GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM OF A TUBERCULOSIS MODEL WITH IMMIGRATION AND TREATMENT

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 1, Spring 2011 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM OF A TUBERCULOSIS MODEL WITH IMMIGRATION AND TREATMENT HONGBIN GUO AND MICHAEL Y. LI ABSTRACT. Mathematical analysis is carried out for a tuberculosis TB model incorporating both latent and clinical stages, immigration and treatment. With immigration into the latent classes, we show that the model always has a unique endemic equilibrium and it is globally asymptotically stable. The global stability of the endemic equilibrium is proved using a global Lyapunov function. 1 Introduction. Tuberculosis TB is an airborne disease caused by infection of bacterium Mycobacterium tuberculosis. Though TB is preventable and curable, the current global TB incidence rate remains at a high level of 139 cases per 100,000 population, according to the 2009 WHO report [24], and there were 9.27 million new TB cases worldwide, with close to 1.77 million TB-related deaths in TB incidence rates are falling slowly in most WHO regions. However, the number of TBrelated deaths and TB cases are still rising, largely due to population growth, emerging antibiotic resistance, increasing migration, and coinfection with HIV [24, 25]. Mathematical models have been developed to improve our understanding of the transmission dynamics of TB, to study the impact of drug-resistance, co-infection with HIV, re-infection and relapses, and to evaluate the effectiveness of various control and prevention strategies [2, 4, 5, 6, 7, 8, 14, 17, 18, 20, 21]. Ziv et al. [21] introduced a TB model with early and late latent stages to discuss effectiveness of treating TB patients at different stages, which has been used as a template for later TB models. In [18], a TB model with a single latent AMS subject classification: 92D30. Keywords: Tuberculosis, endemic equilibrium, global stability, Lyapunov function. Copyright c Applied Mathematics Institute, University of Alberta. 1

2 2 HONGBIN GUO AND MICHAEL Y. LI stage was considered, and immigration was allowed to enter both latent and infective classes. Immigration into the infective class was first considered in a model for the HIV/AIDS [6]. It is known that in such a case, due to the constant influx of infectives, the models will no longer have a disease-free equilibrium, and the basic reproduction number also loses its original meaning. The disease will always be endemic, and a unique endemic equilibrium is expected to be globally stable. In the present paper, we propose and investigate a TB model with both early latent and late latent stages, and immigration into the susceptible and both latent stages. Most of the developed countries have TB screening policies for new immigrants. However, immigrants with early or late latent TB are usually not detected by the TB screening, and TB cases from immigrants or foreign-borns contribute an increasingly significant proportion of the national TB incidence of industrial countries [22, 23]. We also incorporate into the model reinfection and relapses due to treatment failure. So that our model can also be applied to study impact of migration among different regions in high TB incidence countries. Our model is more general than those in [12, 21] in that immigrations are allowed into the latent stages, and it generalizes a class of models in [18] to include two latent stages. We prove that, with immigration into the latent classes, the TB is always endemic in the population, and a unique endemic equilibrium is globally asymptotically stable. The global stability is established with a global Lyapunov function. In the next section, we will formulate the model. In Section 3, we establish the uniqueness of the endemic equilibrium. In Section 4, we prove the main global-stability result. In Section 5, impacts of immigrate levels on equilibrium curves are explored by two figures feathering disease eradication and persistence. 2 Model formulation The TB bacteria can spread in the air from a person with active TB disease to others when they are in close contact. When first infected with TB bacteria, a person typically goes through a latent and asymptomatic period. There are two distinct time periods during the latent TB infection. Within the first two years after infection, the risk of developing active disease is much higher, whereas at the later years of infection, the progression to active disease is much slower. This process is the so-called fast and slow dynamics of TB [4]. Using a compartmental approach, the total host population is partitioned into four compartments: susceptible individuals, early latent E and late latent L individuals, and individuals with active

3 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM 3 pulmonary TB disease T. Let t, Et, Lt and T t denote the density of populations in the four corresponding compartments at time t. Only individuals in compartment T are infectious, and new infections result from contacts between a susceptible and an infectious individual, with an incidence rate βtt t. Once infected, individuals progress through the early latent stage with an average rate ω. A fraction 0 < p < 1 of these individuals progress directly to the active TB stage, and the remaining 1 p fraction progress to the late latent stage. Once there, the rate of progression to active disease is at a lower rate ν. The immigration input to the whole population is assumed to be a constant π, with a fraction q 1 moving into the early latent compartment, q 2 into the late latent compartment, and the remaining fraction 1 q 1 q 2 into the susceptible compartment. Due to strict border screening policies in developed countries for immigrants, we assume that there are no immigrants with active TB can gain entry. The current WHO DOTS plan has a target treatment rate of 85% [24]. However, this target rate is difficult to achieve in most regions with high TB prevalence, partially due to treatment failure or relapse after therapy. We assume that TB patients in compartment T can revert to the susceptible compartment with a rate δ due to complete treatment, and they can revert to the early latent compartment E with rate γ, mostly due to incomplete treatment. The removal rates for the four compartments are d, d E, d L and d T, respectively. The disease induced death rate is α. The dynamical transfer among the four compartments is depicted in the following transfer diagram. Here, all parameters are assumed to be positive. Based on the assumptions and the transfer diagram, the model can be described by FIGURE 1: Transfer diagram of model 1.

4 4 HONGBIN GUO AND MICHAEL Y. LI the following four ordinary differential equations: 1 = 1 q 1 q 2 π βt d + δt, E = q 1 π + βt + γt d E + ωe, L = q 2 π + 1 pωe d L + νl, T = pωe + νl d T + α + δ + γt. Here, we assume that 0 < q 1 + q 2 < 1. When q 1 = q 2 = 0 and δ = γ = 0, model 1 reduces to the one in [12], and its global dynamics was completely established using a global Lyapunov function. When p 1, model 1 has only one latent stage and structurally is similar to one model studied in [18], in which the proportionate incidence is used. Our model allows for different removal rates for all compartments where earlier models typically use equal removal rates. We will establish the global dynamics of model 1 using a global Lyapunov function motivated by those in [1, 12, 15]. Lyapunov functions of this type have been used effectively for other biological and epidemic models [3, 9, 10, 13, 19]. 3 Equilibrium and its uniqueness From 1, in absence of disease, we have 1 q 1 q 2 π d, and thus lim sup t t 1 q 1 q 2 π/d along each solution to 1. Let Nt = t + Et + Lt + T t. Then using 1 we have N π d d E E d L L d T T αt π dn, where d = min{d, d E, d L, d T + α}. This implies lim sup t Nt π/d. The model can be studied in the feasible region Γ defined as Γ = {, E, L, T R q 1 q 2 π, d 0 + E + L + T π d }. It can be verified that the closed set Γ is positively invariant with respect to 1. We denote by Int Γ the interior of Γ in R 4 +. Next we establish the existence of endemic equilibrium in the feasible region Γ.

5 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM 5 For simplification, we use the following notations: 2 Π = 1 q 1 q 2 π, d 2 = d E + ω, d 3 = d L + ν, d 4 = d T + α + δ + γ. An equilibrium, E, L, T of 1 satisfies following equations Π = βt + d δt, d 2 E = q 1 π + βt + γt, 3 d 3 L = q 2 π + 1 pωe, d 4 T = pωe + νl. Combining the last two equations of 3 to cancel the L term, we get 4 νq 2 π + [pd pν]ωe = d 3 d 4 T. Then combining 4 with the second equation of 3 to cancel the E term, we have 5 [pd pν]ω d 2 [q 1 π + βt + γt ] + νq 2 π = d 3 d 4 T. Let 6 a 1 = [pd pν]ω d 2. Then 5 becomes 7 a 1 q 1 π + νq 2 π = d 3 d 4 a 1 γ a 1 β T. From the first equation of 3, we have = Π+δT βt +d. Substituting equation into 7, we obtain a quadratic equation on T 8 ft = AT 2 + BT + C = 0, where A = d 3 d 4 a 1 γ + δ,

6 6 HONGBIN GUO AND MICHAEL Y. LI B = d d 3 d 4 a 1 γ βa 1 q 1 π + νq 2 π + a 1 Π, C = d a 1 q 1 π + νq 2 π. We see that coefficient C < 0 if q q Also 9 A = d 3 d 4 a 1 γ + δ = d L + νd T + α + δ + γ [pd pν]ω d 2 δ + γ = d E + ωd L + νd T + α + δ + γ [pd pν]ωδ + γ /d 2 [ωd L + νδ + γ [pd pν]ωδ + γ] /d 2 = δ + γω [d L + ν pd L + ν] /d 2 = δ + γω [1 pd L ] /d 2 > 0, since 0 < p < 1. Therefore, the quadratic equation 8 has a unique positive solution T. Accordingly, from 3, we know that system 1 always has a unique positive equilibrium P =, E, L, T with > 0, E > 0, L > 0 and T > 0. It can also be concluded that β δ > 0 from the first equation of 3, since Π/d. Proposition 1. Suppose that 0 < q 1 + q 2 < 0. Then system 1 has a unique endemic equilibrium P =, E, L, T, > 0, E > 0, L > 0 and T > 0. Proposition 1 implies that system 1 has no disease-free equilibrium and the TB is always endemic in the population due to the influx of latent TB. In the next section, we show that the TB incidence stabilizes at the endemic equilibrium P. 4 Global stability of the endemic equilibrium Theorem 2. Suppose that 0 < q 1 + q 2 < 0. Then the unique endemic equilibrium P is globally asymptotically stable in the feasible region Γ. Remark. When q 1 = q 2 = 0, there is no immigration into the latent classes. Model 1 in this case reduces to a model in [11], and the

7 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM 7 global dynamics exhibit classical threshold phenomenon: there is a basic reproduction number such that R 0 = π d βωpd L + ν d L + νd E + ωd T + α + δ + γ γωpd L + ν, i If R 0 1, then the disease-free equilibrium P 0 = π/d, 0, 0, 0 is globally asymptotically stable in the feasible region. ii If R 0 > 1, then P 0 is unstable, and a unique endemic equilibrium P is globally asymptotically stable in the interior of the feasible region. We refer the reader to [11] for proofs of these results. Proof of Theorem 2. Denote xt = t, Et, Lt, T t Γ R 4 +. Consider a Lyapunov function 10 V x = a 0 + a 1 ln + a 1 E E E ln E E + a 2 L L L ln L L + a 3 T T T ln T T, where 11 a 0 = a 1δ β δ, a 1 = 1 pa 2 + pd 3 d 2 ω, a 2 = ν, a 3 = d 3 are positive constants and P =, E, L, T is the endemic equilibrium. Constant a 1 is same as in 6. We note that V x is positive definite with respect to x = P. The derivative of V t along a solution of 1 is 12 V = a 0 + a a 1 1 E E E

8 8 HONGBIN GUO AND MICHAEL Y. LI + a 2 1 L L L + a 3 1 T T T. Using the first equation of 3 and the equation in 1 we obtain 13 = Π d βt + δt = Π d βt β T T + d Π + δt T = d βt β T T + δt T = d + βt β δt T. It follows that 14 a 0 = a 0 d + βt 2 a 0 β δt T = a 0 d + βt 2 a 1 δt T. It follows from 1 and the first equation of 3 that 15 1 = Π βt d + δt Π + β T + d δt = β T + d δt 1 βt d + β T + d + δt 1 = d 2 + β T

9 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM 9 βt + β T 1 + δt T 1. Similarly, using 1 and the remaining three equations in 3, we have 16 1 E E = q 1 π + βt + γt d 2 E E q 1 π E E β T E T T E γt T E T E + q 1π + β T + γt, 1 L L = q 2 π + 1 pωe d 3 L L q 2 π L L 1 pωe EL E L + q 2 π + 1 pωe, 1 T T = pωe + νl d 4 T T pωe ET E T νl LT L T + pωe + νl. Substituting 14, 15 and 16 into 12 and using 11, we obtain V = a 1 d 2 a 0 d + βt a 1 β T 2 T E T E + a 1 q 1 π 2 E + a 1 γt 1 T E E T E + a 2 q 2 π 2 L + a 2 1 pωe 1 EL L E L + a 3 pωe 1 ET E T + a 3 νl 1 LT L + T [a 1 β a 3 d 4 + a 1 γ]. T

10 10 HONGBIN GUO AND MICHAEL Y. LI Note that from 7 18 [a 1 β + a 1 γ d 3 d 4 ] + a 1q 1 π + a 2 q 2 π T = 0. Substituting 18 into 17 gives 19 V = a 1 d 2 a 0 d + βt 2 + a 1 β T 2 T E + a 1 q 1 π 2 E E T T + a 1 γt 1 T E T E T E + a 2 q 2 π 2 L L T T + a 2 1 pωe 1 EL E L + a 3 pωe 1 ET + a 3 νl 1 LT. = V 1 + V 2. From the mean inequality E T L T a 1 + a a n n n a 1 a 2 a n for a i 0, i = 1,, n, it follows that V 1 = a 1 d 2 20 a 0 d + βt 2 0 for T 0. The equality in 20 holds if and only if =. The next step is to prove V 2 0. We note that pd L +ν = pd L +ν+1 pν = pd 3 +1 pa 2 and define 21 b 1 = pd 3 pd L + ν, b 2 = 1 pa 2 pd L + ν, b 1 + b 2 = 1, b 1 > 0, b 2 > 0.

11 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM 11 We also define 22 c 1 = q 1π d 2 E, c 2 = β T d 2 E, c 3 = γt d 2 E. Then c i > 0, i = 1, 2, 3, and 3 i=1 c i = 1 from the second equation of 3. Canceling the E term from the second and third equations of 3, we get Define νd 3 L = 1 pνω [q 1 π + β T + γt ] + νq 2 π d 2 = 1 pa 2 pd L + ν pd L + νω [q 1 π + β T + γt ] + a 2 q 2 π d 2 = b 2 a 1 [q 1 π + β T + γt ] + a 2 q 2 π. 23 η 1 = b 2a 1 q 1 π a 2 d 3 L, η 2 = b 2a 1 β T a 2 d 3 L, η 3 = b 2a 1 γt a 2 d 3 L, η 4 = a 2q 2 π a 2 d 3 L. We can show that η i > 0 and 4 i=1 η i = 1. Using 21 23, we can arrange V 2 as 24 V 2 = b 1 + b 2 a 1 β T 2 + a 2 q 2 π 2 L L T T T E T E + b 1 + b 2 a 1 γt 1 T E T E + c 1 + c 2 + c 3 a 2 1 pωe 1 EL E L + b 1 + b 2 a 1 q 1 π 2 E E T T + c 1 + c 2 + c 3 a 3 pωe 1 ET E T + η 1 + η 2 + η 3 + η 4 a 3 νl 1 LT L T

12 12 HONGBIN GUO AND MICHAEL Y. LI = [b 2 a 1 β T 2 T E T E + c 2 a 2 1 pωe 1 EL E L + η 2 a 3 νl 1 LT ] L T + [b 1 a 1 β T 2 T E T E + c 2 a 3 pωe 1 ET ] E T + [b 1 a 1 γt 1 T E T + c 3 a 3 pωe 1 ET ] E E T + [b 2 a 1 γt 1 T E T + c 3 a 2 1 pωe 1 EL E E L + η 3 a 3 νl 1 LT ] L T [ + b 2 a 1 q 1 π 2 E E T T + c 1 a 2 1 pωe 1 EL + η 1 a 3 νl 1 LT ]. = [ + b 1 a 1 q 1 π 2 E [ + a 2 q 2 π 7 I i. i=1 2 L L T T E L L T + c 1 a 3 pωe 1 ET ] E T T E T + η 4 a 3 νl 1 LT ] For the coefficients in I 1, using 22, 23 and 11, we have c 2 a 2 1 pωe = β T d 2 E a 21 pωe = β T 1 pa 2 pd L + ν L T pd L + νω d 2 = b 2 a 1 β T,

13 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM 13 η 2 a 3 νl = b 2a 1 β T a 2 d 3 L a 3 νl = b 2 a 1 β T. By the mean inequality, I 1 = b 2 a 1 β T 4 T E T E EL E L LT 25 L T [ b 2 a 1 β T 4 4 T E T E EL E L LT ] 1/4 L T = 0 for all > 0, E > 0, L > 0, T > 0. The equality in 25 holds if and only if 26 =, E E = L L = T T. Similarly, using 21, 22, 23 and 11, we have the following relations regarding coefficients in I 2,, I 7 c 2 a 3 pωe = β T d 2 E d 3pωE = β T pd 3 pd L + νω pd L + ν d 2 c 3 a 3 pωe = γt c 3 a 2 1 pωe = γt = b 1 a 1 β T, d 2 E a 3pωE = γt pd 3 pd L + ν d 2 E a 21 pωe = γt 1 pa 2 pd L + ν pd L + νω d 2 = b 2 a 1 γt, η 3 a 3 νl = b 2a 1 γt a 2 d 3 L a 3νL = b 2 a 1 γt, c 1 a 2 1 pωe = q 1π d 2 E a 21 pωe = b 2 a 1 q 1 π, η 1 a 3 νl = b 2a 1 q 1 π a 2 d 3 L a 3νL = b 2 a 1 q 1 π, c 1 a 3 pωe = q 1π d 2 E a 3pωE = b 1 a 1 q 1 π, η 4 a 3 νl = a 2q 2 π a 2 d 3 L a 3νL = a 2 q 2 π. pd L + νω d 2 = b 1 a 1 γt,

14 14 HONGBIN GUO AND MICHAEL Y. LI By the mean inequality, we obtain I 2 = b 1 a 1 β T 3 T E T E ET E 0, T I 3 = b 1 a 1 γt 2 T E T E ET E 0, T 27 I 4 = b 2 a 1 γt 3 T E T E EL I 5 = b 2 a 1 q 1 π 4 E I 6 = b 1 a 1 q 1 π 3 E I 7 = a 2 q 2 π 3 L L T T LT L T E L LT L 0, T E T T EL E L LT L 0, T E T T ET E 0, T 0. Substituting 19, 25 and 27 into 17, we conclude that V = V 1 + V 2 0 for all > 0, E > 0, L > 0, T > 0, and V = 0 if and only if relations in 26 hold. To find the largest invariant set in the set where V = 0, we use 26 and set = 0 in the first equation of 1, and we see that T = T, and hence E = E, L = L. Namely, the largest invariant set where V = 0 is the singleton {P }. By LaSalle s Invariance Principle [16], P is globally asymptotically stable with respect to Γ. This establishes Theorem 2. 5 Impact of immigration on endemic levels In this section, we carry out simulations of our model 1 using data from [21] to investigate the effects of immigration level on active TB cases. We choose π = 20000, d = d E = d L = d T = 0.02, p = 0.05, ν = , ω = 1.5, α = and vary the levels of q 1, q 2. The effect of complete or partial treatment is studied numerically in [11], thus we let δ = γ = 0. As a baseline case with q 1 = q 2 = 0, β is set to and , which correspond to R 0 < 1 and R 0 > 1, respectively. Initial condition is set as 0, E 0, L 0, T 0 = , 0, 0, 1 with one infectious case is introduced into the susceptible population. In Figure 2, when q 1 = q 2 varies from 0% corresponding to R 0 = < 1 to 5%, the TB epidemic attains endemic state at different slopes. The higher of q 1 = q 2, the higher levels of endemic state. This shows that constant influx of latent TB can launch a TB epidemic itself irrespective of the initial conditions.

15 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM 15 FIGURE 2: β = and q 1 = q 2 = 0%, 1%, 3%, 5%, respectively. The x-axis coincides with the curve of q 1 = q 2 = 0 and T 0 = 1, which decreases to zero due to R 0 = < 1. In Figure 3, we have chosen β = When q 1 = q 2 = 0, this corresponds to R 0 = > 1. The TB incidence initially increases exponentially and then stabilizes to the endemic level. A higher level of latent immigrants will produce a higher level of TB incidence. FIGURE 3: β = and q 1 = q 2 = 0%, 1%, 3%, 5%, respectively.

16 16 HONGBIN GUO AND MICHAEL Y. LI 6 Acknowledgements. This research is supported in part by grants the Natural Science and Engineering Research Council of Canada NSERC and Canada Foundation for Innovation CFI. Both authors acknowledge the financial support from the NCE MITACS Project Transmission Dynamics and Spatial Spread of Infectious Diseases: Modeling, Prediction and Control. We are especially grateful to an anonymous referee for pointing out a mistake in an earlier version of the manuscript. REFERENCES 1. N. Bame, S. Bowong, J. Mbang, G. Sallet and J. Tewa, Global stability analysis for SEIS models with n latent classes, Math. Biosci. Eng , R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, E. Beretta and V. Capasso, On general structure of epidemic systems: global asysmptotical stability, Comput. Math. Appl. Ser. A , S. M. Blower, A. R. Mclean, T. C. Porco, P. M. Small, P. C. Hopewell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine , S. M. Blower, P. M. Small and P. C. Hopewell, Control strategies for tuberulosis epidemics: new models for old problems, Science , F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci , C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng , Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection, Theor Popul. Biol , B. S. Goh, Global stability in many-species systems, American Naturalist , B. S. Goh, Global stability in a class of prey-predator models, Bull. Math. Bio , H. Guo, Modeling the effects of complete or incomplete treatment, a TB case study, preprint, H. Guo and M. Y. Li, Mathematical analysis of a model for latent Tuberculosis, Canad. Appl. Math. Quart , H. Guo, M. Y. Li and Z. Shuai, A graph-theoretical approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc , H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev , A. Korobinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Bio , J. P. LaSalle, The Stability of Dynamical Systems, Regional Conf. Ser. Appl. Math., SIAM, Philadelphia, M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci , C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Diff. Equ ,

17 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Bios. Eng , T. C. Porco and S. M. Blower, Quantifying the intrisic transmission dynamcis of tuberculosis, Theor. Popul. Biol , E. Ziv, C. L. Daley and S. M. Blower, Early therapy for latent tuberculosis infection, Amer. J. Epidemiol , Public Health Agency of Canada, Tuberculosis in Canada 2004, phac-aspc.gc.ca/publicat/tbcan02/index.html 23. Health Protection Agency, UK, Annual surveillance report, recent trends in tuberculosis: in UK, Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3 Canada address: hguo@mathstat.yorku.ca address: hongbin.guo@gmail.com Department of Mathematical and Statistical Sciences University of Alberta, Edmonton, Alberta, T6G 2G1 Canada address: mli@math.ualberta.ca

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