GLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS

Transcription

1 CANADIAN APPIED MATHEMATICS QUARTERY Volume 13, Number 4, Winter 2005 GOBA DYNAMICS OF A MATHEMATICA MODE OF TUBERCUOSIS HONGBIN GUO ABSTRACT. Mathematical analysis is carried out for a mathematical model of Tuberculosis (TB that incorporates direct progression and latent reactivation. Our analysis establishes that the global dynamics of the model are completely determined by a basic reproduction number R 0. If R 0 1, the TB always dies out. If R 0 > 1, the TB becomes endemic, and a unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region. 1 Introduction Tuberculosis (TB is an ancient disease caused by the infection of bacterium Mycobacterium tuberculosis. Once thought under control using antibiotic therapies, TB made a dramatic come back in the late eighties and early nineties, largely due to the emergence of antibiotic resistant strains and to co-infection with the HIV. Currently, the global per capita incidence rate of TB is growing at approximately 1.1% per year, and the number of cases at 2.4% per year. According to the 2004 WHO report Global Tuberculosis Control [1], there were 8.8 million new cases of TB worldwide in 2002, with close to 2 million TB-related deaths, more than any other infectious diseases. TB remains as one of the most serious health problems facing the world today. Mathematical models have been used to improve our understanding of the basic transmission dynamics of the TB and to evaluate the effectiveness of various control and prevention strategies [2 11]. 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, asymptomatic and noninfectious period during which the body s immune system fights the TB bacteria. We assume that there are two distinct pathogenic mechanisms of the TB infection. One is direct progression or primary progressive Keywords: Tuberculosis(TB, basic reproduction number, endemic equilibrium, global stability, yapunov functions. Copyright c Applied Mathematics Institute, University of Alberta. 313

2 314 HONGBIN GUO TB that the disease develops soon after infection. Another is endogenous reactivation or slow TB that the disease can develop many years after infection. Using a compartmental approach, the total host population can be partitioned into three compartments: susceptible individuals (, latently infected individuals ( and individuals with active TB disease (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 β(tt (t. Here (t, (t, and T (t denote the number of individuals in the three corresponding compartments at time t. Once infected, a fraction p, 0 p 1, of the newly infected individuals develop tuberculosis directly and the remaining 1 p fraction of the newly infected progresses to the latent class. Once there, the rate of progression to active disease is at a lower rate ν. Recruitment to the susceptible population occurs at a constant rate π and removal rates for the three compartments are µ, µ and µ T, respectively. Here removal rate may include natural death, death due to TB. The dynamical transfer among the three compartments is depicted in the following transfer diagram. Here all parameters are assumed to be positive. FIGURE 1: The transfer diagram for model (1. Based on our assumptions and the transfer diagram, the model can be described by three ordinary differential equations as follows: (1 = π βt µ, = (1 pβt (ν + µ, T = pβt + ν µ T T. The model contains an earlier model proposed by Blower et al. [3] to discuss effectiveness of treating TB patients at the early infection stage. A basic reproduction number R 0 is derived in [3], (2 R 0 = β(pµ + νπ µ T (µ + νµ,

3 A MATHEMATICA MODE OF TUBERCUOSIS 315 based on which quantitative analysis was carried out. The parameter R 0 measures the average number of infections caused by one infectious individual throughout the infectious period when introduced into an entirely susceptible population. It is expected that if R 0 < 1, then no TB epidemic can develop in the population, and if R 0 > 1, a TB epidemic can develop and become endemic in the population. In the present paper, we give a rigorous mathematical analysis of model (1, and prove that the global dynamics of the model are completely determined by the parameter R 0 in (2. More specifically, we prove that if R 0 1, then the disease-free equilibrium P 0 = (π/µ, 0, 0 is globally stable in the feasible region; if R 0 > 1, P 0 is unstable, and a unique endemic equilibrium P = (,, T with,, T > 0 exists and is asymptotically stable. Furthermore, all solutions in the interior of the feasible region converge to P. In particular, our results establish that the R 0 in (2 as derived in [3] is a sharp threshold parameter for the global dynamics of (1. In the next section, we discuss the feasible region of the model and its equilibria. The global dynamics when R 0 1 are established in Section 3, and the global results when R 0 > 1 are given in Section 4. 2 Feasible region and equilibria of the system From (1 we have π µ, and thus lim sup t (t π/µ along each solution to (1. et N(t = (t + (t + T (t. Then using (1 we have N = π µ µ µ T T π µn, where µ = min{µ, µ, µ T }. This implies that lim sup t N(t π/µ. Therefore the model can be studied in the feasible region (3 Γ = { (,, T R 3 + : 0 π, T π }, µ µ where R 3 + denotes the non-negative cone of R3 including its lower dimensional faces. It can be verified that Γ is positively invariant with respect to (1. We denote by Γ and Int Γ the closure and the interior of Γ in R 3 +, respectively.

4 316 HONGBIN GUO An equilibrium (,, T of (1 satisfies the following equations (4 π = βt + µ, (1 pβt = (ν + µ, pβt + ν = µ T T. Simplifying these equations we obtain Therefore, [ ] ν(1 pβ µ + ν + pβ µ T T = 0. either T = 0 or = µ T (µ + ν β(pµ + ν. Correspondingly, system (1 has two possible equilibria: the diseasefree equilibrium P 0 = (π/µ, 0, 0 and the endemic equilibrium P = (,, T where (5 = µ T (µ + ν β(pµ + ν. From (2, the basic reproduction number R 0 satisfies R 0 = π µ. This relation can be used to estimate the value of R 0 from data on the susceptible fraction /(π/µ. Using R 0 we have the following expressions for the coordinates of P : (6 = π µ R 0, = (1 pπ (1 1R0, T = µ µ + ν β (R 0 1. It follows from (6 that P exists in IntΓ only when R 0 > 1. The following result is immediate. Proposition 1. System (1 has two possible equilibria. When R 0 1, the disease-free P 0 = (π/µ, 0, 0 is the only equilibrium in Γ; when R 0 > 1, both P 0 and the unique endemic equilibrium P = (,, T exist in Γ, where, and T are given in (6.

5 A MATHEMATICA MODE OF TUBERCUOSIS 317 For epidemic models of this type, it is generally expected that the global dynamics are determined by the basic reproduction number R 0 : if R 0 1, then all solutions converge to the disease-free equilibrium P 0, and the TB dies out from the population irrespective of the initial incidence; while if R 0 > 1, all solutions with positive initial conditions will be persistent and converge to the unique endemic equilibrium P, and any initial TB epidemics will become endemic in the population. In the next two sections, we rigorously establish this threshold behaviour. 3 Stability of the disease-free equilibrium P 0 In this section, we show that the disease-free equilibrium P 0 is globally asymptotically stable with respect to Γ if R 0 1, and P 0 is unstable if R 0 > 1. Theorem 2. If R 0 1, then the disease-free equilibrium P 0 is locally asymptotically stable and all solutions in Γ converge to P 0. If R 0 > 1, then P 0 is unstable. Proof. Consider a yapunov function Direct calculation leads to W = ν + (ν + µ T W = ν + (ν + µ T. = ν(1 pβt + (ν + µ pβt (ν + µ µ T T ( = β(pµ + ν π T. µ R 0 Therefore W 0 in Γ if R 0 1. Furthermore, W = 0 T = 0 or = π µ R 0. Therefore, the largest compact invariant set in G = {(,, T Γ : W = 0}, when R 0 1, is the singleton {P 0 }. asalle s Invariance Principle ([12], Chapter 2, Theorem 6.4 implies that all solutions in Γ converge to P 0. This global convergence also implies that P 0 is locally stable, since otherwise P 0 will have a homoclinic orbit that has to belong entirely to the set G Γ where W = 0, and thus contradicting the fact that the largest compact invariant set in G is the singleton {P 0 }.

6 318 HONGBIN GUO If R 0 > 1, then W > 0 at (,, T Int Γ if is sufficiently close to π/µ, except when T = 0. Solutions in Γ starting sufficiently close to P 0 leave a neighborhood of P 0 except those on the invariant -axis, on which (1 reduces to = π µ and thus (t π/µ, as t. This establishes the theorem. By Theorem 2, the disease-free equilibrium point P 0 is unstable when R 0 > 1. Moreover, the local dynamics near P 0 imply that system (1 is uniformly persistent with respect to R 3 + if R 0 > 1. Namely, there exists constant c > 0 such that lim inf t (t > c, lim inf t (t > c, lim inf t T (t > c, provided ((0, (0, T (0 R 3 +. Here the constant c is independent of initial data in R 3 +. We thus have the following corollary, whose proof is similar to that of Proposition 3.3 of [13]. Corollary 3. System (1 is uniformly persistent if and only if R 0 > 1. 4 Stability of the endemic equilibrium P when R 0 > 1 We have shown in the previous section that system (1 is uniformly persistent if and only if R 0 > 1. In this section, we further establish that all solutions in the interior of the feasible region Γ converge to the unique endemic equilibrium P if R 0 > 1. Therefore, the TB will persist at the endemic equilibrium level. The proof is accomplished by constructing a global yapunov function. yapunov functions of similar type have been used in the literature, see [14 16]. Theorem 4. Assume R 0 > 1. Then the endemic equilibrium P = (,, T is asymptotically stable. Furthermore, all solutions in the interior of Γ converge to P. Proof. Set z = (,, T Γ R 3 +. Consider a yapunov function V = V (z = ( ln + b ( ln + c (T T T ln TT, where z = P = (,, T is the endemic equilibrium and (7 b = ν pµ + ν = βν µ T (µ + ν, c = µ + ν pµ + ν = β µ T.

7 A MATHEMATICA MODE OF TUBERCUOSIS 319 We note that V (z 0, for z Int Γ, the interior of Γ, and V (z = 0 z = z. So the function V is positive definite with respect to the endemic equilibrium z = P. Computing the derivative of V along the solution of system (1, we obtain (8 dv dt = (1 + b (1 + c (1 T Using (1 and π = µ + β T from (4, we have T T. (9 (1 = π βt µ π + β T + µ = 2µ + β T βt µ (10 µ 2 β 2 T = β T + µ ( 2 + β T βt β 2 T + β T. Similarly, b (1 = b(1 pβt b(ν + µ Define b(1 pβt + b(µ + ν, c (1 T T = cpβt + cν cµ T T T (11 q = cpβt cνt T (1 pν pµ + ν, r = p(µ + ν pµ + ν. + cµ T T. Then q + r = 1, q > 0, r > 0. It follows from b(1 p + cp = q + r = 1 and (7 (12 cµ T = β, b(µ + ν = cν.

8 320 HONGBIN GUO Using (9 (12 we can simplify (8 as (13 dv dt = β T + µ (2 From (7, (4 and (5 we have β 2 T cνt T qβt rβt + b(µ + ν + cµ T T. (14 b(µ + ν = qβ T, cµ T T = β T. Using (14, we obtain (15 and qβt = q 2 b(µ + ν βt β T (16 rβt = rβ cµ T β T, cνt T = ν µ T T β T. Substitute (14, (15 and (16 into (13, then we get (17 dv dt = (2 + q β T + µ (2 β 2 T q 2 βt b(µ + ν β T rβ cµ T β T ν µ T T β T ( = µ 2 + β T ((2 + q q 2 βt b(µ + ν rβ ν. cµ T µ T T Since 2 + q = 2(q + r + q = 3q + 2r,

9 A MATHEMATICA MODE OF TUBERCUOSIS 321 we can rewrite (17 as (18 dv dt = µ (2 + β T (2r r ( + β T 3q q q 2 βt b(µ + ν. = I 1 + I 2 + I 3. Applying the inequality rβ cµ T ν µ T T a 1 + a a n n n a 1 a 2 a n, for a i 0, i = 1,, n, we obtain ( (19 I 1 = µ 2 0. Moreover, (20 ( I 2 = β T 2r r rβ cµ T β T (2r 2 r rβ cµ T = β T (2r β 2r = 0, cµ T by the definition of c in (7. Similarly (21 ( I 3 = β T 3q q q 2 βt b(µ + ν ( β T 3q 3 3 q q 2 βt b(µ + ν ( = β T βν 3q 3q 3 = 0, bµ T (µ + ν ν µ T T ν µ T T

10 322 HONGBIN GUO by the definition of b in (7. Using (19 (21 we obtain (22 dv dt = I 1 + I 2 + I 3 0, z Int Γ. Furthermore, dv /dt = 0 if and only if equalities hold in (19 (21, if and only if z = z = P. Therefore dv /dt is negative definite in Int Γ with respect to the endemic equilibrium P. This implies that the basin of attraction of P contains Int Γ. The positive definiteness of V (z with respect to P implies that P is also locally stable. This completes the proof. 5 Summary In this paper, mathematical analysis is carried out for a TB model. Global dynamics of the model are shown to be completely determined by a basic reproduction number R 0, first derived in [3]. More specifically, we have proved that if R 0 1, then the disease-free equilibrium P 0 is asymptotically stable and all solutions in the feasible region converge to P 0. If R 0 > 1, then P 0 becomes unstable, and a unique endemic equilibrium P exists and is asymptotically stable. In this case, all solutions in the interior of the feasible region converge to P. The proof of global convergence uses the method of yapunov functions. 6 Acknowledgement I wish to thank Professor Michael Y. i for direction on this research. REFERENCES 1. Global Tuberculosis Control, WHO Report, R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, Oxford, 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 1 (1995, S. M. Blower, P. M. Small, P. C. Hopewell, Control strategies for tuberculosis epidemics: new models for old problems, Science 273 (1996, C. Castillo-Chavez and Z. Feng, To treat or not to treat: the case of tuberculosis, J. Math. Biol. 35 (1997, C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci. 151 (1998, Z. Feng, C. Castillo-Chavez, and A. F. Capurro, A model for tuberculosis with exogenous reinfection, Theor. Popul. Biol. 57 (2000,

11 A MATHEMATICA MODE OF TUBERCUOSIS C. J.. Murray and J. A. Salomon, Modelling the impact of global tuberculosis control strategies, Proc. Natl. Acad. Sci. 95 (1998, T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis, Theor. Popul, Biol. 54 (1998, H. Waaler, A. Geser, and S. Andersen, The use of mathematical models in the study of the epidemiology of tuberculosis, Am. J. Public Health 52 (1962, Elad Ziv, Charles. Daley, and Sally M. Blower, Early therapy for latent tuberculosis infection, Amer. J. Epidemiol. 153 (2001, J. P. asalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, M. Y. i, J. R. Graef,. Wang, and J. Karsai, Global dynamics of a SEIR model with a varying total population size, Math. Biosci. 160 (1999, H. I. Freedman and J. W. -H. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985, Z. Ma, J. iu, and J. i, Stability analysis for differential infectivity epidemic models, Nonlinear Anal. 4 (2003, A. Korobeinikov and P. K. Maini, A yapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng. 1 (2004, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 address: hguo@math.ualberta.ca

12