Bulletin of Mathematical Biology (2010) 72: 931–952DOI 10.1007/s11538-009-9477-8
O R I G I NA L A RT I C L E
A Tuberculosis Model with Seasonality
Luju Liua,b,∗, Xiao-Qiang Zhaob, Yicang Zhoua
aDepartment of Mathematics, Xi’an Jiaotong University, Xi’an 710049, ChinabDepartment of Mathematics and Statistics, Memorial University of Newfoundland,St. John’s, NL A1C 5S7, Canada
Abstract The statistical data of tuberculosis (TB) cases show seasonal fluctuations inmany countries. A TB model incorporating seasonality is developed and the basic re-production ratio R0 is defined. It is shown that the disease-free equilibrium is globallyasymptotically stable and the disease eventually disappears if R0 < 1, and there exists atleast one positive periodic solution and the disease is uniformly persistent if R0 > 1. Nu-merical simulations indicate that there may be a unique positive periodic solution which isglobally asymptotically stable if R0 > 1. Parameter values of the model are estimated ac-cording to demographic and epidemiological data in China. The simulation results are ingood accordance with the seasonal variation of the reported cases of active TB in China.
Tuberculosis (TB) caused by infection with the Mycobacterium tuberculosis (M. tuber-culosis) is an ancient and chronic infectious disease. It is estimated that one-third of theworld’s population has been infected with the M. tuberculosis (Bleed et al., 2001), whichis a major cause of illness and death worldwide, especially in Asia and Africa (Dye et al.,2008). Furthermore, there were about 9.2 million new TB cases and 1.7 million deathsfrom TB in 2006, of which 0.7 million cases and 0.2 million deaths were in HIV-positivepeople (Dye et al., 2008).
Although TB is not widely recognized as having seasonal trends like measles, diph-theria, chickenpox, cholera, rotavirus, malaria, and even sexually transmitted gonorrhea
∗Corresponding author.E-mail addresses: [email protected] (Luju Liu), [email protected] (Xiao-Qiang Zhao),[email protected] (Yicang Zhou).Research was supported in part by the Chinese Government Scholarship and the Canada–China ThematicProgram on Disease Modeling, funded by the Networks of Centres of Excellence and the InternationalResearch Development Centre (LL); by the NSERC of Canada and the MITACS of Canada (X-QZ); andby the National Natural Science Foundation of China-NSFC10871122 (YZ).
(Grassly and Fraser, 2006; Hethcote and Yorke, 1984), some studies have shown variableperiods of peak seasonality in TB incidence rates in late winter to early spring in SouthAfrica (Schaaf et al., 1996), during summer in UK (Douglas et al., 1996) and Hong Kong(Leung et al., 2005), during summer and autumn in Spain (Rios et al., 2000), and duringspring and summer in Japan (Nagayama and Ohmori, 2006). In northern India, it was in-dicated that TB diagnosis peaked between April and June, and reached a nadir betweenOctober and December, and magnitude of seasonal variation had important positive cor-relation with rates of new smear-positive TB cases (Thorpe et al., 2004). Quite recently,it was demonstrated that there was a spring peak (late April) in TB cases detected amongmigrant workers entering Kuwait from high TB burden countries (Akhtar and Moham-mad, 2008).
The real causes of seasonal patterns of TB remain unknown, but the seasonal trend,with higher incidence rate in winter, may be relevant to the increased periods spent inovercrowded, poorly ventilated housing conditions, these phenomena much more eas-ily seen than in warm seasons (Rios et al., 2000; Schaaf et al., 1996), and/or vita-min D deficiency leading to reactivation of latent/exposed infection, which may havebeen the basic causes for observed TB seasonality (Janmeja and Mohapatra, 2005;Thorpe et al., 2004). Furthermore, in winter and spring, the viral infections like flu aremore frequent and cause immunological deficiency leading to reactivation of the M. tu-berculosis (Rios et al., 2000).
There is a growing awareness that seasonality can cause population fluctuations rang-ing from annual cycles to multiyear oscillations, and even chaotic dynamics (Aron andSchwartz, 1984; Greenman et al., 2004). From an applied perspective, clarifying themechanisms that link seasonal environmental changes to diseases dynamics may aid inforecasting the long-term health risks, in developing an effective public health program,and in setting objectives and utilizing limited resources more effectively (Altizer et al.,2006; Rios et al., 2000). For these reasons, we need to identify possible seasonal patternsin the incidence rate for pulmonary tuberculosis. However, to our knowledge, no studiesso for have described TB seasonality in the mainland of China. But from the monthlyreported data of the Ministry of Health of the People’s Republic of China (2005–2009),there is also a seasonal pattern in new TB cases (see also Table 1 or Fig. 4).
Different mathematical models have been developed incorporating some factors,such as fast and slow progression, drug-resistant strains, coinfection with HIV, re-lapse, reinfection, migration, and vaccination to study the transmission dynamics of TB(Blower, 1995; Blower et al., 1996; Blower and Chou, 2004; Lietman and Blower, 2000;Porco and Blower, 1998; Rodrigues et al., 2007; Sharomi et al., 2008; Zhou et al., 2008;Ziv et al., 2001). These models are described by systems of differential equations. Someother models have also been formulated mainly to investigate the possible seasonaltrends of TB (Akhtar and Mohammad, 2008; Douglas et al., 1996; Leung et al., 2005;Nagayama and Ohmori, 2006; Rios et al., 2000; Schaaf et al., 1996; Thorpe et al., 2004).In the present paper, we develop a mathematical TB model with seasonality to study thepossible seasonal variation in pulmonary TB in the mainland of China.
The rest of the paper is organized as follows. In Section 2, we formulate the TB model,study the existence, uniqueness, and boundedness of solutions, and define the basic repro-duction ratio R0. In Section 3, the long-term behavior of the TB model is analyzed. Thereis a unique disease-free equilibrium and the disease always dies out if R0 < 1; while the
A Tuberculosis Model with Seasonality 933
disease uniformly persists in the population and there is at least one positive periodic so-lution if R0 > 1. The numerical simulations and brief discussion are given in Section 4and Section 5, respectively.
2. The seasonal TB model
In this section, we introduce a simple TB model incorporating periodic coefficients basedon the possible fact that there is a seasonal trend in new TB cases. The whole populationis divided into four classes: the susceptible class, the latent/exposed class, the infectiousclass, and the treated/recovered class. The fast and slow progression was considered ear-lier by some authors to study the transmission of TB (Blower, 1995; Blower et al., 1996;Ziv et al., 2001). In the present paper, we also introduce the fast and slow progressionbased on the real situation of tuberculosis disease. The standard incidence β(t)SI/N isapplied because an infectious individual can contact a finite number of persons in onetime unit in a large population (Ma et al., 2004). The model has the compartmental struc-ture of the classical SEIR epidemic model, and is described by the following system ofnonautonomous differential equations
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
dSdt
= Λ − β(t) SIN
− μS,
dEdt
= (1 − q)β(t) SIN
− (μ + k(t))E,
dIdt
= qβ(t) SIN
+ k(t)E − (μ + d + r)I,
dRdt
= rI − μR,
N = S + E + I + R,
(1)
where S(t), E(t), I (t), and R(t) are the numbers of the susceptible, the latent/exposed,the infectious, and the treated/recovered individuals at time t , respectively. N(t) is thetotal number of the whole population at time t . Λ is the recruitment rate, μ is the naturaldeath rate, q is the fraction of fast developing infectious cases, r is the treatment/recoveryrate, and d represents the disease-induced death rate. These parameters are positive con-stants and independent of time t , and q < 1. In view of the periodic trend of monthly newTB cases (The Ministry of Health of the People’s Republic of China, 2005–2009) and thepossible causes of the seasonal pattern (Janmeja and Mohapatra, 2005; Rios et al., 2000;Thorpe et al., 2004), a model with periodic infection rate β(t) and reactivation rate k(t)
may be a natural choice to describe the TB transmission. Thus, we assume that k(t) andβ(t) are periodic positive continuous functions in t with period ω for some ω > 0 (actu-ally, ω = 12 months).
In order to study the existence and uniqueness of the solutions of system (1), we definea function g ∈ C(R4+,R) by
g(S,E, I,R) ={
0, if (S,E, I,R) = (0,0,0,0),SI
S+E+I+R, if (S,E, I,R) ∈ R
4+\{(0,0,0,0)}.
Clearly, g(S,E, I,R) is continuous on R4+. It is easy to see that g(S,E, I,R) is globally
Lipschitz on R4+ and l = 3 is the corresponding Lipschitz constant.
934 Liu et al.
By a change of variable N = S + E + I + R and the definition of g(S,E, I,R), sys-tem (1) is equivalent to the following one:
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
dSdt
= Λ − β(t)g(S,E, I,N − S − E − I ) − μS,
dEdt
= (1 − q)β(t)g(S,E, I,N − S − E − I ) − (μ + k(t))E,
dIdt
= qβ(t)g(S,E, I,N − S − E − I ) + k(t)E − (μ + d + r)I,
dNdt
= Λ − dI − μN.
(2)
It then follows from Smith (1995, Theorem 5.2.1) that for any (S0,E0, I 0,N0) ∈ R4+,
system (2) has a unique local nonnegative solution (S(t),E(t), I (t),N(t)) through theinitial value (S(0),E(0), I (0),N(0)) = (S0,E0, I 0,N0).
By the fourth equation of system (2), we have
dN
dt= Λ − dI − μN ≤ Λ − μN.
It is easy to see that the linear differential equation dNdt
= Λ−μN has a unique equilibriumN∗ = Λ/μ, which is globally asymptotically stable. The comparison principle (Smithand Walman, 1995, Theorem B.1) implies that N(t) is ultimately bounded, and hencethe solutions of system (2) exist globally on the interval [0,∞). We summarize thesediscussions in the following result.
Theorem 2.1. System (2) has a unique and bounded solution with the initial value
(S0,E0, I 0,N0
) ∈ X := {(S,E, I,N) ∈ R
4+ : N ≥ S + E + I
}.
Further, the compact set
G := {(S,E, I,N) ∈ X : N ≤ Λ/μ
}
is positively invariant and attracts all positive orbits in X.
In what follows, we introduce the basic reproduction ratio R0 for system (2) accordingto the general procedure presented in Wang and Zhao (2008). It is easy to see that system(2) has exactly one disease-free equilibrium P0 = (Λ/μ,0,0,Λ/μ) and the equations forlatent/exposed and infectious compartments of the linearized system of model (2) at P0
are{
dEdt
= (1 − q)β(t)I − (μ + k(t))E,
dIdt
= qβ(t)I + k(t)E − (μ + d + r)I.
We introduce
F(t) =(
0 (1 − q)β(t)
0 qβ(t)
)
, V (t) =(
μ + k(t) 0−k(t) μ + d + r
)
.
A Tuberculosis Model with Seasonality 935
Let ΦV (t) and ρ(ΦV (ω)) be the monodromy matrix of the linear ω-periodic systemdzdt
= V (t)z and the spectral radius of ΦV (ω), respectively. Assume Y (t, s), t ≥ s, is theevolution operator of the linear ω-periodic system
dy
dt= −V (t)y. (3)
That is, for each s ∈ R, the 2 × 2 matrix Y (t, s) satisfies
d
dtY (t, s) = −V (t)Y (t, s), ∀t ≥ s, Y (s, s) = I,
where I is the 2 × 2 identity matrix. Thus, the monodromy matrix Φ−V (t) of (3) is equalto Y (t,0), t ≥ 0.
In view of the periodic environment, we assume that φ(s), ω-periodic in s, is the ini-tial distribution of infectious individuals. Then F(s)φ(s) is the rate of new infectionsproduced by the infected individuals who were introduced at time s. Given t ≥ s, thenY (t, s)F (s)φ(s) gives the distribution of those infected individuals who were newly in-fected at time s and remain in the infected compartments at time t . It follows that
ψ(t) :=∫ t
−∞Y (t, s)F (s)φ(s) ds =
∫ ∞
0Y (t, t − a)F (t − a)φ(t − a)da
is the distribution of accumulative new infections at time t produced by all those infectedindividuals φ(s) introduced at time previous to t .
Let Cω be the ordered Banach space of all ω-periodic functions from R to R2,
which is equipped with the maximum norm ‖ · ‖ and the positive cone C+ω := {φ ∈ Cω :
φ(t) ≥ 0,∀t ∈ R}. Then we can define a linear operator L : Cω→Cω by
Following Wang and Zhao (2008), we call L the next infection operator, and define thebasic reproduction ratio as R0 := ρ(L), the spectral radius of L.
In the special case of β(t) ≡ β , and k(t) ≡ k, ∀t ≥ 0, we obtain F(t) ≡ F , andV (t) ≡ V , ∀t ≥ 0. By Van Den Driessche and Watmough (2002) (see also Wang andZhao, 2008, Lemma 2.2(ii)), we further have
R0 = ρ(FV −1
) = β
μ + d + r
(
q + k
μ + k(1 − q)
)
.
In the periodic case, we let W(t,λ) be the monodromy matrix of the linear ω-periodicsystem
dw
dt=
(
−V (t) + 1
λF(t)
)
w, t ∈ R
with parameter λ ∈ (0,∞). Since F(t) is nonnegative and −V (t) is cooperative, it followsthat ρ(W(ω,λ)) is continuous and nonincreasing in λ ∈ (0,∞), and limλ→∞ ρ(W(ω,λ))
< 1. It is easy to verify that system (2) satisfies assumptions (A1)–(A7) in Wang and Zhao
936 Liu et al.
(2008). Thus, we have the following two results, which will be used in our numericalcomputation of R0 (see Fig. 1 or Fig. 9) and the proof of our main result in Section 3,respectively.
Lemma 2.1 (Wang and Zhao, 2008, Theorem 2.1). The following statements are valid:
(i) If ρ(W(ω,λ)) = 1 has a positive solution λ0, then λ0 is an eigenvalue of operator L,and hence R0 > 1.
(ii) If R0 > 1, then λ = R0 is the unique solution of ρ(W(ω,λ)) = 1.(iii) R0 = 0 if and only if ρ(W(ω,λ)) < 1 for all λ > 0.
Lemma 2.2 (Wang and Zhao, 2008, Theorem 2.2). The following statements are valid:
(i) R0 = 1 if and only if ρ(ΦF−V (ω)) = 1.(ii) R0 > 1 if and only if ρ(ΦF−V (ω)) > 1.
(iii) R0 < 1 if and only if ρ(ΦF−V (ω)) < 1.
Thus, the disease-free equilibrium P0 is locally asymptotically stable if R0 < 1, and un-stable if R0 > 1.
Let [R0] be the basic reproduction number of the autonomous system obtained from theaverage of system (2). We apply Lemma 2.1 to calculate the basic reproduction ratio R0.We take
β(t) = a0
(
1.1 + sinπ(t + 1)
6
)
, and k(t) = b0
(
1.1 + sinπ(t − 1)
6
)
.
If we fix Λ = 0.2, μ = 0.04, d = 0.3, r = 1.1, q = 0.03, and b0 = 0.6, by numericalcomputation, we get the curve of the basic reproduction ratio R0 and the curve of theaverage basic reproduction number [R0] with respect to a0, respectively, in Fig. 1. Wecan see that the average basic reproduction number [R0] is always greater than the basicreproduction ratio R0 when a0 varies ranging from 0.5 to 4. This implies that the risk ofinfection will be overestimated if the average basic reproduction number is used.
3. Global dynamics
In this section, before the following main threshold theorem for system (2) is establishedin terms of R0, some notations and a lemma will be given first in order to better understandthe proof of the following theorem for the readers.
Define
X0 := {(S,E, I,N) ∈ X : E > 0, I > 0
}, ∂X0 := X\X0.
Let P : X→X be the Poincaré map associated with system (2), that is,
P(x0
) = u(ω,x0
), ∀x0 ∈ X,
A Tuberculosis Model with Seasonality 937
Fig. 1 The curves of the basic reproduction ratio R0 and the average basic reproduction number [R0]versus a0.
where u(t, x0) is the unique solution of system (2) with u(0, x0) = x0. It is easy to seethat
P m(S0,E0, I 0,N0
) = u(mω,
(S0,E0, I 0,N0
)), ∀m ≥ 0.
Lemma 3.1. If the basic reproduction ratio R0 > 1, then there exists a σ ∗ > 0, such thatfor any (S0,E0, I 0,N0) ∈ X0 with ‖(S0,E0, I 0,N0) − P0‖ ≤ σ ∗, we have
lim supm→∞
d(P m
(S0,E0, I 0,N0
),P0
) ≥ σ ∗. (5)
Proof: Since R0 > 1, Lemma 2.2 implies ρ(ΦF−V (ω)) > 1. We can choose η > 0 smallenough such that ρ(ΦF−V −Mη(ω)) > 1, where
Mη(t) =(
0 2η
Λ/μ+η(1 − q)β(t)
0 2η
Λ/μ+ηqβ(t)
)
.
Equation dSdt
= Λ − μS has a unique equilibrium S∗ = Λ/μ which is globally attractivein R+. Note that the perturbed system
dS(t)
dt= (
Λ − β(t)σ) − μS(t) (6)
938 Liu et al.
admits a unique solution
S(t, σ ) = e−μt
(
S(0, σ ) +∫ t
0eμs
(Λ − β(s)σ
)ds
)
through the arbitrary initial value S(0, σ ), and has a unique periodic solution
S∗(t, σ ) = e−μt
(
S∗(0, σ ) +∫ t
0eμs
(Λ − β(s)σ
)ds
)
,
where
S∗(0, σ ) =∫ ω
0 eμs(Λ − β(s)σ ) ds
eμω − 1.
Clearly, |S(t, σ ) − S∗(t, σ )| → 0, as t → ∞. Thus, S∗(t, σ ) is globally attractive on R+.From the expression of S∗(0, σ ), it is easy to see that S∗(0, σ ) is continuous in σ . Thecontinuous dependence of the solution S∗(t, σ ) on the initial condition and parametervalue implies that S∗(t, σ ) > S∗ − η holds for sufficiently small σ , and all t ∈ [0,ω].By the periodicity of S∗(t, σ ) and constant S∗ − η, we see that S∗(t, σ ) > S∗ − η holdsfor sufficiently small σ , and all t ≥ 0. By the continuity of the solutions with respectto the initial values, there exists a σ ∗ > 0 such that for all (S0,E0, I 0,N0) ∈ X0 with‖(S0,E0, I 0,N0) − P0‖ ≤ σ ∗, there holds ‖u(t, (S0,E0, I 0,N0)) − u(t,P0)‖ < σ,∀t ∈[0,ω]. We further claim that
lim supm→∞
d(P m
(S0,E0, I 0,N0
),P0
) ≥ σ ∗. (7)
Assume, by contradiction, that (7) does not hold. Then we have
lim supm→∞
d(P m
(S0,E0, I 0,N0
),P0
)< σ ∗
for some (S0,E0, I 0,N0) ∈ X0. Without loss of generality, we assume thatd(P m(S0,E0, I 0,N0),P0) < σ ∗, for all m ≥ 0. It follows that
∥∥u
(t,P m
(S0,E0, I 0,N0
)) − u(t,P0)∥∥ < σ, ∀m ≥ 0, ∀t ∈ [0,ω].
For any t ≥ 0, let t = mω + t ′, where t ′ ∈ [0,ω), and m is the largest integer less than orequal to t
ω. Therefore, we have
∥∥u
(t,
(S0,E0, I 0,N0
)) − u(t,P0)∥∥
= ∥∥u
(t ′,P m
(S0,E0, I 0,N0
)) − u(t ′,P0)∥∥ < σ, ∀t ≥ 0.
Note that (S(t),E(t), I (t),N(t)) = u(t, (S0,E0, I 0,N0)). It then follows that E(t) <
σ, I (t) < σ,∀t ≥ 0. From the first and last equations of system (2), we have
{dSdt
≥ (Λ − β(t)σ ) − μS,
dNdt
≤ Λ − μN.(8)
A Tuberculosis Model with Seasonality 939
Since the periodic solution S∗(t, σ ) of Eq. (6) is globally attractive on R+ and S∗(t, σ ) >
S∗ − η, we have S(t) ≥ S∗ − η, for sufficiently large t . Furthermore, we obtain N(t) ≤Λ/μ + η, for sufficiently large t . From the second and third equations of system (2), forsufficiently large t , we obtain
⎧⎨
⎩
dEdt
≥ (1 − q)β(t)(1 − 2η
Λ/μ+η)I − (μ + k(t))E,
dIdt
≥ qβ(t)(1 − 2η
Λ/μ+η)I + k(t)E − (μ + d + r)I.
(9)
We then consider the following system⎧⎨
⎩
dEdt
= (1 − q)β(t)(1 − 2η
Λ/μ+η)I − (μ + k(t))E,
dIdt
= qβ(t)(1 − 2η
Λ/μ+η)I + k(t)E − (μ + d + r)I .
(10)
By Zhang and Zhao (2007, Lemma 2.1), we know that there exists a positive, ω-periodicfunction (E(t), I (t))T such that (E(t), I (t))T = eζ t (E(t), I (t))T is a solution of system(10), where ζ = 1
ωlnρ(ΦF−V −Mη(ω)). Since ρ(ΦF−V −Mη(ω)) > 1, ζ is a positive con-
stant. Let t = nω and n be nonnegative integer, and we get
(E(nω), I (nω)
)T = eζnω(E(nω), I (nω)
)T →(∞,∞)T
as n→∞, since ωζ > 0 and (E(t), I (t))T ) > 0. For any nonnegative initial condi-tion (E(0), I (0))T of system (9), there exists a sufficiently small m∗ > 0 such that(E(0), I (0))T ≥ m∗(E(0), I (0))T . By the comparison principle (Smith and Walman,1995, Theorem B.1), we have (E(t), I (t))T ≥ m∗(E(t), I (t))T , for all t > 0. Thus, weobtain E(nω) → ∞ and I (nω) → ∞, as n→∞, a contradiction. This completes theproof. �
Theorem 3.1. If the basic reproduction ratio R0 < 1, then the unique disease-freeequilibrium P0 = (Λ/μ,0,0,Λ/μ) is globally asymptotically stable; while if the ba-sic reproduction ratio R0 > 1, then there exists a δ > 0 such that any solution(S(t),E(t), I (t),N(t)) of system (2) with initial value (S0,E0, I 0,N0) ∈ {(S,E, I,N) ∈X : E > 0, I > 0} satisfies
lim inft→+∞ E(t) ≥ δ, and lim inf
t→+∞ I (t) ≥ δ,
and system (2) admits at least one positive periodic solution.
Proof: By Lemma 2.2, we know that if R0 < 1, then P0 is locally asymptotically stable.It is sufficient to prove that P0 is globally attractive if R0 < 1. Clearly, S(t) ≤ N(t), forall t ≥ 0. Then from system (2), we have
{dEdt
≤ (1 − q)β(t)I − (μ + k(t))E,
dIdt
≤ qβ(t)I + k(t)E − (μ + d + r)I.(11)
Consider the following comparison system
dh(t)
dt= (
F(t) − V (t))h(t). (12)
940 Liu et al.
Applying Lemma 2.2, we know that R0 < 1 if and only if ρ(ΦF−V (ω)) < 1. By Zhangand Zhao (2007, Lemma 2.1), it follows that there exists a positive, ω-periodic functionh(t) such that h(t) = eθt h(t) is a solution of system (12), where θ = 1
ωlnρ(ΦF−V (ω)).
Since ρ(ΦF−V (ω)) < 1, θ is a negative constant. Therefore, we have h(t)→0 as t→+∞.This implies that the zero solution of system (12) is globally asymptotically stable. For anynonnegative initial value (E(0), I (0))T of system (11), there is a sufficiently large M∗ > 0such that (E(0), I (0))T ≤ M∗h(0) holds. Applying the comparison principle (Smith andWalman, 1995, Theorem B.1), we have (E(t), I (t))T ≤ M∗h(t), for all t > 0, whereM∗h(t) is also the solution of system (12). Therefore, we get E(t) → 0, and I (t) → 0as t → +∞. By the theory of asymptotic autonomous systems (Thieme, 1992, Theo-rem 1.2), it then follows that N(t)→Λ/μ, and S(t)→Λ/μ as t→ + ∞. We finish theproof of the first part of the theorem.
By Theorem 2.1, the discrete-time system {P m}m≥0 admits a global attractor in X.Now we prove that {P m}m≥0 is uniformly persistent with respect to (X0, ∂X0). For any(S0,E0, I 0,N0) ∈ X0, from the first equation of system (2), we have
S(t) = e− ∫ t0 a(s1) ds1
(
S0 + Λ
∫ t
0e
∫ s20 a(s1) ds1 ds2
)
≥ Λe− ∫ t0 a(s1) ds1
∫ t
0e
∫ s20 a(s1) ds1 ds2 > 0, ∀t > 0, (13)
where a(t) := μ + β(t)I (t)/N(t). Notice that N(t) ≥ S(t) > 0,∀t > 0. By Smith (1995,Theorem 4.1.1) as generalized to nonautonomous systems, the irreducibility of the coop-erative matrix
M(t) =(
−(μ + k(t)) (1 − q)β(t) S(t)
N(t)
k(t) qβ(t) S(t)
N(t)− (μ + d + r)
)
implies that (E(t), I (t))T � 0,∀t > 0. Thus, both X and X0 are positively invariant.Clearly, ∂X0 is relatively closed in X.
Set
M∂ := {(S0,E0, I 0,N0
) ∈ ∂X0 : P m(S0,E0, I 0,N0
) ∈ ∂X0,∀m ≥ 0}.
We now show that
M∂ = {(S,0,0,N) ∈ X : S ≥ 0,N ≥ 0
}. (14)
It suffices to prove that for any (S0,E0, I 0,N0) ∈ M∂ , we have E(mω) = I (mω) = 0,
∀m ≥ 0. If it is not true, there exists an m1 ≥ 0 such that (E(m1ω), I (m1ω))T > 0. Thus,(13) implies
N(t) ≥ S(t) > 0, ∀t > m1ω,
by replacing the initial time 0 with m1ω. Similarly, by Smith (1995), Theorem 4.1.1as generalized to nonautonomous systems, it follows that (E(t), I (t))T � 0, ∀t > m1ω,where the initial value (E(m1ω), I (m1ω))T > 0. Thus, we have
(S(t),E(t), I (t),N(t)
) ∈ X0, ∀t > m1ω,
A Tuberculosis Model with Seasonality 941
which implies that (14) holds. Clearly, there is exactly one fixed point P0 = (Λ/μ,0,0,
Λ/μ) of P in M∂ .Note that the linear nonhomogeneous system
{dNdt
= Λ − μN,
dSdt
= Λ − μS,
admits a global asymptotic stable equilibrium (Λ/μ,Λ/μ). Lemma 3.1 implies thatP0 = (Λ/μ,0,0,Λ/μ) is an isolated invariant set in X and Ws(P0)∩X0 = φ. Note thatevery orbit in M∂ approaches to P0, and P0 is acyclic in M∂ . By Zhao (2003, Theo-rem 1.3.1), it follows that P is uniformly persistent with respect to (X0, ∂X0). By Zhao(2003, Theorem 3.1.1), the solutions of system (2) are uniformly persistent with respectto (X0, ∂X0), that is, there exists a δ > 0 such that any solution (S(t),E(t), I (t),N(t))
of system (2) with initial value (S0,E0, I 0,N0) ∈ X0 satisfies lim inft→+∞ E(t) ≥ δ, andlim inft→+∞ I (t) ≥ δ. Furthermore, Zhao (2003, Theorem 1.3.6) implies that P has afixed point (S∗(0),E∗(0), I ∗(0), N∗(0)) ∈ X0. Then S∗(0) ≥ 0, E∗(0) > 0, I ∗(0) > 0,and N∗(0) ≥ 0. Clearly, N∗(0) > 0 because of N(t) ≥ S(t) + I (t) + E(t), for all t ≥ 0.We further claim that there exists some t ∈ [0,ω] such that S∗(t) > 0. If it is not thecase, S∗(t) ≡ 0, for all t ≥ 0, due to the periodicity of S∗(t). From the first equation ofsystem (2), we get 0 = Λ > 0, which is a contradiction. Then we obtain
S∗(t) = exp
(∫ t
t
(
μ + β(ξ)g(ξ)
S(ξ)
)
dξ
)
×(
S∗(t) + Λ
∫ t
t
exp
(∫ ξ
t
(
μ + β(ζ )g(ζ )
S(ζ )
)
dζ
)
dξ
)
> 0, ∀t ∈ [t , t + ω],
where g(t) := g(S(t),E(t), I (t),N(t) − S(t) − E(t) − I (t)). The periodicity of S∗(t)implies S∗(t) > 0, for all t ≥ 0. Hence, N∗(t) > 0, for all t ≥ 0 due to N(t) ≥ S(t), forall t ≥ 0. By the second and the third equations of system (2) and the irreducibility of thecooperative matrix
(−(μ + k(t)) (1 − q)β(t) S∗(t)
N∗(t)
k(t) qβ(t) S∗(t)
N∗(t)− (μ + d + r)
)
,
it follows that (E∗(t), I ∗(t)) ∈ Int(R2+),∀t ≥ 0. Therefore, (S∗(t),E∗(t), I ∗(t), N∗(t)) isa positive ω-periodic solution of system (2). �
From Theorem 3.1, we see that R0 is a threshold parameter to determine whether or nottuberculosis persists in the population. Furthermore, Fig. 2 supports the theoretical factthat the unique disease-free equilibrium P0 is globally asymptotically stable when R0 =0.6491 < 1. Numerical simulations complete the theoretical analysis that there exists aunique global attractive positive periodic solution when R0 = 1.9474 > 1; see Fig. 3.
942 Liu et al.
Fig. 2 The global asymptotic stability of the disease-free equilibrium P0 when R0 = 0.6491. We choosea0 = 1. Other parameter values are the same as those in Fig. 1.
4. A case study
In this section, we estimate the parameters of model (2) and study the transmission trendof tuberculosis in the mainland of China. Simulation results are given to show that ourmodel with periodic parameters matches the seasonal fluctuation data well.
Tuberculosis is one of the China’s major public health problems. According to WHOestimates, China has the world’s second largest tuberculosis epidemic, behind only In-dia, with more than 1.3 million new cases of tuberculosis every year (Wang et al., 2007;WHO, 2006). Among notifiable communicable diseases in China, tuberculosis ranksfirst or second in terms of notified cases and deaths (Ministry of Health, China, 2006).From the website of China Ministry of Health (2005–2009), we have obtained themonthly numbers of newly reported TB cases from January 2005 to December 2008.The real data and its fitted curve are shown in Table 1 and Fig. 4, respectively. Themonthly reported TB cases in China from 2005 to 2008 show an obvious seasonalfluctuation, with the seasonality peak in late spring to early summer. This seasonaltrend may be mainly attributed to increased times spent in overcrowded, poorly venti-lated housing conditions, much more common than in warm climate (Rios et al., 2000;Schaaf et al., 1996), and/or more frequent viral infections, which cause immunologicaldeficiency leading to reactivation of the M. tuberculosis (Rios et al., 2000).
A Tuberculosis Model with Seasonality 943
Fig. 3 The long-term behaviors of four classes of population when R0 = 1.9474. a0 is fixed as 3. Andother parameter values are the same as those in Fig. 1.
The monthly numbers of new TB cases in Table 1 correspond to the term f (t) :=qβ(t) S(t)I (t)
N(t)+k(t)E(t) in the third equation of system (2). Since variables and parameters
in system (2) are continuous functions of t , we use trigonometric functions to fit f (t) asa periodic function with period 12. Let
f (t) = c0 +5∑
j=1
(cj cos jLt + dj sin jLt)
in order to let the expression of f (t) be simpler and exacter, where L = 2π12 = π
6 is thefundamental frequency. We use the powerful software Mathematica to determine thosecoefficients cj and dj , which yields the function f (t). The comparison of the data with
the curve of f (t) is shown in Fig. 4. The data and the curve match quite well.
f (t) ≈ 124926 − 18215.1 cos(0.523599t) − 2198.5 cos(1.0472t)
+ 9556.16 cos(1.5708t) + 6582.29 cos(2.0944t)
− 1469.43 cos(2.61799t) + 5422.74 sin(0.523599t)
− 9335.46 sin(1.0472t) − 5729.46 sin(1.5708t)
+ 3048.19 sin(2.0944t) + 3402.81 sin(2.61799t).
The statistics show that the total numbers of the whole Chinese population in 2005,2006, 2007, and 2008 are 1299880000, 1307560000, 1314480000, and 1328020000, re-spectively (National Bureau of Statistics of China, 2008). The average population numberof those four years is taken to be initial value, N(0) = 1312480000. In 2000, the TBinfection ratio was 44.5% (The Ministry of Health of the People’s Republic of China,2002), and we take that percentage as the TB infection ratio of January 2005, where theTB infection ratio means the ratio of the individuals infected with the M. tuberculosis tothe whole population. Thus, S(0) = 728430000. In the case of drug-sensitive strain andcomplete compliance during the treatment period, 85% of patients convert from sputumpositive to sputum negative, becoming noninfectious within the first 2 months of treat-ment (Bass et al., 1994).1 We assume the infectious period of an infectious TB case is 2months, thus the total number of the infectious TB cases in the current month is the sumof the numbers of newly reported TB cases in the past 2 months. From the fitted curve off (t), I (0) = 231976. One in every ten of infected people will become sick with activetuberculosis in his/her lifetime (WHO, 2007), and around 5% of them will develop activetuberculosis during the first 2 years of infection (Ziv et al., 2001), and we can chooseq = 0.05. From the average age of the active TB cases, the death rate, and the life expec-tation we can get the estimation that R(0) = 60458024. The direct computation impliesthat E(0) = 523360000.
Fig. 4 The monthly numbers of new TB cases and its fitted curve.
The average life expectancy of Chinese people was 71.4 years in 2000 (National Bu-reau of Statistics of China, 2008). We take it as the current average life expectancy. Thus,μ = 1/(71.4 × 12) = 0.001167. According to Ziv et al. (2001), d = 0.005. Because 85%of patients convert from sputum positive to sputum negative, becoming noninfectiouswithin the first two months of treatment (Bass et al., 1994, and see footnote 1) in thecase of drug-sensitive strain and complete compliance during the treatment period, wetake r = 0.425 in our simulations. The recruitment rate is the product of the birth rateequal to the natural death rate and the total number of the whole population; therefore,Λ = 1531845.
After simulations and comparisons, the infection rate β(t) and the reactivation ratek(t) have been chosen to be β(t) = β0β1(t) and k(t) = k0k1(t) respectively, where β1(t)
and k1(t) are the before following two periodic functions with period 12:
Table 2 The parameter values and the initial condition
Symbol Value Interpretation Reference
Λ 1531845 Recruitment rate See textμ 0.001167 Natural death rate National Bureau of Statistics of China (2008)q 0.05 Proportion of new infections that WHO (2007), Ziv et al. (2001)
develop active TB within 1 yeard 0.005 Disease-induced death rate Ziv et al. (2001)r 0.425 Removal rate of infectious cases See textβ0 0.000033 Seasonality factor of the transmission See text
ratek0 1.65 Seasonality factor of progression rate See textS(0) 728430000 Initial value of susceptible individuals See textE(0) 523360000 Initial value of latent/exposed See text
individualsI (0) 231976 Initial value of infectious individuals See textN(0) 1312480000 Initial value of the total population National Bureau of Statistics of China (2008)R(0) 60458024 Initial value of recovered individuals See text
Note that β0 and k0 are related to the magnitudes of the seasonal fluctuation. In our sim-ulation, we choose β0 = 0.000033 and k0 = 1.65. In our sensitive analysis, those twoparameters are varied to see the influences of the infection rate and the reactivation rateon the new TB case numbers. The parameter values and the initial condition used in oursimulations are summarized in Table 2.
Substituting those values of parameters and functions into system (1), we obtain thefollowing TB transmission model to simulation TB infections in China:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
dSdt
= 1531845 − β0β1(t)SIN
− 0.001167S,
dEdt
= 0.95β0β1(t)SIN
− (0.001167 + k0k1(t))E,
dIdt
= 0.05β0β1(t)SIN
+ k0k1(t)E − (0.001167 + 0.005 + 0.425)I,
dRdt
= 0.425I − 0.001167R,
S(0) = 728430000, E(0) = 523360000,
I (0) = 231976, R(0) = 60458024.
(15)
We take January of 2005 as the start time of simulation. The simulation results areobtained in Figs. 5 and 6. Figure 5 illustrates the comparison of the monthly reported dataand the simulation curve of new TB cases in China. These stars in the curve demonstratethe reported numbers, from January 2005 to February 2009. The simulation result basedon our model exhibits the seasonal fluctuation and matches the data well. Figure 6 givesthe trends of the susceptible, the latent/exposed, the infectious and the treated/recoveredindividuals in the future several years, respectively.
A Tuberculosis Model with Seasonality 947
Fig. 5 New TB cases: reported number and simulation curve.
Fig. 6 The trends of four epidemiological populations.
948 Liu et al.
Fig. 7 The relationship between new TB cases and β0 and k0. In solid-line curve: β0 = 0.000038, andk0 = 1.7. In dashed-line curve: β0 = 0.000033, and k0 = 1.65. In dotted-line curve: β0 = 0.000028, andk0 = 1.6. Other parameter values are given in Table 2. Here, the stars are corresponding to the real datafrom China.
Note that β0 = 0.000033 and k0 = 1.65, as listed in Table 2, are used in our simula-tions. The epidemiological interpretations of β(t) and k(t) are: on average, every TB in-fected individuals will produce 2.23 TB infections, approximately, each month in his/herinfectious period, and about 7.18% of the TB infected, with slow progression, will developactive TB during their life time, respectively.
Sensitivity analysis of parameters is not only critical to model verification and vali-dation in the process of model development and refinement, but also provides insight tothe robustness of the model results when making decisions (Saltelli et al., 2000). Figure 7illustrates the impacts of β0 and k0 on the monthly new TB cases. β0 and k0 have an evi-dent impacts on the number of new TB cases from Fig. 7. The number of new TB casesincreases substantially with a rise in β0 and k0, and falls with a decrease in β0 and k0.
Figure 8 gives the comparison between the sensitivity coefficient of new TB casesagainst β0 and that of new TB cases against k0. The dashed-line curve represents the sen-sitivity coefficient of new TB cases against β0 over time t ; while the solid-line curve iscorresponding to that of new TB cases against k0 over time t . The sensitivity coefficientof new TB cases can be interpreted as the percentage change in the number of new TBcases for a 1% change in the parameter β0 or k0. From Fig. 8, k0 has a greater influence
A Tuberculosis Model with Seasonality 949
Fig. 8 Comparison of sensitivity coefficients of new TB cases on β0 and k0. The other parameter valuescan be found in Table 2.
on new TB cases and new TB cases is more sensitive to k0. Seasonality factor of reac-tivation rate k0 is the more sensitive parameter and plays the more important role in thenumber of new TB cases. More attention should be paid to reduce the reactivation ratefrom latent/exposed individuals to active TB cases.
Applying Lemma 2.1, we plot the curves of the basic reproduction ratio R0 againstβ0, and k0, respectively, in Fig. 9. It seems that the basic reproduction ratio R0 increaseslinearly with respect to β0, but nonlinearly with k0 from Fig. 9. R0 becomes increasing(decreasing) as β0 or k0 is increasing (decreasing).
5. Discussion
We have developed a compartmental model to describe TB seasonal incidence rate byincorporating periodic coefficients. It has been found that there is a seasonal pattern ofnew TB cases, and the numbers of new TB cases peak in late spring to early summer,and reach a nadir in late winter and early spring in the mainland of China (see Fig. 4).This seasonal pattern may be linked to the Chinese Spring Festival, during which thewhole family live in overcrowded, poorly ventilated rooms, and/or to the more frequentviral infections like flu, which can cause immunological deficiency leading to reactiva-tion of the M. tuberculosis. As mentioned in the Introduction, other variable periods ofpeak seasonality in TB incidence rates can be seen in several references (Akhtar and Mo-hammad, 2008; Douglas et al., 1996; Leung et al., 2005; Nagayama and Ohmori, 2006;Rios et al., 2000; Schaaf et al., 1996; Thorpe et al., 2004). The seasonal patterns of TBincidence rates may be also related to vitamin D deficiency (Thorpe et al., 2004).
950 Liu et al.
Fig. 9 The graphs of the basic reproduction ratio R0 versus β0 and k0, respectively. The other parametervalues are given in Table 2.
We define the basic reproduction ratio R0 and prove that the unique disease-free equi-librium P0 is globally asymptotically stable if R0 < 1; while the disease is uniformlypersistent and there exists at least one positive periodic solution if R0 > 1. Numericalsimulations show that there is only one positive periodic solution which is globally as-ymptotically stable in the case where R0 > 1 (see Fig. 3). From Fig. 5, the fitted curve ofnew TB cases matches with the statistical data very well. The number of new TB cases isan increasing function of β0 or k0 from Fig. 7, and is more sensitive to k0 than β0 fromFig. 8. The basic reproduction ratio R0 becomes increasing as β0 or k0 increases (see alsoFig. 9).
From the practical viewpoint, the model formulated in this paper can be used to under-stand the transmission behaviors of the disease and to forecast the disease trends, whichcan help health program planners to implement more preventive interventions in TB con-trol during the period of higher risk of infection.
Acknowledgement
We are grateful to the anonymous referee for his/her careful reading and helpful sugges-tions which led to an improvement of our original manuscript.
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