Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate

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Abstract

This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with a general contact rate. The model exhibits the traditional threshold behavior. We prove that when the basic reproduction ratio is less than unity, then the disease-free equilibrium is globally asymptotically stable and when the basic reproduction ratio is great than unity, a unique endemic equilibrium exists and is globally asymptotically stable under certain conditions. The stability of equilibria is derived through the use of Lyapunov stability theory and LaSalle’s invariant set theorem. Numerical simulations are provided to illustrate the theoretical results.

Introduction

Tuberculosis (TB), one of the most wide spread infectious diseases, is the leading cause of death due to a single infectious agent among adults in the world. According to the World Health Organization, one third of the world’s population is infected with Mycobacterium tuberculosis (M. tuberculosis), leading to between two and three millions death each year. Although between 90% and 95% of infections occur in developing countries [1], emergence of HIV as well as multi-drug-resistant (MDR) strains of M. tuberculosis will dramatically change the dynamics of infection world-wide [2]. Other factors may contribute to the TB epidemic including elimination of TB control programs, drug use, poverty, and immigration [3], [4]. Humans are the natural reservoir for M. tuberculosis, which is spread from person to person via airborne droplets [5]. M. tuberculosis may need only a low infectious dose to establish infection [6]. Factors that affect transmission of M. tuberculosis include the number, viability, and virulence of organisms within sputum droplet nuclei, and most importantly, time spent in close contact with an infectious person [5], [6], [7], [8]. Socio-economic status, family size, crowding, malnutrition, and limited access to health care or effective treatment also influence transmission [9], [10]. Consistent estimates of M. tuberculosis transmission rates do not exist; however, it is known that transmission is rather inefficient for most strains [11]. Infection with M. tuberculosis is dependent on nonlinear contact processes that are determined by population size and density, as well as other factors. Demographic characteristics of a population, therefore, play a significant role in the development and progression of a TB epidemic.

Mathematical models can provide a useful tool to analyze the spread and control of infectious diseases [14], [15]. Mathematical models for tuberculosis are especially useful tools in assessing the epidemiological consequences of medical or behavioral interventions (which may cause many direct and indirect effects) because they contain explicit mechanisms that link individuals with a population-level outcome such as incidence or prevalence. Different mathematical models for tuberculosis have been formulated and studied (see e.g. [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] and references therein). The simplest models include classes of susceptible, exposed and infective individuals, and hence are known as SEI models. However, global stability properties of a nonlinear system are generally a difficult problem. The global stability of SIR or SIRS models, in relation with the basic reproduction ratio, is known since eighties. Since the stability of these systems can be reduced to the study of a two-dimensional system, the Poincaré–Bendixson criterion is used to establish the global stability. Global stability for SEIRS and SEIS has long been conjectured. If the global stability of the disease-free equilibrium was known when the basic reproduction number is less than one, on the other hand the global stability for the endemic equilibrium, when the basic reproduction number is great than one, was an open problem. This was solved in 1995 by Li and Muldowney [28] using the Poincaré–Bendixson properties of competitive systems in dimensions three combined with sophisticated use of compound matrices.

This study extends previous works by formulating, and rigorously analyzing, a deterministic model for tuberculosis transmission dynamics with a general contact rate that incorporate constant recruitment, slow and fast progression, effective chemoprophylaxis (given to latently infected individuals) and therapeutic treatments (given to infectious). To the best of author knowledge, the global analysis of tuberculosis models with general contact rate is not well discussed in the literature. We completely analyze the stability behavior of the model. We compute the basic reproduction ratio R0, and prove the global asymptotic stability of the disease-free equilibrium (DFE) when R01, and that when R0>1 a unique endemic equilibrium exists and is globally asymptotically stable on the non-negative orthant minus the DFE under certain conditions. The global dynamics of the model is resolved through the use of Lyapunov functions which are the same from as those used recently in Refs. [30], [31], [32], [33], [34], [35], [36], [37] to determine the global dynamics of SEIR, SEIS, and SIR models. It should be pointed out that this kind of Lyapunov function has long history of applications to Lotka–Volterra models and was originally discovered by Volterra himself, although he did not use the vocabulary and the theory of Lyapunov functions.

The paper is organized in the following manner. In the next section, we present our motivations. A brief introduction to the epidemiology of TB is provided in Section 3. We formulate a transmission model with a general contact rate to study the dynamics of TB in as simple a setting as possible in Section 4. We use the well-known TB model [20], which, in our opinion, captures the essentials of Mycobacterium tuberculosis transmission. Numerical simulation are presented to illustrate analytical results. Finally, Section 5 contains the conclusion.

Section snippets

Motivation

In most of the models discussed in the literature, the question of contact rate has not been a central one. Nevertheless, the mode of transmission is crucially important for two reasons. First, it determines the probable response of the disease to control. Second, the objective in many models of disease in animals is to predict what will happen when a pathogen is introduced into a system in which it does not currently exist. For example, standard mass action is considered, for human disease,

Epidemiology of tuberculosis

TB was assumed to be on its way ‘out’ in developed countries until the number of TB cases began to increase in the late 1980s. The causes behind recent observed increases of active TB cases are the source of many studies (see e.g., [5], [6], [7], [8], [9], [10], [11], [12] and references therein). TB is an airborne transmitted disease. Mycobacterium tuberculosis droplets are released in the air by coughing or sneezing infectious individuals [13]. Tubercle bacillus carried by such droplets lives

Model formulation

Based on epidemiological status, the population is divided into three classes: susceptible, latently infected (exposed) and infectious with the number in each class denoted by S, E, and I, respectively. The model is represented by the transfer diagram in Fig. 1. All recruitment is into the susceptible class, and occurs at a constant rate Λ. The rate constant for non-disease related death is μ, thus 1/μ is the average lifetime. A fraction p of the newly infected individuals is assumed to undergo

Numerical simulations

To illustrate the theoretical results contained in previous sections, system (2) is simulated with parameter values using real data of Cameroon [46], [47] and summarize in Table 1.

Numerical results are reported in Fig. 2, Fig. 7.

We first consider system (2) with β(N) = β. We choose β = 0.01 so that R01. Fig. 2 presents the trajectories plot and its plane figure for different initial conditions. From this figure, one can see that the trajectories of system (2) converge to the disease-free

Conclusion

This paper has considered a tuberculosis model that incorporate general contact rate, constant recruitment, slow and fast progression, effective chemoprophylaxis and therapeutic treatments. By using the Lyapunov stability theory, the global stability of the proposed model is completely proved. We show that the dynamics of the disease transmission model is governed by a basic reproductive ratio R0. When R01, then all solutions converge to the disease-free equilibrium, while when R0>1, then the

Acknowledgment

Samuel Bowong gratefully acknowledges, with thanks, the support in part of the Alexander von Humbold Foundation and the Postdam Institute for Climate Impact Research, Germany.

References (47)

  • Enarson D, Murray J. In: Rom W, Garay, editors. Global epidemiology of tuberculosis...
  • Adler J, Rose D. In: Rom W, Garay, editors. Transmission and pathogenesis of tuberculosis, vol. 17,...
  • C. Karus

    Tuberculosis: an overview of pathogenesis and prevention

    Nurse Pract

    (1983)
  • J. Chapman et al.

    Social and other factors in intrafamilial transmission of tuberculosis

    Am Rev Respirat Dis

    (1964)
  • E. Nardell et al.

    Transmission of tuberculosis

  • D. Enarson

    Why not the elimination of tuberculosis?

    Mayo Clin Proc

    (1994)
  • H.W. Hetcote

    The mathematics of infectious diseases

    SIAM Rev

    (2000)
  • S. Busenberg et al.

    Analysis of a disease transmission model in a population with varying size

    J Math Biol

    (1990)
  • R.M. Anderson et al.

    Infectious disease of humans, dynamics and control

    (1991)
  • H.W. Hetcote

    The mathematics of infectious diseases

    SIAM Rev

    (2000)
  • F. Brauer et al.

    Mathematical models in population: biology and epidemiology, text in applied mathematics

    (2001)
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