The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.
1. Introduction
Tuberculosis (TB) remains one of the world’s deadliest communicable diseases. In 2013, it was estimated that 9.0 million people developed TB and 1.5 million died from the disease, and TB is slowly declining each year [1]. According to the online global TB data collection system, China alone accounted for 11% of the total cases, which is the second country of 22 TB high-burden countries, only after India. Shaanxi is one of the more serious TB provinces in China, and its reported new cases each year reach about 25,000, which is the second in the number of cases of infectious diseases in Shaanxi, only next to the hepatitis B virus (HBV).
Some researchers have investigated the influence of seasonal variations on the transmission dynamics of infectious diseases [2–4]. And seasonal variation in TB incidence has been described in many countries and cities, such as India, United States, Russia, New York city, and Hong Kong [5–9]. TB is a seasonal disease in China [10, 11], but it remains unknown in Shaanxi.
From 2004 to 2012, there are 273,305 reported notifiable active TB cases in Shaanxi. Monthly reports of notifiable active TB cases from January 2004 to December 2012 in Shaanxi (Table 1) are available on the data-center of China public health science [12]. We apply seasonal filters in MATLAB program, to deseasonalize the time series; then the original time series is decomposed into three components: trend curve and season and irregular noise. The trend curve is the long-term and medium-to-long term movement of the series; it also contains consequential turning points; the seasonal component is within one-year (12 months) fluctuations about the trend that recur in a similar way in the same month or quarter every year; and the irregular component is the residual component that still remains after trend curve and seasonal component are removed from the original series.
Shaanxi TB cases month report from January 2004 to December 2012 [12].
Month/year
2004
2005
2006
2007
2008
2009
2010
2011
2012
January
2761
3781
3727
3751
3250
2696
2586
2290
2309
February
3057
2613
3118
2914
3033
2486
2038
2187
2440
March
3519
4865
3774
3178
3531
2956
2382
2489
2531
April
3284
4739
3254
2996
3005
2606
2435
2260
2276
May
3064
3602
3064
2909
3182
2556
2314
2325
2258
June
3008
3514
2660
2812
2872
2343
2148
2173
1975
July
3076
3142
2644
2508
2674
2317
2091
1957
1923
August
2822
3293
2569
2480
2540
2234
2076
2061
1878
September
2465
3121
2248
2317
2478
2229
1998
1969
1651
October
2517
2554
2328
2204
1602
2133
2036
1966
1662
November
2306
2653
1721
1989
1954
2021
1849
1837
1595
December
1580
1946
1300
1352
1581
1953
1662
1799
1548
Sum
33459
39823
32407
31410
32702
28530
25615
25313
24046
Figure 1 shows the original time series of active TB cases from January 2004 to December 2012 in Shaanxi. Figures 2(a) and 2(b) show the isolated trend curve and the seasonal component, respectively. To show the correctness of this multiplicative decomposition, the original series are compared to a series reconstructed using the component estimates in Figure 3(a) and giving irregular noise component in Figure 3(b). It has shown that the multiplicative decomposition is fit for the TB data of Shaanxi.
The original time series of pulmonary TB cases in Shaanxi of China, January 2004 to December 2012.
(a) The trend component of time series for pulmonary TB cases in Shaanxi of China, January 2004 to December 2012. (b) The seasonal component of time series for pulmonary TB cases in Shaanxi of China, January 2004 to December 2012.
Trend
Seasonal component
(a) Compare the original series to a series reconstructed using the component estimates of time series for pulmonary TB cases in Shaanxi of China, January 2004 to December 2012. (b) The irregular noise component of time series for pulmonary TB cases in Shaanxi of China, January 2004 to December 2012.
TB cases
Irregular component
In Figure 2(a), trend component shows that there is a fast upward trend after 2004 and in 2005 (the data in 2004 is low just because this is the first year for reported data, and some staff may just begin to the report system, so some patients may be omitted), then a downward trend from 2005 to 2007, and then a slowly upward in 2008, a steadily decreasing trend from 2009 to 2012. In Figure 2(b), firstly, seasonal component shows that seasonal amplitude decreases each year from 2004 to 2012; secondly, it illustrates there exists a seasonal period T, 12 months, and along with peak and trough months: the first- and second-peak month of TB notification in March and January, respectively, and between them, there exists a trough, February, between these two months (February may be a peak month, but for Chinese lunar new year, Spring Festival, there exists a special reason that patients may not choose diagnosis for regarding illness as an unlucky thing in this traditional festival); the trough month is in December. TB is a seasonal disease in Shaanxi. In addition, the peak and trough of TB transmission actually are in winter and in autumn, respectively, due to the delay which tends to last 4–8 weeks. Understanding TB seasonality may help TB programs better to plan and allocate resources for TB control activities [8]. Motivated by this, we formulate a seasonality SLIR epidemic model with periodic coefficients for TB and study its global dynamics in this paper.
This paper is organized as follows. In Section 2, a seasonality TB model is formulated. In Section 3, a unique disease-free equilibrium is obtained, and the basic reproduction number RT for the periodic model is given in detail. Furthermore, some numerical simulations are used to compare the average basic reproduction number [RT] to RT in different cases. Threshold dynamics of the TB model is analyzed in Section 4. Meanwhile, some numerical simulations are provided to validate analytical results. Finally, conclusions are given in Section 5.
2. Model Formulation
The total population is divided into four compartments: susceptible (S), latent (L), infectious (I), and recovered (R) individuals.
From Figure 2(b), new TB cases have shown the periodic monthly trend and the possible causes of the seasonal pattern. Shaanxi has a continental monsoon climate and has four distinct seasons. Seasons in Shaanxi are defined as spring (February–April), summer (May–July), autumn (August–October), and winter (November–January). TB is usually acquired through airborne infection from active TB cases; its transmission and progress tend to be effected by the climate within one year (12 months). In particular, the indoor activities are much more in winter than in a warm climate, which improves the probability of susceptible individuals exposed to Mycobacterium tuberculosis (Mtb) from the infectious individuals in a room with windows closed for a longer period of time [13]; thus infection rate may have the periodic influence. In addition, during these months near or in the highest peak month of TB cases, cold weather and lack of sunshine, which may reduce human immunity, cause a higher disease-induced rate. Individuals with lower Vitamin D level may be more reactivated for TB [14]. Thus, disease-induced rate and reactivation rate may also have the periodic influence. To describe and study the TB transmission in Shaanxi, three periodic coefficients are selected: (i) infection rate coefficient β(t) and the bilinear incidence β(t)SI which are applied in this model; (ii) reactivation rate coefficient γ(t), at which an individual leaves the latent compartment for becoming infectious; and (iii) disease-induced rate coefficient α(t), which is the disease-induced death rate coefficients for individuals in compartment I. In view of the periodic trend of monthly, β(t), γ(t), and α(t) are assumed to be positive periodic continuous function of t with period T. In some TB models, the fast and slow progression has been considered [15–17]. Based on those, a seasonality TB model with fast and slow progression and periodic coefficients is formulated in this section. The transfer among compartments is schematically depicted in Figure 4. It leads to the following model of ordinary differential equations:(1)S′=Λ-βtSI-μS,L′=1-pβtSI-γtL-μL,I′=pβtSI+γtL-σI-αtI-μI,R′=σI-μR,with initial condition (S(0),L(0),I(0),R(0))=(S0,L0,I0,R0)∈R+4, and all parameters are positive. Here, Λ is the recruitment rate and parameter μ is the natural death rate coefficient; σ is the rate coefficient at which an infective individual leaves the infectious compartment to the recovered; p(0≤p<1) is the fraction of infected individuals who are fast developing into infected cases and enter the infectious compartment directly, while 1-p is the fraction of infected individuals who are slowly developing into infected cases and transferred to the latent compartment.
The transfer diagram for model (1).
Since R does not appear in the other equations of system (1), system (1) is equivalent to the following system:(2)S′=Λ-βtSI-μS,L′=1-pβtSI-γtL-μL,I′=pβtSI+γtL-σI-αtI-μI.
Theorem 1.
Every forward solution (S(t),L(t),I(t)) of system (2) eventually enters Ω={(S,L,I)∈R+3:S+L+I≤Λ/μ}, and Ω is a positively invariant set for system (2).
Proof.
From system (1), it follows that (3)S+L+I+R′=Λ-μS+L+I+R-αtI≤Λ-μS+L+I+R,and then (4)limsupt→∞S+L+I+R≤Λμ.It implies that region X={(S,L,I,R)∈R+4:S+L+I+R≤Λ/μ} is a positively invariant set for system (1). Then, Ω={(S,L,I)∈R+3:S+L+I≤Λ/μ} is a positive invariant with respect to system (2). Therefore, system (2) is dissipative, and its global attractor is contained in Ω.
In the rest of this paper system (2) will be studied in region Ω.
3. Disease-Free Equilibrium and the Basic Reproduction Number
To study system (2), some notations are introduced.
Let (R,R+n) be the standard ordered n-dimensional Euclidean space with a norm ·. For u,v∈Rn, denote u≥v if u-v∈R+n; u>v if u-v∈R+n∖{0}; and u≫v if u-v∈Int(R+n).
Let A(t) be a continuous, cooperative, irreducible, and n×n matrix function with period T>0, and let ΦA(t) be the fundamental solution matrix of the linear ordinary differential equation: (5)dxdt=Atx.And let ρ(ΦA(T)) be the spectral radius of ΦA(T). By Perron-Frobenius theorem, ρ(ΦA(T)) is the principle eigenvalue of ΦA(T), in the sense that it is simple and admits an eigenvector v∗≫0.
There is a unique disease-free steady state E0, that is (Λ/μ,0,0), for system (2).
In the following, the basic reproduction number RT will be introduced for system (2) according to the general procedure presented in [2].
With χ≔(L,I,S), system (2) becomes (6)χ′t=Fχ-Vχ,where (7)Fχ=1-pβtSIpβtSI0,Vχ=γtL+μL-γtL+σI+αtI+μI-Λ+μS.
Furthermore, here comes (8)Ft=01-pβtΛμ0pβtΛμ,Vt=γt+μ0-γtσ+αt+μ.Then F(t) is nonnegative, and -V(t) is cooperative in the sense that the off-diagonal elements of -V(t) are nonnegative. Thus, it is easy to verify that system (2) satisfies the assumptions (A1)–(A7) in [2].
Define Y(t,s), t≥s, which is a 2×2 matrix, and is the evolution operator of the linear T-periodic system (9)dydt=-Vty.That is, for each s∈R, Y(t,s) satisfies (10)dYt,sdt=-VtYt,s,∀t≥s,Ys,s=E,where E is the 2×2 identity matrix. Thus, the monodromy matrix Φ-V(t) of (9) equals Y(t,0), t≥0. Assume that the population is near the disease-free periodic state E0. And suppose that ϕ(s), T-periodic in s, is the initial distribution of infectious individuals. Then F(s)ϕ(s) is the rate of new infections produced by the infected individuals who were introduced at time s. Given t≥s, Y(t,s)F(s)ϕ(s) gives the distribution of those infected individuals who were newly infected at time s and remain in infected compartments at t. It follows that (11)ψt≔∫-∞tYt,sFsϕsds=∫0∞Yt,t-aFt-aϕt-adais the distribution of accumulative new infections at time t produced by all those infected individuals ϕ(s) introduced at time s(s≤t). Let CT be the ordered Banach space of all T-periodic functions from R to Rn, which is equipped with the maximum norm ∥·∥∞ and the positive cone CT+={ϕ∈CT:ϕ(t)≥0,t∈R}. Define a linear operator H:CT→CT by (12)Hϕt=∫0∞Yt,t-aFt-aϕt-ada,∀t∈R,ϕ∈CT.Then, according to Wang and Zhao [2], the basic reproduction number RT is defined as (13)RT≔ρHfor the periodic epidemic system (2), where ρ(H) denotes the spectral radius of the matrix H.
In the constant case, that is, βt≡β, γ(t)≡γ, δ(t)≡δ, ∀t>0, then F(t)≡F, V(t)≡V, ∀t>0, in which (14)F=01-pβΛμ0pβΛμ,V=γ+μ0-γσ+α+μ.By van den Driessche and Watmough [18], here comes (15)RT=ρFV-1=βμ+α+σp+γμ+γ1-p.
In the periodic case, in order to characterize RT, consider the following linear T-periodic equation: (16)dwdt=-Vt+Ftλw,∀t∈R,with parameter λ∈(0,∞). Let W(t,s,λ) be the evolution operator of system (16) on R2, and RT can be calculated in numerically according to Lemma 2.
For system (2), the following statements are valid:
RT=1 if and only if ρ(ΦF(t)-V(t)(T))=1.
RT>1 if and only if ρ(ΦF(t)-V(t)(T))>1.
RT<1 if and only if ρ(ΦF(t)-V(t)(T))<1.
Thus, the disease-free periodic solution E0 for system (2) is locally asymptotically stable if RT<1 and unstable if RT>1.
Define (17)f≔1T∫0Tftdtas the average for a continuous periodic function f(t) with the period T. Let [RT] be the basic reproduction number of the autonomous systems obtained from the average of system (2); that is,(18)S′=Λ-βSI-μS,L′=1-pβSI-γL-μL,I′=pβSI+γL-σI-αI-μI.
An example is given to show that the basic reproduction number of the time-averaged autonomous systems may underestimate, estimate, or overestimate the infection risk.
Example 4.
Consider the following:(19)βt=b01+k1cosπt+1T/2,γt=g01+k2cosπt-1T/2,αt=a01+k3cosπt-1T/2.
Now Lemma 2 is applied to calculate the basic reproduction number RT of system (2).
Firstly, Λ=0.8, μ=0.008, p=0.08, g0=0.003, k1=k2=k3=1, σ=0.5, T=12, and a0=0.08 in system (2); by numerical computation, it can acquire the curves of the average basic reproduction number [RT] and the basic reproduction number RT when b0 varies, respectively, in Figure 5. It can be seen that [RT] is always greater than RT as b0 is ranging from 0.003 to 0.045. Secondly, when b0=0.015 in system (2), g0 varies from 0.003 to 0.007, and other parameters are the same as those of Figure 5; then the numerical calculations indicate [RT] is greater than RT in Figure 6 as g0 is varying. Thirdly, b0=0.015 in system (2), a0 varies from 0.01 to 0.05, and other parameters are the same as those of Figure 5; then the numerical calculations indicate [RT] is greater than RT in Figure 7 as a0 is varying. Summing up the above, Figures 5, 6, and 7 imply that the risk of infection may be overestimated, if the average basic reproduction number [RT] is used.
But, on the other hand, if T=1, from Figures 8, 9, and 10, the numerical calculations indicate [RT] is less than RT in Figures 8, 9, and 10, respectively. These imply that the risk of infection may be underestimated, if the average basic reproduction number [RT] is used.
Especially, if T=5, from Figures 11, 12, and 13, the numerical calculations indicate [RT] is almost equal to RT in Figures 11, 12, and 13, respectively. These imply that the risk of infection may be estimated by the average basic reproduction number [RT] and [RT] can be used in some conditions.
If b0=0.0165 and k1, k2, k3 vary in [0,1] for system (2), respectively, with other parameters unchanged as Figure 5, numerical computation indicates (see Figures 14, 15, and 16) the average basic reproduction number overestimates the disease transmission risk.
Finally, if b0=0.015 and T varies in [0,22] for system (2), with other parameters unchanged as those shown in Figure 5, numerical computation gives the relation between the basic reproduction number RT and period T in Figure 17, which indicates the average basic reproduction number [RT] may be superior, inferior, or equal to the basic reproduction number RT, which is up to value of period T.
Furthermore, in the next section, we will prove some theoretical results of system (2), in which RT serves as a threshold parameter: if RT<1, then there exists a globally asymptotically stable disease-free periodic state E0(Λ/μ,0,0); if RT>1, then the disease is persistent in the population and there exists at least one positive periodic solution.
For system (2), the graph of the average basic reproduction number [RT] and the basic reproduction number RT with respect to b0 which varies from 0.003 to 0.045, and Λ=0.8, μ=0.008, p=0.08, g0=0.003, k1=k2=k3=1, σ=0.5, T=12, and a0=0.08.
For system (2), the graph of [RT] and RT with respect to g0 which varies from 0.003 to 0.007 when b0=0.015, and other parameter values are the same as those of Figure 5.
For system (2), the graph of [RT] and RT with respect to a0 which varies from 0.01 to 0.05 when b0=0.0165, and other parameter values are the same as those of Figure 5.
For system (2), the graph of [RT] and RT with respect to b0 which varies from 0.003 to 0.045 when T=1, and other parameter values are the same as those of Figure 5.
For system (2), the graph of [RT] and RT with respect to g0 which varies from 0.003 to 0.007 when T=1, and other parameter values are the same as those of Figure 6.
For system (2), the graph of [RT] and RT with respect to a0 which varies from 0.01 to 0.05 when T=1, and other parameter values are the same as those of Figure 7.
For system (2), the graph of [RT] and RT with respect to b0 which varies from 0.003 to 0.045 when T=5, and other parameter values are the same as those of Figure 5.
For system (2), the graph of [RT] and RT with respect to g0 which varies from 0.003 to 0.007 when T=5, and other parameter values are the same as those of Figure 6.
For system (2), the graph of [RT] and RT with respect to a0 which varies from 0.01 to 0.05 when T=5, and other parameter values are the same as those of Figure 7.
For system (2), the graph of [RT] and RT with respect to k1 which varies from 0 to 1 when b0=0.0165, and other parameter values are the same as those of Figure 5.
For system (2), the graph of [RT] and RT with respect to k2 which varies from 0 to 1 when b0=0.0165 and other parameter values are the same as those of Figure 5.
For system (2), the graph of [RT] and RT with respect to k3 which varies from 0 to 1, when b0=0.0165 and other parameter values are the same as those of Figure 5.
For system (2), the graph of [RT] and RT with respect to T which varies from 0 to 22 when b0=0.015, and other parameter values are the same as those of Figure 5.
4. Extinction and Uniform Persistence
The following lemma is useful for our discussion in this section.
Lemma 5 (see [<xref ref-type="bibr" rid="B19">19</xref>], Lemma <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M294"><mml:mrow><mml:mn>2.1</mml:mn></mml:mrow></mml:math></inline-formula>).
Let l=1/Tln(ρ(ΦA(T))), and then there exists a positive T-periodic function v(t) such that eltv(t) is a solution of (5).
Theorem 6.
For system (2), the disease-free periodic state E0(Λ/μ,0,0) is globally stable on set Ω if RT<1; and it is unstable if RT>1.
Proof.
By Lemma 3, if RT>1, then E0(Λ/μ,0,0) is unstable; and if RT<1, then E0 is locally asymptotically stable. Hence, we only need to prove that E0 is globally attractive for RT<1.
Since S(t),L(t),I(t) is a nonnegative solution of system (2) in Ω, we have S≤Λ/μ, and know that(20)L′≤1-pβtΛμI-γtL-μL,I′≤pβtΛμI+γtL-σI-αtI-μI,for t≥0.
Consider an auxiliary system:(21)L~′=1-pβtΛμI~-γtL~-μL~,I~′=pβtΛμI~+γtL~-σI~-αtI~-μI~;that is, (22)L~I~′=Ft-VtL~I~.It follows from Lemma 5 that there exists a positive T-periodic function v1(t), such that el1tv1(t) is a solution of (22), where l1=1/Tln(ρ(ΦF(t)-V(t)(T))). Choose t1≥0 and a real number a1>0 such that (23)Lt1It1≤a1v10.By the comparison principle, we get (24)LtIt≤a1v1t-t1el1t-t1,∀t≥t1.
By Lemma 3, it is easy to know that RT<1 if and only if ρ(ΦF(t)-V(t)(T))<1, thus l1=1/Tln(ρ(ΦF(t)-V(t)(T)))<0. Therefore, L(t)→0, I(t)→0, and S(t)→Λ/μ as t→∞; that is, E0(Λ/μ,0,0) is globally attractive for RT<1. In conclusion, E0 is globally asymptotically stable if RT<1.
Theorem 7.
If RT>1, system (2) is uniformly persistent, and there exists at least one positive periodic solution.
Proof.
Denote Ω0≔{(S,L,I)∈Ω:L>0,I>0} and ∂Ω0≔Ω∖Ω0. And then x0=(S0,L0,I0)∈Ω0. Let P:Ω→Ω be the Poincaré map associated with system (2); that is, P(x0)=u(T,x0), ∀x0∈Ω, where φ(t,x0) is the unique solution of system (2) with φ(0,x0)=x0.
Now it is proved that P is uniformly persistent with respect to (Ω0,∂Ω0).
It is easy to see that Ω and Ω0 are positively invariant, ∂Ω0 is a relatively closed set in Ω, and P is point dissipative from Theorem 1.
Set M∂={(S0,L0,I0)∈∂Ω0:Pm(S0,L0,I0)∈∂Ω0,∀m≥0}.
We claim that(25)M∂=S,0,0:S≥0.
Obviously, {(S,0,0):S≥0}⊆M∂. For any S0,L0,I0∈∂Ω0∖{(S,0,0):S≥0}, if L0>0, I0=0, then L(t)>0,∀t≥0, and then I′=γ(t)L>0. For the other case, L0=0, I0>0, and then I(t)>0, and, from the first equation of system (2), thus(26)St=e-∫0tβsIs+μdsS0+Λ∫0te∫0sβζIζ+μdζds≥Λe-∫0tβsIs+μds∫0te∫0sβζIζ+μdζds>0,for any t>0.
From the second equation of system (2), we have(27)Lt=e-∫0tγs+μdsL0+∫0t1-pβsSsIse∫0sγζ+μdζds>0,∀t>0.It then follows that (S(t),L(t),I(t))∈¯∂Ω0 for 0<t≪1. Thus, the positive invariance of Ω0 implies (25).
Clearly, there is a unique fixed point of P in M∂, which is E0(Λ/μ,0,0).
For system (2), by the continuity solutions with respect to the initial values, ∀α>0, there exists α∗>0 such that, for all x0∈Ω0 with x0-E0⩽α∗, we have ϕ(t,x0)-ϕ(t,E0)<α,∀t∈[0,T].
Then, we will show that(28)limsupm→∞dPmx0,E0≥α∗,∀x0∈Ω0.If not, then(29)limsupm→∞dPmx0,E0<α∗for some x0∈Ω0.
Without loss of generality, we can assume that(30)dPmx0,E0<α∗,∀m≥0.Then, we have(31)ϕt,Pmx0-ϕt,E0<α,∀m≥0,∀t∈0,T.For any t≥0, let t=mT+t1, where t1∈[0,T) and m is the largest integer less than or equal to t/T. Therefore, we have(32)ϕt,Pmx0-ϕt,E0=ϕt1,Px0-ϕt1,E0<α,∀t≥0.Note that x(t)≔(S(t),L(t),I(t))=ϕ(t,x0). It then follows that 0≤S(t),L(t),I(t)≤α,∀t≥0. From the first equation of system (2), we have(33)S′≥Λ-βtSα-μS.Note that the perturbed system(34)S^′=Λ-βtS^α-μS^admits a unique positive T-periodic solution(35)S^t,α=e-∫0tβsα+μdsS^0,α+Λ∫0te∫0sβζα+μdζdswhich is globally attractive in R+, where(36)S^0,α=Λ∫0Te∫0sβζα+μdζds1-e∫0Tβsα+μds>0.Applying Lemma 3, we know that RT>1 if and only if ρ(ΦF(t)-V(t)(T))>1. By continuity of the spectrum for matrices ([20], Section II. 5.8.), we can choose η, which is small enough, such that ρ(ϕF(t)-V(t)-ηM(t)(T))>1, where (37)Mt=01-pβt0pβt.Since S^(0,α) is continuous in α, we can fix α>0 small enough that S^(t,α)>S^(t)-η, ∀t≥0. Furthermore, since the fixed point S^(0,α) of the Poincaré map associated with (34) is globally attractive, there exists t^>0 such that S(t)>S^(t)-η for t≥t^. As a consequence, for t≥t^, it holds that(38)L′≥1-pβtS^t-ηI-γt+μL,I′≥pβtS^t-ηI+γtL-σ+αt+μI.Consider another auxiliary system(39)L~′=1-pβtS^t-ηI~-γt+μL~,I~′=pβtS^t-ηI~+γtL~-σ+αt+μI~.It follows from Lemma 5 that there exists a positive T-periodic function (L~(t),I~(t)) such that (L~(t),I~(t))=elt(L~(t),I~(t)) is a solution of (39), where (40)l=1TlnρϕFt-Vt-ηMtT.Choose t¯≥t^ and a small α2>0 such that (L(t¯),I(t¯))≥(L~(0),I~(0)). By the comparison principle we get (L(t),I(t))≥α2(L~(t-t¯),I~(t-t¯))el(t-t¯), ∀t≥t¯. Since RT>1, ρ(ϕF(t)-V(t)-ηM(t)(T))>1. And thus l>0, which implies that L(t)→∞ and I(t)→∞ as t→∞. This leads to a contradiction.
So suppose (29) is wrong; that is, (28) is right. Furthermore, (28) shows that E0 is an isolated invariant set in Ω, and Ws(E0)∩Ω0=Φ. Every orbit in M∂ converges to E0, and E0 is acyclic in M∂. By the acyclicity theorem on uniform persistence for maps ([21], Theorem 1.3.1 and Remark 1.3.1), it follows that P is uniformly persistent with respect to (Ω0,∂Ω0). Thus ([21], Theorem 3.1.1) implies the uniform persistence of the solutions of system (2) with respect to (Ω0,∂Ω0); that is, there exists ε>0 such that any solution (S(t),E(t),I(t)) of (2) with initial values (S(0),E(0),I(0))∈Ω0 satisfies limt→∞L(t)≥ε and limt→∞I(t)≥ε. Moreover, by Zhao ([21], Theorem 1.3.6), P has a fixed point (S∗(0),E∗(0),I∗(0))∈Ω0. From the first equation of (2), S∗(t) satisfies S∗′≥μA-β(t)S∗Λ/μ-μS∗. By the comparison theorem, we have S∗ ≥ e-∫0t(β(s)Λ/μ+μ)ds(S∗(0)+Λ∫0te∫0s(β(τ)Λ/μ+μ)dτds) > Λe-∫0t(β(s)Λ/μ+μ)ds∫0te∫0s(β(τ)Λ/μ+μ)dτds > 0,∀t>0. The seasonality of S∗(t) implies S∗(0)>0. By the second and third equations of (2) and the irreducibility of the cooperative matrix (41)-γt+μ1-pβtS∗tγtpβtS∗t-σ+αt+μ,it follows that (L∗(t),I∗(t))∈Int(R+2), ∀t≥0. Consequently, (S∗(t),L∗(t),I∗(t)) is a positive T-periodic solution of (2).
Theorems 6 and 7 have shown that RT is a threshold parameter which determines whether or not the disease persists in the population. Now, some numerical simulations (Figures 18 and 19) are presented to demonstrate these results. And in these simulations, T=12, according to the fact that by one year has 12 months. In Figures 18 and 19, the simulations verify Theorems 6 and 7, respectively.
For system (2), β(t),γ(t), and α(t) are listed in Example 4. Λ=0.8, μ=0.008, p=0.08, a0=0.08, g0=0.003, k1=k2=k3=1, σ=0.5, T=12, and b0=0.005, and then RT=0.2814. These figures show that the disease will die out, which is the same as Theorem 6.
For system (2), b0=0.04 and other parameter values are the same as those of Figure 18; then RT=2.251. These figures show that the disease will be asymptotic to a periodic solution, which is the same as Theorem 7.
5. Discussion
Monthly pulmonary TB cases, from January 2004 to December 2012 in Shaanxi province, have been analyzed by the seasonal adjustment program. It has been found that TB cases show seasonal variation in Shaanxi: the peak months (January and March) compare to the lowest trough month (December). It is necessary to study the seasonality TB epidemic model according to the seasonal component of Shaanxi’s data. Considering the regularity of peak TB seasonality may help allocate resources for the prevention and treatment of TB activities, a periodic TB epidemic model has been formulated and studied.
The basic reproduction number RT for the periodic model is given. By numerical simulation, RT has been compared to the average basic reproduction number [RT] in different parameter values. It has shown that [RT] may overestimate or underestimate RT or be equal to RT in different cases. It is also up to the value of periodic T. Furthermore, in theory, the threshold dynamics has been studied for the periodic TB model: if RT<1, then the disease-free equilibrium is globally asymptotically stable; that is, the TB disease will disappear eventually; if RT>1, then there exists at least one positive periodic solution and the disease will be uniformly persistent.
Our numerical simulations are in good accordance with these theoretical results. It should be noted that we have not fit Shaanxi’s data in these simulations since we cannot accurately estimate some parameters’ values in the periodic model according to available data or references. Despite lack of comparing the model results with the Shaanxi’s data, our theoretical results have shown that the basic reproduction number with periodic RT plays a crucial role in determining dynamics of the seasonality TB disease and could be used for controlling the spread of TB epidemic in reality strategies.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by the National Natural Science Foundation of China (nos. 11301320, 11371369, and 11471201), the China Postdoctoral Science Foundation Funded Project (no. 2013M532016), the Postdoctoral Science Foundation in Shaanxi of China, the State Scholarship Fund of China (no. 201407820120), and the International Development Research Centre of Canada.
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