Abstract Bovine Babesiosis (BB) is a tick borne parasitic disease with worldwideover 1.3 billion bovines at potential risk of being infected. The disease, also calledtick fever, causes significant mortality from infection by the protozoa upon exposureto infected ticks. An important factor in the spread of the disease is the dispersion ormigration of cattle as well as ticks. In this paper, we study the effect of this factor. Weintroduce a number, P , a “proliferation index,” which plays the same role as the basicreproduction number R0 with respect to the stability/instability of the disease-freeequilibrium, and observe that P decreases as the dispersion coefficients increase. Weprove, mathematically, that if P > 1 then the tick fever will remain endemic. We alsoconsider the case where the birth rate of ticks undergoes seasonal oscillations. Basedon data from Colombia, South Africa, and Brazil, we use the model to determine theeffectiveness of several intervention schemes to control the progression of BB.
Bovine babesiosis (BB, or tick fever) is a tick-borne parasitic zoonotic disease thatcauses significant morbidity and mortality in cattle. Worldwide, over 1.3 billionbovines are potentially at risk of being infected with tick fever because of frequent ex-posure to infected ticks (Alonso et al. 1990; Aranda Lozano 2011; Cuellar et al. 1999;
A. FriedmanMathematical Bioscience Institute and Mathematics Department, The Ohio State University,Columbus, OH 43210, USAe-mail: [email protected]
A.-A. Yakubu (B)Department of Mathematics, Howard University, Washington, DC 20059, USAe-mail: [email protected]
Petzel 2005; Rickhotso et al. 2005; Rock et al. 2004; Smith 1991; Solorio-Rivera et al.1999; Suarez and Noh 2011; Uribe et al. 1990; Zhang and Zhao 2007). The diseaseresults from infection by protozoa in the genus Babesia. Although several differentspecies of Babesia cause BB, Babesia bovis (B. bovis), Babesia bigemia (B. bigemia)and Babesia divergens (B. divergens) are the three most common cause of infectionin cattle. The disease can be found wherever the tick vectors exist. However, the dis-ease is most common in tropical and subtropical areas. B. bovis and B. bigemia aremore common in Africa, Asia, Australia, Central, and South America, while B. diver-gens is more common in some parts of Europe and North Africa (Alonso et al. 1990;Cuellar et al. 1999).
Inside an infected tick, Babesia zygotes multiply as vermicules, which then in-vade many of the infected tick’s organs including the ovaries. The infected femaletick passes the Babesia species to the next generations of ticks through its eggs.Although the Babesia parasite can be transmitted between animals by direct con-tact, this method of transmission is thought to be of minor importance (De Waal andCombrink 2006; Petzel 2005; Rickhotso et al. 2005; Rock et al. 2004; Smith 1991;Solorio-Rivera et al. 1999; Suarez and Noh 2011).
Alonso et al. (1990) used a mathematical formula to simulate the epizootiologyof babesiosis. More recently, Aranda Lozano (2011) used a system of ordinary dif-ferential equations to model the interactions between the BB tick vector and cattle.Aranda Lozeno’s model assumptions include 100 % vertical BB disease transmis-sion in the bovine population, constant birth rate of tick vector and sedentary tickand bovine populations (no dispersion). To study the combined effect of vertical BBtransmission (in both tick and cattle populations), seasonality, and migration on BBtransmission dynamics, we introduce an extension of Aranda Lozeno’s model basedon a system of partial differential equations (PDEs).
There are several recent infectious disease models with dispersion in the liter-ature. SI epidemic model with variable parameters was considered in Allen et al.(2007). Thieme (2009), developed a general framework for defining the basic repro-duction number R0 for infinite dimensional population structure. This concept wassubsequently used in Vaidya et al. (2012) for an avian influenza model, and furtherextended in Lou and Zha (2011) for a malaria model. An anthrax epizootic modelwith migration was considered in Friedman and Yakubu (2013).
BB can be eradicated by eliminating the host tick populations. For example, in1906, such eradication program led to the creation of a National Cattle Fever TickEradication Program in the USA (Petzel 2005). The program targeted all or partof the following southern US states: Alabama, Arkansas, California, Florida, Geor-gia, Kentucky, Louisiana, Mississippi, Missouri, North Carolina, Oklahoma, SouthCarolina, Tennessee, Texas, and Virginia. By 1943, the tick eradication programhad been declared complete, and all that remains today is a permanent BB quar-antine zone along the Rio Grande River in South Texas. The existing BB quar-antine zone is an approximately 500-mile long swath of land stretching from DelRio to Brownsville, Texas, ranging in width from several hundred yards to about10 miles. Within the quarantine zone, BB surveillance and control activities arecarried out by personnel from US Department of Agriculture (USDA), AnimalHealth and Plant Inspection Service (APHIS) and Veterinary Services (VS). Prior
100 A. Friedman, A.-A. Yakubu
to movement out of the quarantine zone, all cattle must be inspected for ticks, de-clared tick-free, and dipped in coumaphos or organophosphate acaricides. WhenBB ticks are found on cattle, the premise of origin is considered infested andplaced under quarantine. Cattle on all adjacent premises are then also inspectedand traceback work begins in order to locate and treat any livestock that have leftthe herd and/or to find the source of livestock added to the herd (Petzel 2005;Zhang and Zhao 2007).
Reintroduction of BB is a significant threat. In some countries, attenuated strainsof B. bovis, B. bigemia, and B. divergens are used to vaccinate cows. A number ofdrugs are known to be effective against Babesia but most of them have been with-drawn due to safety or residue concerns (Lou and Zha 2011; Pascual and Dobson2005; Petzel 2005; Rickhotso et al. 2005; Zhang and Zhao 2007).
To understand when an initial infection of BB disease into a population of sus-ceptible tick and bovine will either disappear or become enzootic, we introduce a“proliferation index,” P , and compute it explicitly in terms of the BB model’s param-eters. P plays the same role as the basic reproduction number R0 in SIR epidemicmodels in the sense that P< 1 (> 1) if R0 < 1 (> 1). Indeed using rigorous mathe-matical analysis, we show that P < 1 implies BB extinction while P > 1 implies thepersistence of the disease. We also show that our analysis can be applied to other in-fectious disease models such as anthrax epizootic and malaria (Dembele et al. 2009;Friedman and Yakubu 2013).
As a case study, we use our model with data from the BB enzootic regions ofNorth Colombia in South America to demonstrate the impact of the profile of theinitial BB infections on the BB epidemic in the region (Aranda Lozano 2011; Suarezand Noh 2011). We illustrate the difference in the total number of infections when BBinfection starts at the boundary versus when it starts in the center of the region. Ourmodel suggests that effective control measures need to include more biosurveillanceprograms at the boundary regions.
Our model suggests that “non-aggressive” form of BB in the sense of small ver-tical transmission, small infectivity rate, or large recovery rate, can be eradicated, orat least stabilized within less than 10 years. We also use the model to study othereradication methods, such as inoculation and dipping.
The paper is organized as follows. In Sects. 2 and 3, we introduce the PDE com-partmental BB epidemic model and a simplified version of the model, respectively.In Sect. 4, we introduce the proliferation index P of the BB epidemic model withdiffusion and constant tick birth rate, μT . In Sect. 5 we show that P < 1 implies BBextinction, and in Sect. 6, we show that P > 1 implies BB persistence. In Sect. 7,we consider the case where μT is a τ -periodic function. The persistence results ofSect. 6 can be applied to other disease models. We illustrate this in Sect. 8 where themethod is applied to an anthrax epizootic model. As a case study, in Sects. 9 and 10we use data from Colombia, South Africa, and Brazil to illustrate the effect of variouscontrol measures of BB. Concluding remarks are presented in Sect. 11.
1.1 Life Cycle of Rhipicephalus Microplus
Rhipicephalus microplus (Boophilus microplus) is a vector for the two more com-mon agents of BB in Colombia and South Africa, B. bigemia and B. bovis. This
A Bovine Babesiosis Model with Dispersion 101
one-host tick vector spends all its life stages on one animal host. The eggs ofBoophilus microplus are hatched in the environment, and the larvae may either beblown away by the wind or they crawl up grass and other plants to find a host.In the summer, Boophilus microplus can survive without feeding for up to 3 to4 months, and up to 6 months in cooler temperatures. Boophilus microplus lar-vae that do not find a host eventually die of starvation. Each developmental stageof Boophilus microplus (larva, nymph, and adult) feeds only once, but the feed-ing takes place over several days. After feeding and mating, the adult female tickdetaches from the host and deposits a single batch of many eggs in the environ-ment. The life cycle of Rhipicephalus annulatus, another vector for both B. bigemiaand B. bovis, is similar to that of Boophilus microplus (Aranda Lozano 2011;De Waal and Combrink 2006; Smith 1991; Solorio-Rivera et al. 1999; Zhang andZhao 2007).
1.2 Life Cycle of Ixodes Ricinus
Ixodes ricinus (castor bean tick) is a BB vector for Babesia divergens in Europe.Unlike the one-host Boophilus microplus, the castor bean tick is a “three-host tick.”A newly hatched larva of Ixodes ricinus feeds on a host, drops off to the ground,and molts to a nymph which seeks out and feeds on a second host, then drops offto the ground, and molts to an adult. The ticks can be found on the host for severaldays while they feed, then drop to the ground to develop to the next stage. The lifecycle of Ixodes ricinus takes two to four years. Ixodes ricinus feeding generally peaksin spring and early summer, with a second active season in autumn in some areas(Aranda Lozano 2011; De Waal and Combrink 2006; Smith 1991; Solorio-Riveraet al. 1999; Zhang and Zhao 2007).
1.3 Model Focus on Babesia Bovis
There are important heterogeneities among the three main species of BB, namelyB. bigemia, B. bovis, and B. divergens. For example, among the more com-mon species in Colombia and South Africa, B. bigemia is known to be morepathogenic while B. bovis has wider distribution. BB is maintained in cattle pop-ulations by asymptomatic carriers that have recovered from the disease. Typi-cally, B. bovis persists in cattle for years, but B. bigemia survives only for a fewmonths. For simplicity, we will use a susceptible-infected-recovered (SIR) compart-mental model with no asymptomatic class to focus primarily on B. bovis. How-ever, by adding the asymptomatic class and making a suitable change of param-eters, our model is easily adaptable for studying any of the other main speciesof BB.
2 Bovine Babesiosis Mathematical Model
We use a non-autonomous system of a continuous-time compartmental modelwith spatial dispersion to study the bovine babesiosis disease epidemic. The to-
102 A. Friedman, A.-A. Yakubu
tal bovine population density per unit area at location x and time t , NB(x, t),is divided into three compartments: susceptible, SB(x, t), infected, IB(x, t), andrecovered, RB(x, t). Susceptible bovine hosts have no BB parasite in their blood-stream, but can become infected with BB after receiving bites from babesiosis in-fected ticks. Infected bovines have the Babesia parasite in their bloodstream, andthe recovered bovine are infected bovines that have been treated for the disease. Atlocation x and time t , the tick total population density per unit area, NT (x, t), isdivided into two compartments: susceptible, ST (x, t), and infected, IT (x, t). Suscep-tible ticks do not carry the disease but may become infected after a blood meal oninfected bovine hosts.
The bovine (respectively, tick) birth and death rates are denoted, respectively, byμB and γB (respectively, μT and γT ). We assume that μB , γB , and γT are con-stants. BB parasite transmission in both bovine and tick populations are assumed tobe frequency dependent, rather than density dependent, due to homogeneous mix-ing. We assume that the disease transmission rates in the bovine population, βB ,and in the tick population, βT , are constants. To include conservation of bites inour model, we make the simplifying assumption that the average number of tick-biting rate is equal to the average number of bites that a bovine receives per unittime.
In the bovine population, we assume vertical disease transmission with very highprobability (1 − q), where q ∈ (0,1) is the small probability of “no vertical” trans-mission of BB. Similarly, in the tick population, vertical transmission of the Babesiaparasite occurs with probability (1 − p), where p ∈ (0,1) is the small probabil-ity of “no vertical” transmission of BB. A constant fraction λB ∈ (0,1) of treatedinfected bovine population is assumed to have recovered, and a constant fractionαB ∈ (0,1) of the recovered bovine population can return to the susceptible compart-ment.
Endemic BB shows recurrent annual seasonal patterns. In the Venda district ofLimpopo Province in South Africa, for example (Rickhotso et al. 2005), BB is mainlytransmitted during the rainy season (October–May) when tick numbers are higher. Tocapture the observed seasonal effects in our model, we assume that the tick birth rateis τ -periodic, that is, μT (t + τ) = μT (t) for t ≥ 0. For simplicity, we do not makeany distinction between ticks on the ground and ticks on the bovine. Grazing cattletend to disperse over a given area from high concentration to low concentration inorder to feed more effectively. For this reason, we shall include a dispersion termin the dynamics of bovine, with constant coefficient dB . It will be shown later onthat increased dispersion increases the probability of BB prevalence. Ticks on cattledisperse with the constant dispersal coefficient of the cattle, while the dispersion ofticks on the ground varies very little seasonally (Hegeman 2003). Consequently, weassume that the tick population is dispersing with a constant dispersal coefficient, dT ,where dT is smaller than dB .
For simplicity, we do not include asymptomatic BB transmission in our model, al-though it is known that Babesia are maintained in cattle population by asymptomaticcarriers (Smith 1991). The following system of equations then describes the spread
A Bovine Babesiosis Model with Dispersion 103
of the BB disease:
∂SB
∂t− dB�SB = μB(SB + RB) + αBRB + qμBIB − βB
SB
NBIT − γBSB,
∂IB
∂t− dB�IB = (1 − q)μBIB + βB
SB
NBIT − (λB + γB)IB,
∂RB
∂t− dB�RB = λBIB − (αB + γB)RB,
∂ST
∂t− dT �ST = μT (ST + pIT ) − βT ST
IB
NB
−γT ST ,
∂IT
∂t− dT �IT = (1 − p)μT IT + βT ST
IB
NB
− γT IT ,
∂NB
∂t− dB�NB = (μB − γB)NB,
∂NT
∂t− dT �NT = (μT − γT )NT ,
NB(x, t) = SB(x, t) + IB(x, t) + RB(x, t),
NT (x, t) = ST (x, t) + IT (x, t),
(1)
where x = (x1, x2) varies in a region Ω , and
�=2∑
j=1
∂2
∂x2j
;
the model parameters are defined in Table 1.
Table 1 Model parametersParameter Definition
αB Rate of treated bovine
βB Bovine per tick infectivity rate
βT Tick per bovine infectivity rate
λB Rate of recovered bovine
γB Bovine death rate
γT Tick death rate
μB Bovine birth rate
μT Tick birth rate
dB Bovine diffusion coefficient
dT Tick diffusion coefficient
p Probability of “no vertical” transmission in ticks
q Probability of “no vertical” transmission in bovines
104 A. Friedman, A.-A. Yakubu
When μT is constant, μB = γB , μT = γT and q = dB = dT = 0, Model (1) re-duces to the following ODE BB model of Aranda Lozano:
dSB
dt= (αB + μB)RB − βB
SB
NB
IT ,
dIB
dt= βB
SB
NB
IT − λBIB,
dRB
dt= λBIB − (αB + μB)RB,
dST
dt= μT pIT − βT ST
IB
NB
,
dIT
dt= −pμT IT + βT ST
IB
NB
.
(2)
Setting
P(L) = βBβT
λBpμT
(3)
for the case NT
NB= 1, Aranda Lozano proved in Aranda Lozano (2011) that the
disease-free equilibrium (DFE) of (2) is stable if P(L) < 1 and unstable if P(L) > 1.The threshold parameter P(L) is related to the basic reproduction number, R0,
used in epidemiology. For system (1) with no dispersion (dB = dT = 0) and con-stant μT . R0 is defined as follows:
An average of 1 infected bovine gives birth to (1−q)μB
(λB+γB)infected bovine and infects
βT
(λB+γB)NT
NBticks. Similarly, average of 1 infected tick gives birth to (1−p)μT
γTinfected
ticks and infects βB
γTbovines. R0 is the spectral radius of the matrix
⎛
⎝(1−q)μB
(λB+γB)βB
γT
βT
(λB+γB)NT
NB
(1−p)μT
γT
⎞
⎠ ,
and the DFE is stable if R0 < 1 and unstable if R0 > 1. For more on how R0 iscomputed for models with vertical transmission, see Diekmann et al. (2012).
One can verify that for model (2), P(L) < 1 if R0 < 1 and P(L) > 1 if R0 > 1.Thus, for the purpose of stability/instability of the DFE, P(L) is a useful index. Weshall refer to P(L) as a “proliferation index,”, and later on extend this definition to anappropriate parameter P for the system (1), showing that the DFE for system (1) isstable (unstable) if the proliferation index P <1 (> 1).
3 The Simplified BB Model
We study Model (1) in a bounded domain Ω in R2 with smooth boundary ∂Ω ; the
mathematical analysis, however, is the same for a domain Ω in Rn for any n ≥ 1. We
A Bovine Babesiosis Model with Dispersion 105
assume that the bovine has the same birth and death rates,
μB = γB. (4)
In Sects. 5 and 6, we assume that the birth rate of the tick population does not changewith seasonality, and that it is equal to their death rate,
μT (t) = constant = γT , (5)
but in Sect. 7, we take μT (t) to be τ -periodic in t with
1
τ
∫ τ
0μT (t) dt = γT . (6)
We assume for simplicity that
NB(x,0) = constant = NB0 for x ∈ Ω (7)
and take the boundary conditions, for bovine,
∂SB
∂ν+ α(SB − NB0) = 0,
∂IB
∂ν+ αIB = 0,
∂RB
∂ν+ αRB = 0 for x ∈ ∂Ω, t > 0,
(8)
where α is a positive constant and ∂∂ν
is the derivative in the direction of the outwardnormal. Then
∂NB
∂t− dB�NB = 0 if x ∈ Ω, t > 0,
∂NB
∂ν+ α(NB − NB0) = 0 if x ∈ ∂Ω, t > 0;
hence NB(x, t) ≡ NB0 for x ∈ Ω , t ≥ 0.Let NT 0(t) denote the solution of
∂NT
∂t− (
μT (t) − γT
)NT = 0,
where
NT (x,0) = constant = N0T 0
for some positive constant N0T 0. In view of (5), NT 0(t) is τ -periodic. We take the
boundary conditions
∂ST
∂ν+ α
(ST − NT 0(t)
) = 0,
∂IT
∂ν+ αIT = 0 if x ∈ ∂Ω, t > 0,
(9)
106 A. Friedman, A.-A. Yakubu
so that
∂NT
∂ν+ α
(NT (t) − NT 0
) = 0 if x ∈ ∂Ω, t > 0.
Assuming also that
NT (x,0) = N0T 0 for x ∈ Ω,
we find upon recalling the differential equation for NT (x, t), that
NT (x, t) = NT 0(t),
and if μT = constant = γT , then NT 0 = constant.Setting
ρB = αB + μB,
in Models (1) and (4)–(9) and introducing the variables
sB = SB
NB
, iB = IB
NB
, rB = RB
NB
, sT = ST
NT
, iT = IT
NT
, (10)
so that
sB + iB + rB = 1 and sT + iT = 1,
we arrive at the following system of equations for x ∈ Ω and t ≥ 0:
∂sB
∂t− dB�sB = ρBrB + qμBiB − βB
NT 0(t)
NB0sBiT , (11)
∂iB
∂t− dB�iB = βB
NT 0(t)
NB0sBiT − (λB + qμB)iB, (12)
∂rB
∂t− dB�rB = λBiB − ρBrB, (13)
∂sT
∂t− dT �sT = (μT − γT )sT + pμT (t)iT − βT sT iB − N ′
T 0
NT 0sT , (14)
∂iT
∂t− dT �iT = (μT − γT )iT − pμT (t)iT + βT sT iB − N ′
T 0
NT 0iT , (15)
with the boundary conditions for x ∈ ∂Ω and t ≥ 0:
∂sB
∂ν+ α(sB − 1) = ∂iB
∂ν+ αiB = ∂rB
∂ν+ αrB = 0, (16)
and
∂sT
∂ν+ α(sT − 1) = ∂iT
∂ν+ αiT = 0. (17)
A Bovine Babesiosis Model with Dispersion 107
Since sB ≤ 1, iB ≥ 0 and rB ≥ 0 in Ω , the conditions in (16) mean that sB(x, t) isincreasing toward the boundary while iB(x, t) and rB(x, t) are decreasing toward theboundary. We interpret these conditions as an attempt from outside Ω to control thebovine population so as to decrease their infection. Similarly, the condition (17) canbe we interpreted as an attempt to decrease the infected tick population.
We finally prescribe initial conditions:
(sB, iB, rB, sT , iT )|t=0 = (sB0(x), iB0(x), rB0(x), sT 0(x), iT 0(x)
), (18)
and assume that the initial values are in C1(Ω), they satisfy the boundary conditions(16), (17), nonnegative, and
sB0 + iB0 + rB0 = 1, sT 0 + iT 0 = 1 in Ω. (19)
For simplicity, we assume that the boundary ∂Ω is in C2+α . Then by standardparabolic theory (Friedman 1964, 1992), there exists a unique solution of (11)–(19)and all the derivatives are continuous for x ∈ Ω , t ≥ 0.
The disease-free equilibrium points (DFE) of the System (1) and (4)–(9) is
DFE ≡ (NB0,0,0,NT 0(t),0
),
which corresponds in the transformed system (11)–(18) to
DFE ≡ (s0B, i0
B, r0B, s0
T , i0T
) = (1,0,0,1,0).
Eradication of BB can be achieved by elimination of the tick vector population.In what follows, we study the impact of the model parameters on BB transmission inregions where eradication of the tick vector is not feasible or desirable.
4 Preliminary Results
In the sequel, we shall use a comparison theorem for two systems of k parabolicpartial differential inequalities in Ω × (0, t0),
∂Vi
∂t− di�Vi ≥ fi(x, t,V1,V2, . . . , Vk) + ε,
∂Wi
∂t− di�Wi ≤ fi(x, t,W1,W2, . . . ,Wk) − ε′,
(20)
for 1 ≤ i ≤ k and di > 0, with boundary conditions on ∂Ω × (0, t0),
∂Vi
∂ν+ αVi ≥ ∂Wi
∂ν+ αWi + (
ε + ε′), (21)
where α > 0, ∂∂ν
is the derivative in the direction of the outward normal, and initialconditions
Vi(x,0) ≥ Wi(x,0) + (ε + ε′), (22)
108 A. Friedman, A.-A. Yakubu
where ε ≥ 0 and ε′ ≥ 0. We assume that the fi satisfy the following monotonicitycondition for x ∈ Ω and t ∈ (0, t0):
We also assume that if ε′ = 0, then (20)–(22) hold for all 0 ≤ ε < ε0 for some ε0 > 0and the Vi depends continuously on ε as ε → 0, whereas if ε = 0, then the Wi dependscontinuously on ε′ as ε′ → 0.
Theorem 4.1 Under the foregoing conditions, for the limit case ε = ε′ = 0 thereholds:
Vi(x, t) ≥ Wi(x, t) in Ω × [0, t0), 1 ≤ i ≤ k. (24)
Proof It suffices to prove that
Vi(x, t) > Wi(x, t) in Ω × [0, t0), 1 ≤ i ≤ k (25)
in the case where either ε′ = 0, ε > 0 or ε = 0, ε′ > 0, for then (24) will follow bytaking ε → 0 or ε′ → 0. We suppose for definiteness that ε′ = 0, ε > 0, and proceedby contradiction. If (25) is not true, then there is a smallest value of t , t = t , and apoint x ∈ Ω such that (25) holds for all x ∈ Ω , t < t , but
Vi(x, t) = Wi(x, t) for some i = i0.
By (22), t > 0. Since
Vi0(x, t) ≥ Vi0(x, t),
if x ∈ ∂Ω , then we have
∂
∂ν(Vi0 − Wi0) ≤ 0 at (x, t)
which is a contradiction to (21). Hence, x ∈ Ω , and then
∂
∂t(Vi0 − Wi0) ≤ 0, �(Vi0 − Wi0) ≥ 0 at (x, t).
Using these inequalities in (20), we get
ε + fi0(x, t,V1,V2, . . . , Vk) ≤ f
i0(x, t,W1,W2, . . . ,Wk)
with Vi0 = Wi0 , Vj ≥ Wj if j �= i0 at the point (x, t), thus contradicting the assump-tion (23). �
A Bovine Babesiosis Model with Dispersion 109
Theorem 4.2 There holds, for x ∈ Ω and t ≥ 0:
(i)
sB(x, t) ≥ 0, iB(x, t) ≥ 0, rB(x, t) ≥ 0;(ii)
sB(x, t) + iB(x, t) + rB(x, t) ≡ 1;(iii)
sT (x, t) ≥ 0, iT (x, t) ≥ 0;(iv)
sT (x, t) + iT (x, t) ≡ 1.
Proof The assertions (i), (iii) follow from Theorem 4.1 by comparing (sB, iB, rB,
sT , iT ) with the zero solution (0,0,0,0,0). To prove (ii), we note that the functionnB = sB + iB + rB satisfies
∂nB
∂t− dB�nB = 0 for x ∈ Ω and t ≥ 0,
∂nB
∂ν+ α(nB − 1) = 0 for x ∈ ∂Ω and t ≥ 0,
and by (19), nB(x,0) = 1 in Ω . Hence, by uniqueness, nB(x, t) ≡ 1 for all x ∈ Ω
and t ≥ 0. The proof of (iv) is similar. �
We introduce the principal eigenpair (λ∗, φ):
�φ + λ∗φ = 0 in Ω,
∂φ
∂ν+ αφ = 0 on ∂Ω,
λ∗ > 0, φ(x) > 0 in Ω,
(26)
and take
1 ≤ φ(x) ≤ φ1, (27)
where φ1 is a constant.For clarity, we shall first consider the case where
μT (t) = constant = μT .
Then both NT 0 and NB0 are constants, and we let
N = NT 0
NB0. (28)
110 A. Friedman, A.-A. Yakubu
The case where μT is τ -periodic will be considered in Sect. 7.We define as proliferation index, P , of the disease model (1) and (6)–(19) the
number
P = NβBβT
(λB + qμB + dBλ∗)(pμT + dT λ∗). (29)
Notice that P reduces to P(L) for model (2) of Aranda Lozano. In the next section,we shall prove the following result.
Theorem 4.3 If P < 1, then the disease-free equilibrium point, DFE = (1,0,0,1,0),is globally asymptotically stable.
In Sect. 6, we shall prove that if P > 1 then any small infection in bovines or tickswith either iB(x,0) not identically equal to 0 or iT (x,0) not identically equal to 0will develop into an endemic BB disease. More precisely, we have the following.
Theorem 4.4 If P > 1, then there exist positive numbers δ1 and δ2 such that if eitheriB(x,0) is not identically equal to zero or iT (x,0) is not identically equal to zero,then
iB(x, t) ≥ δ1 and iT (x, t) ≥ δ2 for all x ∈ Ω, t > T (30)
where T depends on the initial values in (13).
In proving Theorems 4.3 and 4.4, we shall use the comparison theorem, Theo-rem 4.1, and following Lemmas 4.5 and 4.6.
Lemma 4.5 Let u(x, t) be a function satisfying
∂u
∂t− d�u ≥ −γ u in Ω × (0, T ),
∂u
∂ν+ αu = 0 on ∂Ω × (0, T ),
where d > 0, α > 0 and u(x,0) ≥ 0 for x ∈ Ω . If u(x0,0) ≥ η0 for some point x0 ∈ Ω
and η0 > 0, then
u(x,T ) ≥ δη0 for all x ∈ Ω, (31)
where δ is a positive constant depending on η0 and T .
Proof The function U = eγ tu satisfies
∂U
∂t− d�U ≥ 0 in Ω × (0, T ),
∂U
∂ν+ αU = 0 on ∂Ω × (0, T )
A Bovine Babesiosis Model with Dispersion 111
and
U(x,0) ≥ 0 for x ∈ Ω.
Furthermore, by regularity,
U(x,0) = u(x,0) ≥ 1
2η0
if x ∈ Ω and |x −x0| < ε1, where ε1 sup |�u(x,0)| = cη0 and c is a sufficiently smallpositive constant.
Let G(x,y, t) denote the Green function for vt −d�v = 0 in Ω with ∂v∂ν
+αv = 0on ∂Ω . Then G(x,y, t) > 0 for x, y ∈ Ω and
U(x,T ) ≥∫
Ω
G(x,y,T )U(y,0) dy ≥∫
Ω∩{|y−y0|<ε1}G(x,y,T ) dy ≥ δ0η0 > 0,
where δ0 = δ0(η0, T ). Since u(x, t) = e−γ tU(x, t), the assertion (30) follows withδ = e−γ T δ0. �
Lemma 4.6 Let a and b be any positive numbers.
(i) If
dz(t)
dt+ az(t) ≤ b for 0 < t < ξ
and 0 ≤ z(0) ≤ z0, z0 > ba
, 0 < ε < z0 − ba
, then z(t) ≤ ba
+ ε for all T0 < t < ξ ,where
T0 = 1
aln
(z0 − b
a
ε
).
(ii) If
dz(t)
dt+ az(t) ≥ b for 0 < t < ξ
and z(0) ≥ z0 ≥ 0, z0 < ba
, 0 < ε < ba
− z0, then z(t) ≥ ba
− ε for all T1 < t < ξ ,where
T1 = 1
aln
( ba
− z0
ε
).
The proof follows from the integrated form of the differential inequalities.
4.1 Bovine Babesiosis Control Based on Theorems 4.3 and 4.4
From (28), we see that P is an increasing function of the transmission rates βB
and βT . Consequently, high BB transmissions among susceptible bovines and ticks
112 A. Friedman, A.-A. Yakubu
can lead to disease invasion while low transmissions lead to disease extinction. Anobvious way to decrease P is to increase the BB transmission rates. That is, trans-mission rates in susceptible cattle and ticks are critical parameters for the persistenceor control of bovine babesiosis.
It is interesting to note that if we consider Model (1) and (6)–(10) with μT constantbut without diffusion, then dB = dT = 0 and the corresponding proliferation index is
P = βBβT
(λB + qμB)pμT
,
so that P < P if N = 1. We conclude that dispersion of the bovines and ticks in-creases the stability of the DFE and decreases the probability that initial infectionwill become endemic. This can be explained by the fact that dispersion of small ini-tial infection does not allow a “critical mass” of infections to settle in the region.
If we write P = P(q) and P = P(q) to indicate their dependence on the proba-bility of the non-vertical transmission q , then clearly these functions are monotonedecreasing in q . Hence, as the probability of vertical transmission increases so doesthe stability of the DFE. This is rather a surprising result. One possible explanationis that when iB is small, a decrease in q will result in a decrease in the growth ofsB and, therefore, a decrease in infected bovines, which means more stable DFE. If,however, iB is not initially small, then increasing q leads to reduced growth in theinfected bovine population, as illustrated in Fig. 8 in Sect. 9.
Recall that in (29), N is the relative number of the tick population to the bovinepopulation. If N is increased, then the proliferation index will increase and the sta-bility of the DFE will decrease (see Fig. 8).
5 Extinction of Bovine Babesiosis: P < 1
Proof of Theorem 4.3 Since sB ≤ 1 and sT ≤ 1, from (12) and (15) we get
∂iB
∂t− dB�iB ≤ NβBiT − (λB + qμB)iB,
∂iT
∂t− dT �iT ≤ −pμT iT + βT iB.
We shall compare iB , iT with solutions iB , iT satisfying the equations
∂iB
∂t− dB�iB = NβBiT − (λB + qμB)iB,
∂iT
∂t− dT �iT = −pμT iT + βT iB,
(32)
and the same boundary conditions as iB , iT . We take
iB(x, t) = IB(t)φ(x), iT (x, t) = IT (t)φ(x),
A Bovine Babesiosis Model with Dispersion 113
so that
dIB
dt= NβBIT − (λB + qμB + dBλ∗)IB,
dIT
dt= (−pμT + dT λ∗)IT + βT IB.
(33)
Choosing IB(0) = IT (0) = 1, we conclude by the comparison theorem (Theorem 4.1)that
iB(x, t) ≤ IB(t)φ(x) and iT (x, t) ≤ IT (t)φ(x) (34)
Setting
X =(
IB
IT
),
we can rewrite System (33) in the vector form
dX
dt= AX
where
A =(−(λB + qμB + dBλ∗) NβB
βT −(pμT + dT λ∗)
). (35)
Then, for any 0 < t < ∞,
X(t) = exp(tA)X(0), (36)
Both eigenvalues of A have negative real part if and only if the determinant of A ispositive, i.e., if and only if P < 1. Hence, if P < 1, then from (35), (36) we concludethat
iB(x, t) ≤ Ce−δt and iT (x, t) ≤ Ce−δt
for some positive constants C and δ. It then easily follows that rB(x, t) is alsobounded by (constant)·e−δt , and thus the DFE is globally asymptotically stable. �
6 Enzootic Bovine Babesiosis: P > 1
The proof of Theorem 4.4 depends on several lemmas.
Lemma 6.1 Let η0 be any positive number such that
η0 ≤ 1
2
(ρB + dBλ∗
λB
)
114 A. Friedman, A.-A. Yakubu
and set
T0 ≡ T0(η0) = 1
(ρB + dBλ∗)ln
(λB + dBλ∗
2λBη0
). (37)
If iB(x, t) ≤ η0φ(x) for x ∈ Ω , 0 < t < ξ where ξ ≥ T0(η0), then
rB(x, t) ≤ 2λBη0
ρB + dBλ∗for x ∈ Ω, T0(η0) ≤ t ≤ ξ. (38)
Proof From (13), we get
∂rB
∂t− dB�rB ≤ λBη0 − ρBrB.
We compare rB with rB(x, t) = RB(t)φ(x), where
∂rB
∂t− dB�rB = λBη0 − ρBrB,
that is,
dRB
dt= λBη0 − (ρB + dBλ∗)RB.
If RB(0) = 1 then, by the comparison theorem,
rB(x, t) ≤ RB(t)φ(x).
We now apply to RB(t) Lemma 4.6(i) with
a = ρB + dBλ, b = λBη0, ε = b
aand z0 = 1
and conclude that
RB(t) ≤ 2b
a= 2λBη0
ρB + dBλ∗
provided T1 ≤ t ≤ ξ ; using the assumed estimate on η0 we find that T1 < T0(η0), andthe proof of (37) follows. �
Notation For any two vectors, z = (z1, z2, . . . , zn) and y = (y1, y2, . . . , yn), we writez ≥ y if zi ≥ yi for all 1 ≤ i ≤ n.
Lemma 6.2 Let η0 and η1 be small positive numbers such that
iB(x, t) ≤ η0φ(x), iT (x, t) ≤ η1φ(x) for x ∈ Ω, 0 ≤ t ≤ ξ, (39)
where ξ > T0(η0) with T0(η0) as in (37). Then, for any T0(η0) < t < ξ , there holds:
X(t) ≥ exp{t(A + O(η0 + η1)
)}X(t0), (40)
A Bovine Babesiosis Model with Dispersion 115
where A is the matrix defined in (35) and O(η0 + η1) is a matrix with elementsbounded by K(η0 + η1), K a positive constant.
Proof By our assumptions and Lemma 6.1,
sB(x, t) = 1 − iB(x, t) − rB(x, t) ≥ 1 − Cη0 if x ∈ Ω, T0(η0) ≤ t ≤ ξ,
and
sT (x, t) = 1 − iT (x, t) ≥ 1 − η1φ1 if 0 ≤ t ≤ ξ.
Hence, by comparison, as in the proof of Theorem 4.3, iB ≥ iB and iT ≥ iT , whereiB and iT satisfy modified versions of System (32), and where −Cη0 and −η1φ1 areadded, respectively, to the right-hand sides of the first and second equations. TakingiB(x, t) = IB(t)φ(x), iT (x, t) = IT (t)φ(x) and proceeding as in the proof of Theo-rem 4.3, we obtain inequality (40). �
Set
i+(x, t) = max{iB(x, t), iT (x, t)
},
i−(x, t) = min{iB(x, t), iT (x, t)
}.
Since P > 1, at least one of the eigenvalues of the matrix A has a positive real part.Hence, we obtain from (40) the following result.
Corollary 6.3 If (39) holds and η0, η1 are sufficiently small then, for any, T0(η0) <
t < ξ ,
i+(x, t) ≥ C1eσ t i−(x, t) for x ∈ Ω, (41)
where C1 and σ are positive constants independent of η0, η1.
Lemma 6.4 For any η0 > 0, set
η1 = 1
2
βT η0
pμT + βT η0φ1 + dT λ∗
and
T1 = ln 2
pμT + βT η0φ1 + dT λ∗.
If ξ ≥ T1 and
iB(x, t) ≥ η0φ(x) for x ∈ Ω, 0 < t ≤ ξ, (42)
then
iT (x, t) ≥ η1φ(x) for x ∈ Ω, T1 ≤ t ≤ ξ. (43)
116 A. Friedman, A.-A. Yakubu
Proof From (15), we get
∂iT
∂t− dT �iT ≥ −pμT iT − βT (1 − iT )η0φ(x)
= βT η0φ(x) − (pμT + βT η0φ1)iT .
We estimate iT from below by comparing it with the solution iT (x, t) = IT (t)φ(x)
of
∂iT
∂t− dT �iT = βT η0φ(x) − (pμT + βT η0φ1)iT ,
so that
dIT
dt= βT η0 − (pμT + βT η0φ1 + dT λ∗)IT ,
where IT (0) = 0. Applying Lemma 4.6(ii) with z0 = 0, a = pμT + βT η0φ1 + dT λ∗,b = βT η0 and ε = 1
2ba
, the assertion (43) follows. �
Lemma 6.5 For any η1 > 0, set
η0 = 1
2
pσBη1
λB + qμB + dT λ∗
and
T2 = ln 2
ρB + βBη1φ1 + dBλ∗+ ln 2
λB + qμB + λ∗,
where σB is defined below by (46) and (48). If
iT (x, t) ≥ η1φ(x) for x ∈ Ω, 0 < t ≤ ξ, (44)
then
iB(x, t) ≥ η0φ(x) for x ∈ Ω, T2 ≤ t ≤ ξ. (45)
Proof We wish to estimate iB from Eq. (8), but we first need to derive a lower boundon sB . From Eq. (11), we have
∂sB
∂t− dB�sB = ρB(1 − sB − iB) + qμBiB − βBNsBiT
= ρB + (qμB − ρB)iB − (ρB + βBNiT )sB.
Noting that, for any 0 ≤ iB ≤ 1,
ρB + (qμB − ρB)iB ≥ min{qμB,ρB} ≡ σB, (46)
A Bovine Babesiosis Model with Dispersion 117
we get, after using (44),
∂sB
∂t− dB�sB ≥ 1
φ1σBφ(x) − (ρB + βBNη1φ1)sB. (47)
We estimate sB from below by comparing it with a solution sB(x, t) = SB(t)φ(x)
which satisfies (47) with equality, so that
dSB
dt= σB
φ1− (ρB + βBη2φ1 + dBλ∗)SB
where η2 = η1N and SB(0) = 0. Applying Lemma 4.6(ii) with z0 = 0, a = ρB +βBη1φ1 + dBλ∗, b = σB
φ1and ε = 1
2ba
, we conclude that
sB(x, t) ≥ 1
2
b
a= 1
2φ1
σB
ρB + βBη2φ1 + dBλ∗≡ σB (48)
if T ∗ ≤ t ≤ ξ where
T ∗1 = 1
aln 2 = ln 2
ρB + βBη2φ1 + dBλ∗. (49)
Substituting (48) into (12) and using (44), we get
∂iB
∂t− dB�iB ≥ βBσBη2φ(x) − (λB + qμB)iB. (50)
We estimate iB from below by comparing it with a function iB(x, t) = IB(t)φ(x)
satisfying (50) with equality. Thus,
dIB
dt= βBσBη2 − (λB + qμB + dBλ∗)IB
for T ∗1 < t < ξ and IB(0) = 0. Using Lemma 4.6(ii) again, with z0 = 0, a = λB +
qμB + dBλ∗, b = βBσBη2 and ε = 12
ba
, we get
IB(t) ≥ b
2a= 1
2
βBσBη2
λB + qμB + dBλ∗if T ∗
1 + T ∗2 < t < ξ,
where
T ∗2 = ln 2
λB + qμB + dBλ∗.
Since, iB(x, t) ≥ iB(x, t) for T ∗1 < t < ξ where T ∗
1 is defined by (49), the assertion(45) follows. �
Lemma 6.6 There exist positive numbers η0 and η1 such that if iB(x,0) is not iden-tically equal to zero or iT (x,0) is not identically equal to zero, then there is a time t1depending on these initial data such that
iB(x, t1) ≥ η0φ(x) and iT (x, t1) ≥ η1φ(x) for x ∈ Ω. (51)
118 A. Friedman, A.-A. Yakubu
Furthermore, if
iB(x,0) ≥ σ1, iT (x,0) ≥ σ1 for x ∈ Ω, (52)
where σ1 > 0, then t1 ≤ K and K is a constant depending only on σ1.
Proof Since iB(x,0) is not identically equal to zero or iT (x,0) is not identicallyequal to zero, by Lemma 4.5 we easily see that there is a positive constant δ0 suchthat both
iB(x,T0) ≥ δ0φ(x) and iT (x, T0) ≥ δ0φ(x) for x ∈ Ω, (53)
where we take T0 as defined in (37) for the special choice
η0 = 1
2
(ρB + dBλ∗)λB
.
It will be useful to note that
δ0 → 0 if maxx∈Ω
{iB(x,0) + iT (x,0)
} → 0, (54)
whereas
δ0 = δ0(σ1) > 0 is independent of any initial data that satisfy (52). (55)
Thus, to prove (51), we need to show that δ0 in (53) can be replaced by η0 when T0is replaced by some large time t1.
Suppose next that
iB(x, t) ≤ η0 and iT (x, t) ≤ η0 for x ∈ Ω, 0 < t < ξ, (56)
where η0 is so small that Corollary 6.3 holds. Using (53) we conclude, from Corol-lary 6.3, that if ξ is large enough so that
C1eσ(ξ−T0)δ0 > η0
then one of the inequalities in (56) is not valid for some point x ∈ Ω , that is, thereexists a time t = t(δ0) and a point x0 ∈ Ω such that
either iB(x0, t) > η0 or iT (x0, t) > η0. (57)
Recalling (54), (55), we also conclude that
t(δ0) → ∞ if maxx∈Ω
{iB(x,0) + iT (x,0)
} → 0, (58)
but, if the initial data satisfy (52) then
t(δ0) ≤ K(σ1), (59)
A Bovine Babesiosis Model with Dispersion 119
where K(σ1) is a constant depending on σ1.Consider the case where iB(x0, t) > η0 in (57), and set θ = (λB + qμB). Since
∂iB
∂t− dB�iB ≥ −θiB,
Lemma 4.5 yields the estimate
iB(x, t) ≥ δη0 if x ∈ Ω, t = t + 1,
where δ > 0 depends only on η0. By comparison, we then have
iB(x, t) ≥ δη0e−θ(t−t−1) if x ∈ Ω, t + 1 ≤ t ≤ ξ,
and, in particular,
iB(x, t) ≥ η∗0φ(x) for x ∈ Ω, t + 1 ≤ t ≤ ξ,
if
η∗0 = δ
φ1η0e
−θ(ξ−t−1).
Applying Lemma 6.4, we deduce that
iT (x, t) ≥ η∗1φ(x) if x ∈ Ω, t + 1 + T ∗
1 ≤ t ≤ ξ,
where η∗1 and T ∗
1 are, respectively, defined as η1 and T1 in Lemma 6.4 when η0 = η∗0 .
We have thus proved that if iB(x0, t) ≥ η0 in (57), then
iB(x, t) ≥ η∗0φ(x) and iT (x, t) ≥ η∗
1φ(x)
for some t where t ≤ t + 1 + T ∗1 . Similarly, using Lemma 6.5, we draw the same
conclusion with a different T ∗1 , if iT (x0, t) ≥ η0.
This completes the proof of (51). A review of the proof, based on (54), (55), (57),and (58), shows that
t1 → ∞ if maxx∈Ω
{iB(x,0) + iT (x,0)
} → 0,
while t1 ≤ K(σ1) is a constant independent of any initial data satisfying (52). �
Proof of Theorem 4.4 By Lemma 6.6, there exists a first time t1 such that
iB(x, t1) ≥ η0φ(x) and iT (x, t1) ≥ η1φ(x) for x ∈ Ω.
Repeating the proof for t > t1 and using (55) with
σ1 = min{η0, η1}φ1
,
120 A. Friedman, A.-A. Yakubu
we deduce that there is a first time t2, t2 > t1, such that
iB(x, t2) ≥ η0φ(x) and iT (x, t2) ≥ η1φ(x) for x ∈ Ω
and t2 − t1 ≤ K(σ1). Repeating this process, step by step, we construct a sequence{tj } such that
0 < tj+1 − tj ≤ K(σ1),
and
iB(x, tj ) ≥ η0φ(x) and iT (x, tj ) ≥ η1φ(x) for x ∈ Ω.
Since
∂iB
∂t− dB�iB ≥ −θiB,
where θ = (λB + qμB), we conclude that
iB(x, t) ≥ η0e−θK(σ1) for tj ≤ t ≤ tj+1.
Similarly, from
∂iT
∂t− dT �iT ≥ −pμT iT
we deduce that
iT (x, t) ≥ η1e−ζ K(σ1) for tj ≤ t ≤ tj+1,
where ζ = pμ1. The lower bounds on iB and iT then complete the proof of Theo-rem 4.4. �
7 Extension to the Case of τ -Periodic Tick Birth Function
Epidemic models with periodic parameters, but with no dispersion, were consideredin Bacaër (2007), Bacaër and Guernaoui (2006), Dembele et al. (2009), Friedman(2013), Greenhalgh and Moneim (2003), Liu et al. (2010), Wang and Zhao (2008),and Zintl et al. (2003). In this section, we consider extensions of Theorems 4.3 and 4.4to the case where μT (t) is a τ -periodic function. In this case,
X(t) =(
IB
IT
)
satisfies the equation
dX
dt= A(t)X (60)
A Bovine Babesiosis Model with Dispersion 121
where
A(t) =(−(λB + qμB + dBλ∗) NβB
βT −(pμT (t) + dT λ∗ + N ′T 0(t)
NT 0(t))
). (61)
We write X(τ) in the form
X(τ) = eτAX(0) (62)
and, more generally,
X(nτ) = enτAX(0) (63)
for any positive integer n. However, it is not clear how to compute analytically theelements of the matrix A or, more importantly, the eigenvalues λ1, λ2 of A. Forthis reason, we are unable to define a proliferation index, P , directly in terms of themodel’s coefficients. If we denote by λ1, λ2 the eigenvalues of A, with Reλ1 ≤ Reλ2,then we expect that the DFE will be stable if Reλ2 < 0 and unstable if Reλ2 > 0.
It will be convenient to adopt the following language: We say that the proliferationindex is less than 1, and write P <1, if Reλ2 < 0, and that the proliferation index isbigger than 1, and write P >1, if Reλ2 > 0. Using this language, we can easily seethat if P <1, then
∣∣enτAX(0)∣∣ ≤ C1e
−nστ∣∣X(0)
∣∣
for any initial value X(0) and some positive constants C1, σ , whereas, if P > 1,
∣∣enτAX(0)∣∣ ≥ C2e
nσ0τ∣∣X(0)
∣∣
for some initial value X(0) �= 0 and positive constants C2, σ0.If P < 1, then the assertion of Theorem 4.3 holds with the same proof, except for
replacing (36) by
X(nτ + t0) = enτAX(t0).
If P > 1, then the assertion of Theorem 4.4 holds with the following minor changes:In Eq. (40), we take t = nτ + t0 < ξ , and
X(t) = X(nτ + t0) ≥ exp{nτ
(A + O(η0 + η1)
)}X(t0),
and in Eq. (41) we take t = nτ + t0 < ξ , t0 > T0(η0) and write
i+(x,nτ + t0) ≥ C1eσnτ i−(x,nτ + t0) for x ∈ Ω,
We also need to replace μT (t) by μ1 or μ2 in some of the inequalities in the proof ofLemma 6.4, where μ1 ≤ μT (t) ≤ μ2, and in (47), (48) take N = N(t) = NT 0(t)
NB0and
η2 = η1 max0≤t≤τ N(t).
122 A. Friedman, A.-A. Yakubu
Remark 7.1 We were kindly informed by Nicolas Bacaër that in articles (Dembeleet al. 2009) and (Friedman 2013) with τ -periodic matrix A(t), the matrix A wascomputed incorrectly as
1
τ
∫ τ
0A(t) dt;
in fact this is correct only in the case where the matrices A(t) commute among them-selves. The situation in Dembele et al. (2009) and Friedman (2013) is similar to thatof the present paper, hence one cannot compute A explicitly. However, we can esti-mate P by using constant matrices Ai , noting that if
A1 ≤ A(t) ≤ A2
and
dXi
dt= AiXi, Xi(0) = X(0)
then
X1(t) ≤ X(t) ≤ X2(t)
so that
enτA1X(0) ≤ enτAX(0) ≤ enτA2X(0).
P , or actually R0, was computed numerically for some special disease models inBacaër (2007), Liu et al. (2010), and Wang and Zhao (2008).
8 Anthrax Epizootic: An Extension of Theorem 4.4
The method of proof of Theorem 4.4 can be applied to other infectious disease mod-els to prove that the disease is endemic if R0 > 1. We illustrate this in the case ofour recent paper (Friedman and Yakubu 2013), of an anthrax epizootic model in abounded domain Ω with boundary ∂Ω . The model equations are:
∂s
∂t= d �2 s + rn
(1 − n
K
)− as − ηcsc − ηi
si
n− μs, (64)
∂i
∂t= d �2 i + as + ηcsc +
(ηi
s
n− (γ + μ)
)i, (65)
∂a
∂t= −αa + βc, (66)
∂c
∂t= (γ + μ)i − δ(s + i)c, (67)
where at location x ∈ Ω and time t ≥ 0, s = s(x, t) and i = i(x, t) are, respectively,the susceptible and infected animals, a = a(x, t) and c = c(x, t) are, respectively, the
A Bovine Babesiosis Model with Dispersion 123
anthrax spores and carcases, and n = n(x, t) = s(x, t) + i(x, t) is the total animalpopulation. On ∂Ω × (0 < t < ∞),
∂s
∂ν+ σ(s − s0) = 0, (68)
∂i
∂ν+ σ i = 0, (69)
where
s0 = K
(r
μ− 1
)(70)
and it is assumed that r > μ, and the initial conditions satisfy
s(x,0) ≥ 0, i(x,0) ≥ 0, s(x,0) + i(x,0) ≤ s0,
a(x,0) ≥ 0 and c(x,0) ≥ 0.(71)
Note that the DFE of Model (64)–(69) is
(s0,0,0,0).
It was proved in Friedman and Yakubu (2013), that the basic reproduction number is
R0 ≡ (ηc + βα)
δλ∗(γ+μ)
, (72)
where (λ∗, φ(x)) is the solution of
d �2 φ + [(ηi − (γ + μ) + λ∗
]φ = 0 in Ω,
∂φ
∂ν+ σφ = 0 on ∂Ω,
φ(x) > 0 in Ω.
(73)
We can choose φ so that
1 ≤ φ(x) ≤ φ1
for some constant φ1.
Lemma 8.1 There exists a sufficiently small positive number η0 such that if iB(x,0)
is not identically equal to zero and iT (x, t) < η0 for x ∈ Ω , t < ξ then ξ = t1, sothat i(x0, t1) = η0 for some x0 ∈ Ω , where t1 depends on i(x,0); furthermore, ifi(x,0) > δ for x ∈ Ω and some 0 < δ < η0, then t1 depends only on δ.
Proof As in Lemma 4.5, we can prove that
i(x, T ) ≥ c∗ for all x ∈ Ω and some T > 0, c∗ > 0. (74)
124 A. Friedman, A.-A. Yakubu
By adding Eqs. (64), (65), we get
∂n
∂t= d �2 n + rn
(1 − n
K
)− μn − γ i.
Hence,
∂n
∂t≥ d �2 n + g(n)
where
g(n) = rn
(1 − n
K
)− μn − γ η0
for x ∈ Ω and 0 < t < ξ . The function g(n) has two zeros,
A1η0 and s0 − A2η0,
where A1, A2 are positive constants, and
g′(A1η0) > 0, g′(s0 − A2η0) < 0.
Hence, any solution of
dN
dt= g(N), t > T
with N(T ) > A1η0 converges to s0 −A2η0 as t → ∞. Taking, in particular, a solutionwith N(T ) = c∗ and noting that, on ∂Ω ,
∂N
∂ν+ σ(N − s0) < 0 = ∂n
∂ν+ σ(n − s0),
we conclude by the comparison theorem that
n(x, t) > N(t) > s0 − 2A2η0 if t ≥ t1.
We next solve for c and a (from Eqs. (66), (67)), using estimate
n(x, t) > s0 − 2A2η0 if t1 < t < ξ,
and plug the expressions for c and a into Eq. (65) for i, as we did in Friedmanand Yakubu (2013). Then using the assumption that R0 > 1, we deduce that i(x, t)
increases exponentially in t , for t > t1. This completes the first part of Lemma 8.1The proof of the second part follows by reviewing the previous proof and noting thatboth T and t1 depend only on δ. �
Using Lemma 8.1, we can now repeat the arguments used in the proof of Theo-rem 4.4 and thus derive the following disease persistence result.
A Bovine Babesiosis Model with Dispersion 125
Table 2 Parameters for BB inColombia Parameter Values per day
αB 0.001 (Aranda Lozano 2011)
βB 0.00061 (Aranda Lozano 2011)
βT 0.00048 (Aranda Lozano 2011)
λB 0.000265 (Aranda Lozano 2011; Uribe et al. 1990)
γB 0.0002999 (Aranda Lozano 2011)
γT 0.001609 (Aranda Lozano 2011)
p 0.1 (p has no unit) (Aranda Lozano 2011)
μB 0.0002999 (Aranda Lozano 2011)
μT 0.001609 (Aranda Lozano 2011)
Theorem 8.2 If R0 > 1 in Model (64)–(69), then there exists a positive number δ1
such that if i(x,0) is not identically equal to zero then
i(x, t) ≥ δ1 for all x ∈ Ω, t > T ,
where T depends on the initial data i(x,0).
9 Simulations of BB Model
In this section, we simulate the BB dynamics first using the parameters of Table 2, andthen we investigate various scenarios when some of the parameters are changed. Thebase parameters for the simulations were deduced by Aranda Lozano (2011) usingdata from the northern regions of Colombia (South America); with these parametersand with q = 0 and no diffusion, R0 = 6.8 > 1, so that the disease is enzootic. Thequalitative results of our simulation can be extended to other BB enzootic regions.
As an example, we shall compare the progress of the disease in the case the initialinfection begins near the boundary of Ω versus the case when the disease starts inthe middle of Ω . For simplicity, we take Ω to be one-dimensional, namely,
Ω = {0 ≤ x ≤ 1}.
For a BB infection that starts near the boundary x = 0, we take the initial populationsizes, (sB, iB, rB, sT , iT )t=0, as
(0.3816 + 0.5184 ∗ (
1 − e− x20.001
),0.5184 ∗ e− x2
0.001 ,0.10,
0.40 + 0.6 ∗ (1 − e− x2
0.001),0.60 ∗ e− x2
0.001). (75)
When the initial BB infection starts near the center x = 0.5 of Ω , to make theinitial population sizes the same as for the case where the infection starts at x = 0,that is, we take the population sizes, (sB, iB, rB, sT , iT )t=0, as
126 A. Friedman, A.-A. Yakubu
Fig. 1 Constant Tick Birth rate: Profiles of iB (x, t0), the infected northern Colombia region cattle popu-lation, where initial infection is near x = 0 and x = 0.5, and where t0 = 5,10 and 50 years
(0.1908 + 0.2592 ∗ (
1 − e− (x−0.5)20.001
),0.2592 ∗ e− (x−0.5)2
0.001 ,0.05,
0.20 + 0.3 ∗ (1 − e− (x−0.5)2
0.001),0.30 ∗ e− (x−0.5)2
0.001). (76)
We also take
q = 0, α = 10−4, dB = 0.1, dT = 0.01.
With our choice of parameters, μT = constant. We take the combined number of tickson the ground and on cattle per unit area to be 5 times the number of cattle. That is,
N = NT 0(t)
NB0= 5.
In Figs. 1–6 and 8, the vertical axis is the proportion of the infected bovine; in Fig. 7we also display along the vertical axis the proportion of the infected tick population.
Figure 1 shows how initial infections spread due to dispersion of the bovine pop-ulation. For the bovine population, infection that starts at the boundary of the regionspreads slower, and takes nearly 50 years to become uniformly distributed over theentire region. The infected tick population spreads much slower due to a smaller dif-fusion coefficient (not shown here).
Figure 2 profiles the total infected bovine population as a function of time, for10 years, using the same initial infections as in Fig. 1.
When BB infection starts near the boundary, movement away from the initialsource of BB infection is unidirectional, whereas when BB infection starts in thecenter of the region, movement away from the initial source of BB infection is bidi-rectional. As a result, the concentration of BB infected bovines and ticks is largerwhen the initial infection starts at the boundary. High dispersion of bovines spreadsthe disease fast, while the very low dispersion of the ticks spreads it much slower.Consequently, we can expect the total BB infected bovine population to increasemore rapidly when the infection starts near the boundary, while for the ticks the to-tal infection should not depend significantly on the location of the initial infection.
A Bovine Babesiosis Model with Dispersion 127
Fig. 2 Constant Tick Birth rate:IB(t) = ∫ 1
0 iB(x, t) dx, the sumof infected Colombia cattlepopulation, increasesmonotonically as the BBepidemic time increases, whereinitial BB infection is near x = 0(blue) and near x = 0.5 (red)(Color figure online)
Fig. 3 Constant Tick Birth Rate: In the cattle and tick populations of Colombia, increasing the rate of re-covered bovines, λB , shifts the system from BB persistence when λB = 0.000265 per day to BB extinctionin 5 years when λB = 1 per day and all the other parameters are as in Fig. 2
Figure 2 indeed shows a significant increase in the total infected bovine populationwhen the disease starts at the boundary; for ticks the difference is negligible (notshown here).
In Figs. 1, 2, we assumed that λB = 0.000265 per day, of the treated BB infectedcattle of the Northern Colombia region have recovered from the disease. To studythe impact of successful treatment of infected cattle against the Babesia parasite, weassume that the BB infection starts at x = 0 and take in Fig. 3 the recovery rates,λB = 0.000265 per day, λB = 0.08 per day, λB = 1 per day while keeping all theother parameters fixed at their current values in Figs. 1, 2. When λB = 0.000265 perday, P > 1 and Northern Colombia is a BB enzootic region (see Fig. 3). However,when λB = 0.08 per day (and then P < 1) the BB infected cattle are successfullytreated against the Babesia parasite, and the total number of BB infected cattle andticks decline, but the region remains BB enzootic even after 10 years of the epidemic.However, when λB = 1 per day the epidemic goes extinct in the cattle population
128 A. Friedman, A.-A. Yakubu
Fig. 4 Constant Tick BirthRate: In the cattle population ofColombia, increasing the rate ofnon-vertical transmission, q ,leads to smaller numbers of thetotal population of infectedcattle, where q = 0, 0.3, 0.6 andall the other parameters are as inFigs. 2, 3
after 5 years, although it remains enzootic in the tick population. Figure 3 thus showsthat a successful cattle treatment program against the Babesia parasite of Colombiacan lead to a dramatic BB extinction within 5 years.
In Figs. 1–3, as in Aranda Lozano (2011), we assumed 100 % vertical transmis-sion of BB (q = 0) in the bovine population of the Northern Colombia region. Tostudy the impact of reducing vertical transmission in the cattle population, we as-sume that the BB infection starts at x = 0 and take in Fig. 4, increasing values ofnon-vertical transmission rates, q = 0.0, q = 0.3, and q = 0.6 while keeping all theother parameters fixed at their current values in Figs. 1–3. When q = 0.0, P > 1 andNorthern Colombia is a BB enzootic region (see Figs. 2, 3). P = P(q) decreases withincreasing values of q and Fig. 4 shows that increasing values of q leads a decreasein BB infection in the cattle population, while no significant changes occur in the tickinfections (not shown here).
In Figs. 1–4, we assumed that the bovine infectivity rate βB = 0.00061 per day. Tostudy the impact of reducing bovine infectivity rate, we assume that the BB infectionstarts at x = 0 and take in Fig. 5 βB = 0.00061 per day, βB = 0.0003 per day andβB = 0.0001 per day, while keeping all the other parameters fixed at their currentvalues in Figs. 1–4. When βB = 0.00061 per day, P > 1 and Northern Colombia is aBB enzootic region (see Figs. 3, 4). However, when βB = 0.0001 per day (and thenP < 1) the BB bovine infectivity rate is below the critical bovine infectivity rate, thetotal number of BB infected cattle declines (Fig. 5) while the BB is not significantlyaffected (not shown here), and the region remains BB enzootic even after 50 years ofthe epidemic.
9.1 Tick Control
Eradication programs of the host tick population should clearly reduce BB infections.In the United States, this has been done by treating all cattle every two or three weekswith acaricides. An eradication program will reduce the proportion of tick popula-tion to cattle population, that is, the parameter N = NT 0
NB0in our model. In Figs. 1–5,
A Bovine Babesiosis Model with Dispersion 129
Fig. 5 Constant Tick BirthRate: In the cattle population ofColombia, reducing theinfectivity rate, βB , shifts thesystem from increasing totalpopulation of infected cattlewhen βB = 0.00061 per day todecreasing total population ofinfected cattle whenβB = 0.0001 per day and all theother parameters are as inFigs. 2, 3
Fig. 6 Constant Tick Birth rate: Profile of IB(t) = ∫ 10 iB(x, t) dx, with initial BB infection at x = 0 and
x = 0.5, using the data (75) from Aranda Lozano (2011), with N = 5, 10, 15 and 20
we have chosen N = 5. However, in BB endemic countries, such as Colombia andSouth Africa, the proportion N may be substantially larger. In Fig. 6, we comparethe profiles of IB for larger values of N , namely N = 10, 15 and 20 while keepingall the other parameters fixed at their current values in Fig. 1. We see in Fig. 6 that,independently of the source of initial BB infections, IB significantly increases as N
is increased.
9.2 Fluctuating Tick Burden
In Figs. 1–6, we assumed constant tick birth rate (μT = 0.001609 per day). However,tick life is cyclic and the tick birth rate is non-constant. To study the impact of sea-sonality on both the infected tick and bovine host populations (simulated in Fig. 2),we assume that μT and NT 0 are periodic functions with a period of one year (raining
130 A. Friedman, A.-A. Yakubu
Fig. 7 Periodic Tick Birth rate: As time increases, IB(t) = ∫ 10 iB(x, t) dx, the sum of infected Colombia
cattle population increases while IT (t) = ∫ 10 iT (x, t) dx, the total population of infected Colombia tick,
fluctuates where initial BB infection is near x = 0 (blue) and near x = 0.5 (red) (Color figure online)
and dry seasons) and let
μT (t) = a + b sin 2πt, NT 0(t) = K + L sin 2πt and NB0 = 1,
where
a = 0.001609, b = 0.00156, K = 5, L = 1,
and keep all the other parameter values fixed as in Fig. 2. With this choice of param-eters,
μT (t + 1) = μT (t) and∫ 1
0μT (t) dt = 0.001609 as in Aranda Lozano (2011),
while
NT 0(t + 1) = NT 0(t) and∫ 1
0
NT 0(t)
NB0dt = 5 as in Figs. 1–4.
In Fig. 2, the environment is constant and there are no fluctuations in the infectedpopulations. However, Fig. 7 shows that seasonal variations can force strong fluctua-tions in the infected tick but not in the infected bovine.
Seasonal tick fluctuations neither enhance nor diminish BB cattle infections (notshown here). Thus, climate change that increases the seasonal oscillation of μT (t)
while keeping the same average μT does not seem to increase the BB infected pop-ulation. However, there is no data on how this average might be affected by climatechange (Smith 1991).
A Bovine Babesiosis Model with Dispersion 131
Fig. 8 Period Tick Birth rate: Profile of IB(t) = ∫ 10 iB (x, t)dx, with initial BB infection at x = 0 and
x = 0.5, using the data in Fig. 7, with control programs 1, 2, 3 and no control. Notice that the verticalscale of the graph on the left (x = 0) is extended to show more fully the distinct results of interventions
10 BB Control Programs
In this section, we use data from Colombia, South Africa and Brazil to study theeffect of the following three BB control strategies on the BB incidence as profiled inFig. 7:
1. Inoculation only,2. Dipping only,3. Combination of inoculation and dipping.
In the following subsections, we describe each of these control strategies and thencompare the BB incidence incurred under each strategy to the disease incidence inthe case of no control (Fig. 7). The results are shown in Fig. 8.
10.1 Control Program 1: Inoculation Only
To control BB, most cattle producers practice an integrated approach that includestick control by chemical means (acaricides); cattle vaccination with attenuated strainsof B. bovis, B. bigemina, or B. divergens, and treatment of clinically-ill cattle. Weillustrate the results of cattle inoculation using data from Smith (1991). Accordingly,we calculate the risk of BB among susceptible cattle from the inoculation rate by theequation
Q = 1 − e−ht ,
where Q is the proportion of susceptible cattle likely to contract infection within t
days when exposed to inoculation rate h.Using data from Campo Grande, Brazil, Smith calculated a range for h, h =
0.0009–0.0605 per day with mean h = 0.0123 per day. To adopt control program1 and use it to study the response of inoculation in Model (6)–(9) with seasonal fluc-
132 A. Friedman, A.-A. Yakubu
tuations in the growth rate of ticks, μT , we replace βB by
QβB = (1 − e−0.0123t
)βB, (77)
and keep all the other parameter values fixed at their current values in Fig. 7. Figure 8shows that after each year of BB, independently of whether the infection starts atx = 0 or x = 0.5, the proportion of BB infected total cattle population under controlprogram 1 is smaller than the corresponding number of infected cattle under the nocontrol program of Fig. 7.
10.2 Control Program 2: Dipping Only
Typically, cattle farmers use regular short-interval dipping to keep their cattle vir-tually tick-free in BB endemic regions. In Rickhotso et al. (2005), Rikhosto et al.studied the impact of reduced acaricides application on endemic stability to BB inthe local cattle population in the Bushbuckridge region of the Limpopo Province inSouth Africa. In the 12-month study, Rikhosto et al. observed an increase in sero-prevalence to B. Bovis in the strategically dipped group and a decrease in seropreva-lence to B. Bovis in the intensively dipped group. To compare program 1 to 2, weadopt a dipping program that reduces the initial tick infection to one-eighth its valuesin the no control program of Fig. 7 and control program 1. That is, under program 2,when the BB infection that starts near the boundary x = 0, we take the initial pop-
ulation sizes, (sB, iB, rB, sT , iT )t=0, as (0.3816 + 0.5184 ∗ (1 − e− x20.001 ),0.5184 ∗
e− x20.001 ,0.10,0.40 + 0.075 ∗ (1 − e− x2
0.001 ),0.075 ∗ e− x20.001 ).
Similarly, when the initial BB infection starts near the center x = 0.5 of Ω , tomake the initial population sizes the same as for the case where the infection startsat x = 0, we take the population sizes, (sB, iB, rB, sT , iT )t=0, as (0.1908 + 0.2592 ∗(1 − e− (x−0.5)2
0.001 ).Under program 2, we keep all the parameter values fixed at their current values
in Fig. 7 (no control). Figure 8 shows that after each year of BB, independently ofwhether the infection starts at x = 0 or x = 0.5, the proportion of BB infected totalcattle population under control program 2 (dipping only) is smaller than the corre-sponding number of infected bovines under control program 1.
10.3 Control Program 3: Combine Program of Inoculation and Dipping
To adopt program 3, we use the inoculation effect as in (77) and keep all the otherparameter values fixed at their current values in Fig. 7, where the initial conditionsare as in program 2 (dipping). Figure 8 shows that independently of whether the in-fection starts at x = 0 or x = 0.5, the proportion of BB infected total cattle populationunder control program 3 (inoculation and dipping) is smaller than the correspondingnumber of infected bovines under control programs 1 and 2. That is, for effective BBcontrol, program 3 is the best strategy.
A Bovine Babesiosis Model with Dispersion 133
11 Conclusion
Aranda Lozano (2011) introduced a BB model by a system of ordinary differentialequations. Our BB model, is based on a system of partial differential equations, wherewe have introduced dispersion of both cattle and ticks. In the first part of the paper,we introduced the concept of “proliferation index,” P , and proved that if P < 1 thenthe DFE is asymptotically stable, whereas if P > 1 then the BB will become en-demic. Thus, P plays the same role as the more familiar basic reproduction number,R0. The explicit formula (28) that we derived for P shows that P decreases as thedispersion coefficients increase. Thus, enhanced dispersion decreases the probabilityof BB prevalence arising from initial small infection in healthy cattle.
In the second part of the paper, we simulated the model with different choice ofparameters, and different treatments of BB, taking into account seasonal variationswhich affect the birth rate of ticks. There are several advantages of the PDE modelover the ODE model. One of them, already stated above, is that the PDE model showsthat the disease is less likely to become prevalent if the dispersion coefficient is in-creased. Another advantage of the PDE model is that it allows us to study how thelocation of the initial infection affects the progression of the disease. For example,we have shown that when an initial BB infections start at the boundary region, thetotal population of cattle infections is larger than the corresponding population whenthe initial infections start in the interior of the region. This has important implicationsfor effective BB control programs. We considered also other factors that may affectthe progression of the disease. For example, we showed that seasonal fluctuations donot affect the population of BB infected cattle over the entire season. We have alsoquantified the effect of increased number of ticks on the population of BB infectedbovine (see Fig. 6).
Finally, we addressed the effects of various BB intervention programs on BB inci-dence. First, we considered the effect of inoculation. A mathematical formula for theinoculation rate, h, was introduced by Alonso et al. (1990). Smith (1991) computed h
using age-class spreadsheet model of BB with fluctuating tick population to estimatethe risk of BB outbreak in Brazil and Paraguay. Using the value of h of Smith, butadjusting it to the data of Aranda Lozano (2011) from Colombia, we computed thereduction in the growth of BB achieved by inoculation, over a period of 10 years. Wefound that although this reduction is significant, it falls quite short of eradicating BB.
Next, we considered a second intervention method: dipping cattle in acaricides.This method of intervention was studied by Rickhotso et al. (2005) using data fromfour communal grazing fields in South Africa to analyze the impact of intensive andstrategic dipping systems on BB enzootic stability. We used our model to simulate theeffect of dipping and we found that our dipping program leads to a smaller number ofinfected bovine than the inoculation program. A combination program of inoculationand dipping leads to slightly improved results compared to dipping alone.
Our model shows that combined inoculation and dipping stabilizes the disease,and even decreases the number of infected bovines, but does not quite eradicate it. Itthus suggests that additional control steps need to be taken, for example, aggressivetreatment of sick cattle (to increase recovery rate and reduce BB transmission) or abetter age-dependent inoculation program.
134 A. Friedman, A.-A. Yakubu
Acknowledgements This research has been supported in part by the Mathematical Biosciences Instituteof The Ohio State University, Department of Homeland Security, DIMACS and CCICADA of RutgersUniversity and the National Science Foundation under grant DMS 0931642, 0832782, and 1205185.
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