 �2.4 – �1.4d 2.9 2.4 – 3.4

G d 2.4 1.8 – 3.0 85.7

differently. Using the rule of thumb for 4AICc values, model B hasstrong support, models C;D; E, and F have less support but shouldstill remain in the pool of possible models, and model G should bediscarded. However, it is important to point out that models A andB were sensitive to the initial guess for the parameter values in thefitting procedure, whereas the remaining model results were fairlyrobust to the initial guess. This means that the relationship be-tween models C;D; E;F , and G remain the same for different initialparameter guesses while A and B find different positions in the listdepending on the initial guess. We arrived at the ordering pre-sented in Table 8 by repeating the fitting procedure for 3 differentinitial guesses for each of the seven models, and choosing the esti-mates corresponding to the smallest confidence intervals.

In general, adding the estimation of d, mosquito death rate, to amodel increased the AICc value, suggesting that changes in mos-quito death rate do not explain the decline in cases beginning in1997. Similarly, since model C ¼ fav ; af g performed better thanD ¼ fcv ; cf g and likewise E ¼ fav ; af ; dg performed better thanF ¼ fcv ; cf ; dg, we conclude that changes in mosquito biting ratebetter explain the decline in malaria prevalence than do changesin treatment recovery rates. Moreover, a smaller change in bitingrate (roughly half) is required to yield the same results as changingthe treatment recovery rate.

Some of the results are more surprising and difficult to inter-pret. For example, the results of model A suggest that treatmentrecovery rates in 1997–2010 were worse, particularly for treat-ment of vivax malaria, than in 1987–1996. This outcome of themodel could be a consequence of increased parasite resistance todrugs.

ModelsA through F can also provide some insight into how en-hanced control measures affect the competition between P. falcipa-rum and P. vivax. Using the parameter estimates yielded by eachcandidate model and the analytic expressions for the reproductionnumbers RCv and RCf along with analytic expressions for the inva-sion numbers Rf

v and Rvf , we can determine in which region of the

competitive outcome graph the point fRCv ;RCf g lies. The set of can-didate models fA;Bg yields a set of reproduction numbers lying ina region where P. vivax outcompetes P. falciparum (Fig. 5). On theother hand, the set of models fC;D; E;Fg yields a set of reproduc-tion numbers lying within corresponding invasion boundarieswhere P. falciparum outcompetes P. vivax. A summary of the isola-tion reproduction numbers resulting from each model candidate isgiven in Table 9. The result that model A predicts P. vivax will out-compete P. falciparum is surprising given that the data suggests the

RCv

RC

f

Rfv=1

Rvf =1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(a)RCv

RC

f

0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 1.0250.999

1

1.001

1.002

1.003

1.004

1.005

(RCv

, RCf

) 1987−1996

(RCv

, RCf

) 1997−2010

(b)Fig. 5. (a) Graph of Rf

v ¼ 1 and Rvf ¼ 1 for the period 1987–1996 (green and blue lines, respectively) and 1997–2010 (grey lines) as a function of RCv and RCf ; (b) Plot of the

point RCv ;RCf� �

for India before and after 1997. Prior to 1997, India was in the coexistence region. During period of enhanced control measures (1997–2010), India is in theregion where P. vivax will eventually outcompete P. falciparum.

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996

0.8

1

1.2

1.4

1.6

1.8

2

Year

Indi

vidu

als

(in

mill

ions

)

Vivax Cases

Falciparum/Mixed Cases

Vivax Data

Falciparum/Mixed Data

(a)

1985 1990 1995 2000 2005 2010

0.8

1

1.2

1.4

1.6

1.8

2

Year

Indi

vidu

als

(in

mill

ions

)

Vivax CasesFalciparum/Mixed CasesVivax DataFalciparum/Mixed Data

(b)Fig. 4. (a) Best fit of model to 1987–1996 case data; (b) best fit of model to 1997–2010 data. Data from [11].

44 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

opposite. However, extending the solution corresponding to modelA (Fig. 6)) to the year 2200 confirms that the data is potentiallymisleading. Although the proportion of cases due to falciparumhas been increasing, model A reveals that this observation is insuf-ficient to draw conclusions about the longterm competitive out-come of the two species.

While this type of analysis has the potential to unveil informa-tion regarding the future of malaria in a region, the reproductionnumbers in each case lie very close to the invasion boundaries,and consequently it is difficult to draw definitive conclusions aboutthe outcome of malaria. To address this concern, we again carriedout a parametric bootstrap procedure to not only estimate confi-dence intervals for the reproduction numbers, but to also deter-mine what the probability is that the reproduction number willlie in any one of the four possible competitive-outcome regions.For each of the six candidate models, we also calculated the AICcfor every 1000 runs in the bootstrap routine to determine whatthe most frequent ordering of the set of candidate models is. Tomake sure that the results of the bootstrap method betweenmodels is comparable, we draw 1000 sets of data from a Poisson

distribution with mean equal to the solution curve associated withmodel A in Table 8 – the best fitting model based on AICc values.

The bootstrapping procedure allowed us to compile 1000 sets ofparameter estimates for each of the six models (ignoring model Gbecause of the poor fit), from which we computed 1000 pairs ofreproduction numbers ðRCv ;RCf Þ. The new parameter sets andreproduction number pairs were used to compute the invasionnumbers for the 1000 runs, allowing us to determine what theprobability is that a model will land in a particular competitive-outcome region. The results are listed in Table 10. The competitiveoutcomes vary the most for models A and B, which is consistentwith our earlier observation that these two models were the mostsensitive to the initial parameter guess used for fitting.

Although the 1000 bootstrapped samples resulted in 123 differ-ent orderings, 5 orderings made up more than half of the samples.The original ordering fA;B; C;D; E;F ;Gg occurred 10.1% of thetime. 29.5% of the runs led to the ordering fA; C;D; E;F ;B;Gg.9.7% of the samples yielded the ordering fC;D; E;F ;A;B;Gg. Theordering fC;D; E;F ;B;A;Gg appeared 8.4% of the time, while 4.7%of the samples resulted in the ordering fB;A; C;D; E;F ;Gg. The AICc

2000 2020 2040 2060 2080 2100 2120 2140 2160 2180 22000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Year

Indi

vidu

als

(in

mill

ions

)

Vivax CasesFalciparum/Mixed CasesVivax DataFalciparum/Mixed Data

Fig. 6. Model A solution curve extended to the year 2200.

Table 10Percentage of bootstrap runs in which vivax and falciparum will coexist (I), vivax willoutcompete falciparum (II), falciparum will outcompete vivax (III), and the percentageof runs in which both will become extinct (IV).

I II III IV

A 14.3 44.0 19.2 22.5B 4 11.3 53.3 31.4C 0.2 0 77.4 22.4D 0 0 74.3 25.7E 0 0 77.9 22.1F 0.4 0 83.40 16.2

Table 91997–2010 estimates of RCv and RCf for each candidate model.

A B C D E F

RCv 1.00111 1.00221 0.98891 0.98757 0.98874 0.98820RCf 0.99990 0.99997 1.00018 1.00014 1.00015 1.00022

O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50 45

values selected model A as the top model 54.4% of the time. Ofthese 544 samples for which model A was selected as the top

0.98 0.985 0.99 0.995 1 1.005 1.01 1.0150

50

100

150

200

250

300

Freq

uenc

y

RCf

Rvf

Fig. 7. Subfigure (a) presents a histogram of the Rfv and RCv bootstrap data for mode

model, roughly 15.1% yielded the outcome that vivax and falcipa-rum would continue to coexist, 44.9% yielded that vivax wouldoutcompete falciparum, 18.4% yielded that falciparum would out-compete vivax, and finally 21.7% yielded that both species wouldbecome extinct.

Determining confidence intervals for the reproduction numbersfor each model was not as straightforward as it was for the 1987–1996 time period. Histograms of the reproduction numbers foreach model revealed that not all of the samples of RCf were sym-metric. In fact, the collection of reproduction numbers RCf for mod-el B exhibits a bimodal distribution. Since, models A and C werethe most common models taking ‘‘first place’’, and because theircorresponding reproduction numbers exhibited fairly symmetricdistributions (see Fig. 7), we present the confidence intervals forthese two models. The 95% confidence intervals for RCv and RCf ,respectively, corresponding to model A are (0.99133–1.00640)and (0.99890–1.00075). For model C, the confidence intervals are(0.98676–0.99079) and (0.99979–1.00054). Ultimately, the per-centages in Table 10 provide more meaningful information thanthe confidence intervals derived for the post-1996 reproductionnumbers. Fig. 7 illustrates that the spread of the RCf data resultingfrom the bootstrap procedure was always less than the spread ofRf

v . Conversely, the variance in RCv is greater than that of Rvf . This

observation, which was consistent across all six candidate modelssuggests that RCf is less sensitive than Rf

v , and RCv is more sensitivethan Rv

f , to changes is parameter values.

4. Discussion

India, as is true for many other countries, has struggled with thecontrol of malaria, experiencing several ups and downs. While morerecent efforts have been successful in dramatically decreasing thenumber of cases, India is still far from reaching its goal. Conse-quently, knowing which of the control strategies India’s successcan be attributed to is valuable to India’s future success and couldhelp India use their resources more efficiently. The presence oftwo malaria parasites in India makes this a challenging problem,both in practice and in terms of mathematical modeling. To ourknowledge, attempts to model both vivax and falciparum ([35,36])at the population level do not include the possibility of co-infection.Chiyaka et al. [7] address co-infection in their falciparum–malariaemalaria model, however the symmetric nature of this model doesnot lend itself well to the application to falciparum and vivax. Ourfalciparum–vivax model addresses the need for a model that

0.985 0.99 0.995 1 1.005 1.01 1.0150

50

100

150

200

250

Freq

uenc

y

RCv

Rfv

l A. Subfigure (b) is a histogram of the Rvf and RCf bootstrap data for model A.

46 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

considers not only the possibility of co-infection, but also the char-acteristics of vivax that differentiate it from falciparum.

Competition between species can have a profound effect on sur-vival. We have shown with our model, by studying the invasionboundaries, that two species can coexist, even if the isolated repro-duction number of one of the species is less than one. This hasimportant consequences for malaria control, since reducing oneof the reproduction numbers below one may not be sufficient toeradicate either pathogen or the disease.

The emergence of parasite resistance to drug therapies is also ofgreat concern since this foreboding obstacle poses a threat to thesuccess of malaria control. While we do not address parasite resis-tance directly in our model, the fitting of several models to the en-hanced malaria control period suggested that sufficient use ofbednets may be able to counteract the negative effects of increasedresistance to the treatment of malaria. In fact, the model selectedas the best model for the majority of the bootstrapped samples(54.4% of the time) in Section 3.3, was one in which both biting rateand treatment recovery rates decreased after 1996. A decrease inrecovery rate increases the average time to recovery followingthe administration of anti-malarial drugs. As expected, our topmodel indicates that decreasing biting rate and increasing the timeto recovery following treatment have opposing effects on thereproduction number.

Incorporating both P. falciparum and P. vivax malaria into ourmodel provided us with a way to determine what the most likelyoutcomes are for malaria in India. Bootstrapping of the best post-1996 model (model A) yielded that P. vivax outcompeting P. falci-parum is the most likely outcome, while the probability of extinc-tion is only slightly more probable than the probability thatfalciparum will outcompete vivax malaria (22.5% versus 19.2%).The remaining candidate models predicted that P. falciparum out-competing vivax is the most likely outcome. A side-by-side com-parison of the histograms of the reproduction numbers and theinvasion numbers revealed that the variance in RCf was always lessthan the variance in Rf

v . Conversely, the variance in RCv is greaterthan that of Rv

f . This means that estimating the reproduction num-bers alone may not be a good predictor of the outcome of thedisease.

The application of our mathematical model to data suggestedthat the future of malaria in India is uncertain. Although we ad-dressed the uncertainty in the model predictions, it’s importantfor us to note that applying the same methods to data sets forsmaller regions is likely to produce very different results. In the fu-ture, we hope to use the framework we have developed here tomake more confident predictions about the outcome of malariain various regions of India.

Acknowledgments

Olivia Prosper and Maia Martcheva would like to thank Dr.Christina Chiyaka for her input during the early stages in the devel-opment of the model presented here, as well as the reviewers andeditor for their valuable feedback. Olivia Prosper would also like toacknowledge support from IGERT Grant NSF DGE-0801544.

Appendix A. Appendix

A.1. Finding RC using the next-generation approach

RC is a threshold criterion that determines whether P. vivax or P.falciparum will be able to invade the disease-free equilibrium.

Following the approach of Diekmann et al. [13], we consider asubset of our system comprising only of equations for the infectedstate variables. We order these equations as follows:

fj0v ; L0;C0v ; I

0v ; j0f ;C

0f ; I0f ;C

0vf ;C

0fv ; I

0cg. Next, we write the Jacobian

(evaluated at the disease-free equilibrium) J of the subsystem asthe difference of two matrices F and V (J ¼ F � V). We choose thesematrices such that the elements of F include only new infectionsand the remaining transitions (recovery, relapse, death, or progres-sion to a new disease state) appear in the V matrix, giving us

F ¼ F1 F2

F3 0

� �where,

F1 ¼

0 0 0 f1;4 0f2;1 0 � � � � � � 0

0 ... . .

. ...

0 ... . .

. ...

0 0 � � � � � � 0

0BBBBBBB@

1CCCCCCCA; F2 ¼

0 0 f1;8 0 f1;10

0 0 � � � � � � 0

0 ... . .

. ...

0 0 0 00 f5;7 0 f5;9 f5;10

0BBBBBB@

1CCCCCCA;

F3 ¼

0 0 0 0 f6;5

0 0 � � � � � � 0

0 ... . .

. ...

0 ... . .

. ...

0 0 � � � � � � 0

0BBBBBBB@

1CCCCCCCAand V ¼ V1 V2

0 V3

� �where,

V1 ¼

v1;1 0 0 0 0

0 v2;2 0 �v2;4...

0 �v3;2 v3;3 0 ...

0 �v4;2 v4;3 v4;4 00 � � � � � � 0 v5;5

0BBBBBBBB@

1CCCCCCCCA;

V2 ¼

0 0 0 0 00 0 � � � � � � 0

0 ... . .

.0

0 ... . .

.�v4;10

0 0 � � � � � � 0

0BBBBBBB@

1CCCCCCCA

V3 ¼

v66 0 0 0 0�v7;6 v7;7 0 0 �v7;10

0 0 v8;8 0 00 0 0 v9;9 00 0 �v10;8 �v10;9 v10;10

0BBBBBB@

1CCCCCCAThe nonzero elements of F are f1;4 ¼ bv=K̂; f 2;1 ¼ bv=K̂; f 1;8 ¼bv=K̂; f 1;10 ¼ fbv=K̂; f 5;7 ¼ bf =K̂; f 5;9 ¼ bf =K̂; f 5;10 ¼ ð1� fbf =K̂Þ;f 6;5 ¼ rf bf K̂.

The nonzero elements of V are v1;1 ¼ d; v2;2 ¼ kþ l; v2;4 ¼d; v3;2 ¼ rvk; v3;3 ¼ mv þ l; v4;2 ¼ ð1� rv Þk; v4;3 ¼ mv ; v4;4 ¼ dþqv þ l; v5;5 ¼ d; v4;10 ¼ gf ; v6;6 ¼ mf þ l; v7;6 ¼ mf ; v7;7 ¼ qf þ l;v7;10 ¼ gv ; v8;8 ¼ mvf þ l; v9;9 ¼ mfv þ l; v10;8 ¼ mvf ; v10;9 ¼ mfv ;

v10;10 ¼ gv þ gf þ l.If F is nonnegative and V is a nonsingular M-matrix (a Z-matrix

whose eigenvalues have positive real part), then qðFV�1Þ < 1 if andonly if all eigenvalues of J ¼ F � V have negative real part (Lemma 2in [12]). This is equivalent to saying that if F and V satisfy theseproperties, then the disease free equilibrium is locally asymptoti-cally stable only when the spectral radius (or dominant eigen-value) of FV�1 is less than one. Furthermore, the inverse of anM-matrix is nonnegative [12], so that FV�1 is also nonnegative.FV�1 nonnegative implies that FV�1 has a positive real eigenvaluewith modulus greater than or equal to all other eigenvalues ofFV�1 [4]. In other words, qðFV�1Þ > 0. Since qðFV�1Þ is positive, itmakes sense to define RC to be precisely qðFV�1Þ.

0Þ

O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50 47

Thus, to derive an expression for RC , we must first check that Fand V satisfy the appropriate conditions. Clearly F is a nonnegativematrix and V is a Z-matrix, that is, a matrix with nonpositive off-diagonal elements. One can show that a Z-matrix A is an M-matrixby showing that there exists a nonnegative vector v such that Av ispositive [15].

We claim that for v ¼ ð1; � � � ;1ÞT ;VTv is positive. Since V is a Z-matrix, it is clear that VT is also a Z-matrix. Furthermore, since Vand VT have the same eigenvalues, if VT is an M-matrix, then sois V. Showing that VTv > 0 is equivalent to showing that all rowsums of VT are positive, or equivalently that all column sums ofV are positive.

It is simple to show that Sj :¼P10

i¼1v i;j > 0 for eachj 2 f1;2; � � � ;10g:

S1 ¼ v1;1 > 0S2 ¼ v2;2 � v3;2 � v4;2

¼ kþ l� rv � ð1� rvÞ ¼ l > 0S3 ¼ v3;3 � v4;3

¼ mv þ l� mv ¼ l > 0S4 ¼ �v2;4 þ v4;4

¼ �dþ dþ qv þ l ¼ qv þ l > 0S5 ¼ v5;5 > 0S6 ¼ v6;6 � v7;6

¼ mf þ l� mf ¼ l > 0S7 ¼ v7;7 > 0S8 ¼ v8;8 � v10;8

¼ mvf þ l� mvf ¼ l > 0S9 ¼ v9;9 � v10;9

¼ mf v þ l� mf v ¼ l > 0S10 ¼ �v4;10 � v7;10 þ v10;10

¼ �gf � gv þ gv þ gf þ l ¼ l > 0:

Thus, V is an M-matrix, and consequently, RC ¼ qðFV�1Þ. To deter-mine the expression for RC , we first compute the inverse of V usingthe formula V�1 ¼ 1

detðVÞAdjðVÞ, where AdjðVÞ is the adjugate of V.

ci;j :¼ ð�1ÞðiþjÞVi;j is called the ði; jÞ cofactor of V. The matrix C whoseelements are the cofactors of V is called the cofactor matrix of V. Theadjugate of V is defined to be the transpose of the cofactor matrix of

V, that is, AdjðVÞ :¼ CT . We find that C�1 ¼ 1detðVÞ

C1 C2

0 C3

� �, where Ci

are defined as follows for i ¼ 1;2;3:

C1 ¼

c1;1 0 0 0 0

0 c2;2 c3;2 c4;2 0

0 c2;3 c3;3 c4;3 0

0 c2;4 c3;4 c4;4 0

0 0 0 0 c5;5

0BBBBBBB@

1CCCCCCCA;

C2 ¼

0 0 0 0 0

0 0 c8;2 c9;2 c10;2

0 0 c8;3 c9;3 c10;3

0 0 c8;4 c9;4 c10;4

0 0 0 0 0

0BBBBBBB@

1CCCCCCCA; and

C3 ¼

0 0 0 0 0c6;7 c7;7 c8;7 c9;7 c10;7

0 0 c8:8 0 00 0 0 c9;9 00 0 c8;10 c9;10 c10;10

0BBBBBB@

1CCCCCCA

Thus, FV�1 ¼ 1detðVÞ

K1 0 K2

0 K3 K4

0 0 0

0@ 1A, where

K1 ¼

0 f1;4c2;4 f1;4c3;4 f1;4c4;4

f2;1c1;1 0 0 00 0 0 00 0 0 0

0BBB@1CCCA

K2 ¼

f1;4c8;4 þ f1;8c8;8 þ f1;10c8;10 f1;4c9;4 þ f1;10c9;10 f1;4c10;4 þ f1;10c10;10

0 0 00 0 00 0 0

0BBB@1CCCA

K3 ¼0 f5;7c6;7 f5;7c7;7

f6;5c5;5 0 0f7;5c5;5 0 0

0B@1CA and

K4 ¼f5;7c8;7 þ f5;10c8;10 f5;7c9;7 þ f5;9c9;9 þ f5;10c9;10 f5;7c10;7 þ f5;10c10;10

0 0 00 0 0

0B@1CA

The nonzero eigenvalues of FV�1 are precisely the eigenvalues ofdFV�1 , where

dFV�1 ¼ 1detðVÞ

K̂1 00 K̂2

!

and K̂1 ¼0 f1;4c2;4

f2;1c1;1 0

� �and K̂2 ¼

0 f5;7c6;7 f5;7c7;7

f6;5c5;5 0 0f5;7c5;5 0 0

0@ 1A.

Since dFV�1 is block triangular, its eigenvalues are the eigenvalues

of K̂1=detðVÞ and K̂2=detðVÞ.Using the formula for the elements ci;j of the cofactor matrix C,

we find that

c1;1 ¼ ½v2;2v3;3v4;4 � v2;4ðv3;2v4;3 þ v3;3v4;2Þ�ðv5;5v6;6v7;7v8;8v9;9v10;1

c2;4 ¼ v1;1ðv32v43 þ v33v42Þv5;5v6;6 � � � v10;10

c5;5 ¼ v1;1½v2;2v3;3v4;4 � v24ðv32v43 þ v33v42Þ�v6;6 � � �v10;10

c6;7 ¼ v1;1v5;5v7;6½v2;2v3;3v4;4 � v24ðv32v43 þ v33v42Þ�v8;8 � � � v10;10

c7;7 ¼ v1;1v5;5 � � �v10;10½v2;2v3;3v4;4 � v24ðv32v43 þ v33v42Þ�

The determinant of V is detðVÞ ¼ v1;1v2;2 � � �v10;10½1� k2;4

ðk3;2k4;3 þ k4;2Þ�, where we define the ki;j’s as follows:

ki;j ¼ v i;j=v j;j

k1;j ¼ f1;j=v j;j

ki;1 ¼ fi;1=v1;1

ð10Þ

for i; j – 1.By finding the roots of the characteristic polynomials of K̂1 and

K̂2, we arrive at the analytic expression for RC : RC ¼ maxfRCv ;RCfg,

where RCv and RCfare as described in Section 2.6.

A.2. Finding Rfv using the next-generation approach

The procedure for finding an analytic expression for the inva-sion reproduction numbers Rf

v and Rvf , although more challenging

to carry-out, is identical to the procedure presented in A.1 forderiving RC . The infected subsystem will now consist only of equa-tions for state variables infected with P. vivax, since we want todetermine the stability of the falciparum-only equilibrium when vi-vax attempts to invade. We first find the Jacobian of our infectedsubsystem, evaluated at the falciparum-only equilibrium, with

48 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

the equations ordered as follows: fj0v ; L0;C0v ; I

0v ;C

0vf ;C

0fv ; I

0cg. We write

J ¼ F � V , where F and V are 7� 7 square matrices and

F ¼

0 0 0 f1;4 f1;5 0 f1;7

f2;1 0 � � � � � � � � � � � � 0

0 ... . .

. ...

0 ... . .

. ...

0 ... . .

. ...

f6;1... . .

. ...

0 0 � � � � � � � � � � � � 0

0BBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCAand

V ¼

v1;1 0 � � � � � � 0 � � � 0

0 v2;2 0 �v2;4... . .

. ...

..

.�v3;2 v3;3 0 0 � � � 0

..

.�v4;2 �v4;3 v4;4 0 0 �v4;7

..

.�v5;2 0 �v5;4 v5;5 0 0

0 � � � � � � � � � 0 v6;6 00 � � � � � � 0 �v7;5 �v7;6 v7;7

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCAwhere the elements of F are: f1;4 ¼ bvð1� j�vÞ=K̂; f 1;5 ¼ f1;4;

f 1;7 ¼ ff1;4; f 2;1 ¼ bvS�; andf6;1 ¼ af bv I�f .The elements of V are: v1;1 ¼ d; v2;2 ¼ avbf j�f þ kþ l; v2;4 ¼ d;

v3;2 ¼ rvk; v3;3 ¼ mv þ l; v4;2 ¼ ð1� rvÞk; v4;3 ¼ mv ; v4;4 ¼ avbf j�fþ dþ qv þ l; v4;7 ¼ gf ; v5;2 ¼ avbf j

�f ; v5;4 ¼ v5;2; v5;5 ¼ mvf þ l;

v7;5 ¼ mvf ; v7;6 ¼ mfv ; v7;7 ¼ gv þ gf þ l.Again, it is clear that V has the Z-sign pattern. So, as described in

A.1, it is straightforward to show that V is an M-matrix by verifyingthat all column-sums of V are positive. The inverse of V has the fol-lowing form, where the dots represent zeros:

V�1 ¼

dv1;1 � � � � � �� dv2;2 dv2;3 dv2;4 dv2;5 dv2;6 dv2;7

� dv3;2 dv3;3 dv3;4 dv3;5 dv3;6 dv2;7

� dv4;2 dv4;3 dv4;4 dv4;5 dv4;6 dv4;7

� dv5;2 dv5;3 dv5;4 dv5;5 dv5;6 dv5;7

� � � � � dv6;6 �� dv7;2 dv7;3 dv7;4 dv7;5 dv7;6 dv7;7

0BBBBBBBBBBB@

1CCCCCCCCCCCA:

So,

FV�1 ¼

f1;4dv4;2 f1;4dv4;3 f1;4 dv4;4 f1;4dv4;5 f1;4 dv4;6 f1;4dv4;7

� þf1;5dv5;2 þf1;5dv5;3 þf1;5 dv5;4 þf1;5dv5;5 þf1;5dv5;6 þf1;5 dv5;7

þf1;7dv7;2 þf1;7dv7;3 þf1;7 dv7;4 þf1;7dv7;5 þf1;7dv7;6 þf1;7 dv7;7

f2;1dv1;1 � � � � � �� � � � � � �� � � � � � �� � � � � � �

f6;1dv1;1 � � � � � �� � � � � � �

0BBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCAThe nonzero eigenvalues of FV�1 are precisely the nonzero eigen-values of the 3 � 3 matrix

dFV�1 ¼

f1;4dv4;2 f1;4dv4;6

� þf1;5dv5;2 þf1;5dv5;6

þf1;7dv7;2 þf1;7dv7;6

f2;1dv1;1 � �f6;1dv1;1 � �

0BBBBBB@

1CCCCCCAThe characteristic polynomial of dFV�1 is given by pðsÞ ¼

jdFV�1 � sIj ¼ s½k6;1ðf1;4dv4;6 þf1;5dv5;6 þ f1;7dv7;6Þ þ k2;1ðf1;4dv4;2 þ f1;5dv5;2 þ f1;7dv7;2Þ � s2�, where the ki;j’s are defined as in (10). Since V

is an M-matrix, we know that the inverse of V has only nonnegativeelements. Thus, dv5;6 ; dv7;6 ;dv5;2 ; anddv7;2 are nonnegative, and thelargest positive root of pðsÞ is

s� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik6;1ðf1;4dv4;6þ

qf1;5dv5;6 þ f1;7dv7;6Þ þ k2;1ðf1;4dv4;2 þ f1;5dv5;2þ

f1;7dv7;2Þ�.Recall that cv i;j are the elements of V�1. Thus, cv i;j ¼ ð�1Þiþj

cj;i=detðVÞ, where C :¼ ðci;jÞ is the cofactor matrix of V. To determinedv4;6 ; dv5;6 ; dv7;6 ; dv4;2 ; dv5;2 ; and dv7;2 , we need only calculatec6;4; c6;5; c6;7; c2;4; c2;5; c2;7; and detðVÞ.

c6;4 ¼ v1;1v2;2v3;3v5;5v4;7v7;6

c6;5 ¼ v1;1v7;6v4;7v3;3ðv2;2v5;4 þ v2;4v5;2Þc6;7 ¼ v1;1v7;6v5;5½v2;2v3;3v4;4 � v2;4ðv3;2v4;3 þ v3;3v4;2Þ�c2;4 ¼ v1;1v6;6½v7;5v5;2v3;3v4;7 þ v7;7v5;5ðv3;2v4;3 þ v3;3v4;2Þ�c2;5 ¼ v1;1v6;6v7;7½v3;2v4;3v5;4 þ v3;3ðv4;2v5;4 þ v4;4v5;2Þ�c2;7 ¼ v1;1v6;6v7;5½v3;2v4;3v5;4 þ v3;3ðv4;2v5;4 þ v4;4v5;2Þ�

and

detðVÞ ¼ v1;1v2;2 � � � v7;7½1� k2;4ðk3;2k4;3 þ k4;2Þ

� k7;5k4;7ðk5;4 þ k2;4k5;2Þ� ¼ 1� d

avbf j�f þ dþ qv þ l

rvk

avbf j�f þ kþ l

� mv

mv þ lþ ð1� rvÞk

avbf j�f þ kþ l

!

� mvf

mvf þ l�

gf

gv þ gf þ lavbf j

�f

avbf j�f þ dþ qv þ l

þ

d

avbf j�f þ dþ qv þ l�

avbf j�f

avbf j�f þ kþ l

!The invasion reproduction number Rf

v :¼ s�. Making the appropriatesubstitutions into s�, we arrive at the expression for Rf

v in Section2.8, Eq. (6).

A.3. Finding Rvf using the next-generation approach

To derive Rvf , we find the Jacobian of the falciparum-infected

subsystem, with the order:fj0f ;C

0f ; I0f ;C

0vf ;C

0fv ; I

0cg. The Jacobian evaluated at the vivax-only

equilibrium is given by J ¼ F � V , where

F ¼

0 0 f1;3 0 f1;5 f1;6

f2;1 0 � � � � � � � � � 0

f3;1... . .

. ...

f4;1... . .

. ...

0 ... . .

. ...

0 0 0 0 0 0

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA; and

V ¼

v1;1 0 0 0 0 00 v2;2 0 0 0 00 �v3;2 v3;3 0 0 �v3;6

0 0 0 v4;4 0 00 0 �v5;3 0 v5;5 00 0 0 �v6;4 �v6;5 v6;6

0BBBBBBBB@

1CCCCCCCCAwhere f1;3 ¼ bf ð1� j�vÞK̂; f 1;5 ¼ f1;3; f 1;6 ¼ ð1� fÞf1;3; f 2;1 ¼ rf bf S

�;

f 3;1 ¼ ð1� rf Þbf S�; f 4;1 ¼ avbf ðI�v þ L�Þ, and v1;1 ¼ d; v2;2 ¼ mf þ l;

v3;2 ¼ mf ; v3;3 ¼ af bv j�v þ qf þ l; v3;6 ¼ gv ; v4;4 ¼ mvf þ l; v5;3 ¼af bv j�v ; v5;5 ¼ mfv þ l; v6;4 ¼ mvf ; v6;5 ¼ mfv ; v6;6 ¼ gv þ gf þ l.

F is nonnegative and V is a nonsingular Z-matrix. We show, aswe did in appendices A.1 and A.2, that V is also an M-matrix by

O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50 49

showing that the column sums of V are positive. Since each v i;j > 0and v2;2 > v3;2;v3;3 > v5;3; v4;4 > v6;4;v5;5 > v6;5, and v6;6 > v3;6,each column sum is positive, and hence V is a nonsingularM-matrix. Thus, V�1 is nonnegative (hence FV�1 is also nonnega-tive) and all eigenvalues of J have negative real part if and only ifRv

f :¼ qðFV�1Þ < 1. Using the notation that C :¼ ðci;jÞ is the cofactormatrix of V, we have that

V�1 ¼ 1detðVÞ �

c1;1 � � � � �� c2;2 � � � �� c2;3 c3;3 c4;3 c5;3 c6;3

� � � c4;4 � �� c2;5 c3;5 c4;5 c5;5 c6;5

� c2;6 c3;6 c4;6 c5;6 c6;6

0BBBBBBBB@

1CCCCCCCCASo,

FV�1¼ 1detðVÞ �

f1;3c2;3 f1;3c3;3 f1;3c4;3 f1;3c5;3 f1;3c6;3

0 þf1;5c2;5 þf1;5c3;5 þf1;5c4;5 þf1;5c5;5 þf1;5c6;5

þf1;6c2;6 þf1;6c3;6 þf1;6c4;6 þf1;6c5;6 þf1;6c6;6

f2;1c1;1 0 � � � � � � � � � 0

f3;1c1;1... . .

. ...

f4;1c1;1... . .

. ...

0 ... . .

. ...

0 0 0 0 0 0

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCAObserve that the nonzero eigenvalues of FV�1 are exactly the non-zero eigenvalues of

dFV�1 ¼ 1detðVÞ �

f1;3c2;3 f1;3c3;3 f1;3c4;3

0 þf1;5c2;5 þf1;5c3;5 þf1;5c4;5

þf1;6c2;6 þf1;6c3;6 þf1;6c4;6

f2;1c1;1 0 � � � 0

f3;1c1;1... . .

. ...

f4;1c1;1 0 � � � 0

0BBBBBBBBB@

1CCCCCCCCCA; where

c1;1 ¼ v2;2 � � �v6;6ð1� k3;6k5;3k6;5Þc2;3 ¼ v1;1v4;4v5;5v6;6v3;2

c3;3 ¼ v1;1v2;2v4;4v5;5v6;6

c4;3 ¼ v1;1v2;2v5;5v3;6v6;4

c2;5 ¼ v1;1v4;4v6;6v3;2v5;3

c3;5 ¼ v1;1v2;2v4;4v6;6v5;3

c4;5 ¼ v1;1v2;2v5;3v3;6v6;4

c2;6 ¼ v1;1v4;4v3;2v5;3v6;5

c3;6 ¼ v1;1v2;2v4;4v5;3v6;5

c4;6 ¼ v1;1v2;2v3;3v5;5v6;4; anddetðVÞ ¼ v1;1 � � � v6;6ð1� k3;6k5;3k6;5Þ

The only positive root of the characteristic polynomial pðsÞ ¼

jdFV�1 � sIj is s� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1 þ a2 þ a3p

, where (using the ki;j notation in(10))

a1 ¼k2;1

1� k3;6k5;3k6;5� k3;2k1;3 þ k3;2k5;3k1;5 þ k3;2k5;3k6;5k1;6ð Þ

a2 ¼k3;1

1� k3;6k5;3k6;5k1;3 þ k5;3k1;5 þ k5;3k6;5k1;6ð Þ

a3 ¼k4;1

1� k3;6k5;3k6;5� k6;4k1;6 þ k6;4k3;6k1;3 þ k6;4k3;6k5;3k1;5ð Þ

Since the invasion number Rvf is defined to be the dominant eigen-

value of FV�1;Rvf ¼ s� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1 þ a2 þ a3p

, the expression presented inSection 2.8, Eq. (5).

References

[1] T. Adak, V.P. Sharma, V.S. Orlov, Studies on the Plasmodium vivax relapsepattern in Delhi, India, Am. J. Trop. Med. Hyg. 59 (1998) 175.

[2] N.T.J. Bailey, The Biomathematics of Malaria, Charles Griffin & Company Ltd.,London, 1982.

[3] T. Barik, B. Sahu, V. Swain, A review on Anopheles culicifacies: from bionomicsto control with special reference to Indian subcontinent, Acta Trop. 109 (2009)87.

[4] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,Academic, New York, 1970.

[5] K.P. Burnham, D.R. Anderson, Multimodel inference: understanding AIC andBIC in model selection, Sociol. Methods Res. 33 (2004) 193.

[6] Malaria facts, CDC.[7] C. Chiyaka, Z. Mukandavire, P. Das, F. Nyabadza, S.D. Hove-Musekwa, H.

Mwambi, Theoretical analysis of mixed plasmodium malariae andplasmodium falciparum infections with partial cross-immunity, J. Theor.Biol. 263 (2010) 169.

[8] G. Chowell, C. Ammon, N. Hengartnera, J. Hyman, Transmission dynamics ofthe great influenza pandemic of 1918 in geneva, Switzerland: assessing theeffects of hypothetical interventions, J. Theor. Biol. 241 (2006) 193–204.

[9] A. Das, A.R. Anvikar, L.J. Cator, R.C. Dhiman, A. Eapen, N. Mishra, B.N. Nagpal, N.Nanda, K. Raghavendra, A.F. Read, S.K. Sharma, O.P. Singh, V. Singh, P. Sinnis,H.C. Srivastava, S.A. Sullivan, P.L. Sutton, M.B. Thomas, J.M. Carlton, N. Valecha,Malaria in India: the center for the study of complex malaria in India, ActaTrop. 121 (2012) 267.

[10] A.P. Dash, N. Valecha, A.R. Anvikar, A. Kumar, Malaria in India: challenges andopportunities, J. Biosci. 33 (2008) 583.

[11] National Vector Born Disease Control Programme.[12] O. Diekmann, J.A.P. Heesterbeek, Mathematical epidemiology, Further Notes

on the Basic Reproduction Number, Springer, 2008.[13] O. Diekmann, J.A.P. Heesterbeek, M.G. Roberts, The construction of next-

generation matrices for compartmental epidemic models, J. R. Soc. Interface(2009).

[14] Ralph D. Feigin, Textbook of Pediatric Infectious Diseases, 5th ed., Saunders,Philadelphia, PA, 2004.

[15] M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics,Martinus Nijhoff, Dordrecht, 1986.

[16] A.K. Githeko, W. Ndegwa, Predicting malaria epidemics in the Kenyanhighlands using climate data: a tool for decision makers, Global ChangeHuman Health 2 (2001) 54.

[17] B. Gupta, P. Gupta, A. Sharma, V. Singh, A.P. Dash, A. Das, High proportion ofmixed-species plasmodium infections in india revealed by pcr diagnosticassay, Trop. Med. Int. Health 15 (July 2010).

[18] D. Hamer, M. Singh, B. Wylie, K. Yeboah-Antwi, J. Tuchman, M. Desai, V.Udhayakumar, P. Gupta, M. Brooks, M. Shukla, K. Awasthy, L. Sabin, W.MacLeod, A. Dash, N. Singh, Burden of malaria in pregnancy in Jharkhand state,India, Malaria J. 8 (2009) 21.

[19] M.B. Hoshen, A.P. Morse, A weather-driven model of malaria transmission,Malaria J. 3 (2004) 1.

[20] H. Joshi, S.K. Prajapati, A. Verma, S. Kang’a, J.M. Carlton, Plasmodium vivax inIndia, Trends Parasitol. 24 (2008) 228.

[21] A. Kumar, N. Valecha, T. Jain, A.P. Dash, Burden of malaria in India:retrospective and prospective view, Am. J. Trop. Med. Hyg. 77 (2007) 69.

[22] S. Lawpoolsri, E. Klein, P. Singhasivanon, S. Yimsamran, N. Thanyavanich, W.Maneeboonyang, L. Hungerford, J. Maguire, D. Smith, Optimally timingprimaquine treatment to reduce plasmodium falciparum transmission inlow endemicity Thai–Myanmar border populations, Malaria J. 8 (2009) 15.

[23] Life expectancy at birth, total (years), The World Bank.[24] S.W. Lindsay, W.J.M. Martens, Malaria in the African highlands: past, present,

and future, Bull. World Health Organiz. 76 (1998) 33.[25] P. Martens, R.S. Kovats, S. Nijhof, P. de Vries, M.T.J. Livermore, D.J. Bradlet, J.

Cox, A.J. McMichael, Climate change and future populations at risk of malaria,Global Environ. Change 9 (1999) S89.

[26] M. Mayxay, S. Pukritrayakamee, K. Chotivanich, M. Imwong, S. Looareesuwan,N. White, Identification of cryptic coinfection with plasmodium falciparum inpatients presenting with vivax malaria, Am. J. Trop. Med. Hyg. 65 (2001) 588.

[27] M. Mayxay, S. Pukrittayakamee, P.N. Newton, N.J. White, Mixed-speciesmalaria infections in humans, Trends Parasitol. 20 (2004) 233.

[28] Malaria in india, Malaria Site.[29] M. Nacher, U. Silachamroon, P. Singhasivanon, P. Wilairatana, W.

Phumratanaprapin, A. Fontanet, S. Looareesuwan, Comparison of artesunateand chloroquine activities against plasmodium vivax gametocytes,Antimicrob. Agents Chemother. 48 (2004) 2751.

[30] Estimation of true malaria burden in india, A Profile of National Institute ofMalaria Research.

[31] Guidelines for diagnosis and treatment of malaria in india – 2009, NationalVector Born Disease Control Programme.

[32] J.A. Patz, K. Strzepek, S. Lele, M. Hedden, S. Greene, B. Noden, S.I. Hay, L.Kalkstein, J.C. Beier, Predicting key malaria transmission factors, biting and

50 O. Prosper, M. Martcheva / Mathematical Biosciences 242 (2013) 33–50

entomological inoculation rates, using modeled soil moisture in Kenya, Trop.Med. Int. Health 3 (1998) 818.

[33] G.T. Phan, P.J. De Vries, B.Q. Tran, H.Q. Le, N.V. Nguyen, T.V. Nguyen, S.H.Heisterkamp, P.A. Kager, Artemisinin or chloroquine for blood stageplasmodium vivax malaria in vietnam, Trop. Med. Int. Health 7 (2002) 858.

[34] W. Phimpraphi, R. Paul, S. Yimsamran, S. Puangsa-art, N. Thanyavanich, W.Maneeboonyang, S. Prommongkol, S. Sornklom, W. Chaimungkun, I. Chavez, H.Blanc, S. Looareesuwan, A. Sakuntabhai, P. Singhasivanon, Longitudinal studyof plasmodium falciparum and plasmodium vivax in a karen population inthailand, Malaria J. 7 (2008) 99.

[35] P. Pongsumpun, I.M. Tang, The transmission model of P. falciparum and P.vivax malaria between Thai and Burmese, Int. J. Math. Models Methods Appl.Sci. 3 (2004) 19.

[36] P. Pongsumpun, I-Ming Tang, Impact of cross-border migration on diseaseepidemics: case of the P. falciparum and P. vivax malaria epidemic along theThai–Myanmar border, Journal of Biological Systems 18 (2010) 55–73.

[37] International data base (IDB), U.S. Census Bureau.

[38] R.N. Price, E. Tjitra, C.A. Guerra, S. Yeung, N.J. White, N.M. Anstey, Vivaxmalaria: neglected and not benign, Am. J. Trop. Med. Hyg. 77 (Suppl 6) (2007)79.

[39] G. Snounou, N.J. White, The co-existence of plasmodium: sidelights fromfalciparum and vivax malaria in thailand, Trends Parasitol. 20 (2004)333.

[40] P. van den Driessche, J. Watmough, Reproduction numbers and sub-thresholdendemic equilibria for compartmental models of disease transmission, Math.Biosci. 180 (2002) 29.

[41] 10 facts on malaria, WHO.[42] P. Wilairatana, N. Tangpukdee, S. Kano, S. Krudsood, Primaquine

administration after falciparum malaria treatment in malaria hypoendemicareas with high incidence of falciparum and vivax mixed infection: pros andcons, The Korean Journal of Parasitology 48 (2010) 175.

[43] Y. Zhang, Q.-Y. Liu, R.-S. Luan, X.-B. Liu, G.-C. Zhou, J.-Y. Jiang, H.-S. Li, Z.-F. Li,Spatial-temporal analysis of malaria and the effect of environmental factors onits incidence in Yongcheng, China, 2006–2010, BMC Public Health 12 (2012) 1.