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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 293747, 31 pages
Stability of a Two-Strain Tuberculosis Model with General Contact Rate
1Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
2Department of Pediatrics, First Hospital of Lanzhou University, Lanzhou, Gansu 730000, China
Received 1 November 2010; Accepted 31 December 2010
Academic Editor: D. Anderson
Copyright © 2010 Hai-Feng Huo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A two-strain tuberculosis model with general contact rate which allows tuberculosis patients with the drug-sensitive Mycobacterium tuberculosis strain to be treated is presented. The model includes both drug-sensitive and drug-resistant strains. A detailed qualitative analysis about positivity, boundedness, existence, uniqueness and global stability of the equilibria of this model is carried out. Analytical results of the model show that the quantities and , which represent the basic reproduction numbers of the sensitive and resistant strains, respectively, provide the threshold conditions which determine the competitive outcomes of the two strains. Numerical simulations are also conducted to confirm and extend the analytic results.
Tuberculosis (TB), an infectious disease caused by Mycobacterium tuberculosis (M. tuberculosis), is one of the world's leading causes of infectious mortality. According to the World Health Organization(WHO), one third of the world's population is infected with M. tuberculosis, leading to between two and three millions death each year. At present, about 95% of the estimated 8 million new cases of TB occurring each year are in developing countries, where 80% occur among people between the ages of 15 and 59 years . For the time being, TB is becoming a world-wide problem. War, famine, homelessness, and a lack of medical care all contribute to the increasing incidence of tuberculosis among disadvantaged persons. Since TB is easily transmissible between persons, the increase in TB in any segment of the population represents a threat to all segments of the population. Sub-Saharan Africa remains the epicenter of the epidemic, but India, China, Indonesia, Bangladesh and Pakistan together account for more than half of the cases in the world . TB was assumed to be on its way out in developed countries until the number of TB cases began to increase in the 1980s. With this return, we face the paradox of a well-known bacteria, fully treatable with efficient and affordable drugs according to internationally recommended guidelines, which yet causes increasing human suffering and death. Active TB cases may be pulmonary or extrapulmonary, but pulmonary cases are more infectious and form the bulk of most cases of active TB. The usual symptoms of active TB include fatigue, high fever, night sweats, loss of appetite, and a cough, but confirmation of active TB requires a positive sputum culture. Extrapulmonary accounts for between 5% and 30% of the total cases and may affect any part of the body. M. tuberculosis droplets are released into the air by sneezing and coughing of infectious individuals. Tubercle bacillus carried by such droplets live in the air for a short period of time ; it is believed that occasional contacts with an infectious individual rarely lead to transmission. TB is described as a slow disease because of its long and variable latency period distribution and its short and relatively narrow infectious period distribution. Individuals who are latently infected are neither clinically ill nor capable of transmitting TB . Most latently infected individuals do not become infectious (active TB). About 5%–10% of the latently infected individuals develop active TB, that is, about 90%–95% remain latently infected.
Antituberculosis drugs are a two-edged sword. While they destroy pathogenic M. tuberculosis, they also select for drug-resistant bacteria against with those drugs which are then ineffective. Global surveillance has shown that drug-resistant TB is widespread and is now a threat to TB control programs in many countries. The WHO distinguishes between two types of resistance: acquired resistance—resistance among previously treated patients; and primary resistance—resistance among new cases. In all regions studied, prevalence of acquired resistance is higher than prevalence of primary resistance, but the size of this difference varies between regions . Treatment of TB consists of a combination of different drugs to avoid acquisition of resistance. Despite these precautions, drug resistance continues to emerge being favoured by the long duration of treatment and improper use of the antibiotics. Drug resistant TB has higher rates of treatment failure and longer periods of infectiousness in part due to the time lapse between TB diagnosis and obtaining drug-sensitivity test results . In general, tuberculosis can be treated with antibiotics. However, most worrisome is resistance to the two first-line drugs, isoniazid and rifampicin, defined as multidrug resistance(MDR)-TB, which is an emerging problem of great importance to public health, with higher mortality rates than drug-sensitive TB, particularly in immunocompromised patients. MDR-TB patients require treatment with more toxic second line drugs and remain infectious for longer than patients infected with drug-sensitive strains, incurring higher costs due to prolonged hospitalization.
Mathematical models can provide useful tools to analyze the spread and control of infectious diseases . Different mathematical models for TB have been formulated and studied [3, 5, 8–16]. In most of the epidemics models for TB 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 . Even more important, several laboratory studies have found that the mass action incidence rate is inadequate for describing pathogen transmission and standard incidence rate is considered, for human disease, more accurate than mass action incidence rate . Moreover, epidemic models with nonlinear incidence rate have recently attracted a great deal of attention from mathematical models [18–22]. In fact, the incidence of a disease which is the number of new cases per unit time plays an important role in the study of mathematical epidemiology. Thieme and Castillo-Chavez  argued that the general form of a population-size-dependent incidence should be written as , where and are the numbers of susceptible and infective at time , respectively, is the probability of transmitting the infection between two individuals taking part in a contact per unit time, is the probability for an individual to take part in a contact and is the size of the total population. In , Zhang and Ma think the above incidence frequently takes two forms in most of the literature. When , the corresponding incidence is the bilinear form . When , the corresponding incidence is called standard form. When the total population size is not quite large, since the number of contacts made by an individual per unit time should increase as the total population size increases, the bilinear form would be suitable, but when the total population size is too large, since the number of contacts made by an infective per unit time should be limited, or should grow less rapidly as the total population size increases, the bilinear form is not suitable and the standard form may be more realistic . Therefore, the two forms of incidence mentioned above are actually two extreme cases for the total population size being very small and very large, respectively. Moreover, Heesterbeek and Metz  derived the expression for the saturating contact rate of individuals contacts in a population that mixes randomly, that is, where is nondecreasing and is nonincreasing.
In Bowong and Tewa , a SEI type of tuberculosis model with a general contact rate is proposed to study the global asymptotically stability of disease-free equilibrium and endemic equilibrium. The stability of equilibria is derived through the use of Lyapunov stability theory and LaSalle's invariant set theorem. Nevertheless, the effect of drug-resistant strain of tuberculosis is not taken into account in their article. In this paper, we study a tuberculosis model with general contact rate that includes explicitly both drug-sensitive strain and drug-resistant strain. We adapt the approach of Bowong and Tewa  for modeling the effective chemoprophylaxis (given to latently infected individuals) and therapeutic treatments (given to infectious). We extend their model by including the latently infected with resistant strain class and infectious with resistant strain class. The new model allows us to examine the effects of drug treatment on the prevalence of both drug-sensitive and drug-resistant strains. Mathematical properties of the model system are studied both analytically and numerically. It is shown that the system has three possible equilibrium points including an endemic equilibrium at which both strains are present. A detailed analysis of global asymptotically stability is conducted, which shows that the dynamic behaviors of the system are determined by two quantities, and , which represent the basic reproduction numbers of the sensitive and resistant strains. Without regard to disease-induced death rate, we prove that the disease-free equilibrium is globally asymptotically stable when , , the unique drug-resistant-TB-strain-only equilibrium exists and is globally asymptotically stable when . As for the unique interior equilibrium, existence, uniqueness and global stability can be proved under certain conditions.
The paper is organized as follows. In Section 2, we formulate a two-strain tuberculosis model with general contact rate. The threshold conditions for the existence and uniqueness equilibria are derived and the global asymptotically stability of equilibria are proved in Sections 3 and 4 is devoted to numerical simulations. In Section 5, we summarize the findings and conclusions.
2. Model Description
According to the transmitted features of TB, we subdivide the population into susceptible individuals , those exposed to drug-sensitive TB , individuals with symptoms of TB and drug-sensitive , those exposed to drug-resistant TB and those displaying symptoms of TB and drug-resistant , the total number of population at time is given by The model is represented by the transfer diagram in Figure 1. Here, is the recruitment rate, is nondisease related death rate. Since exposed individuals are not capable of transmitting the disease, we assume that susceptible may become infected only through contacts with active infectious individual at a rate , , where and represent rates of infection by drug-sensitive strain and drug-resistant strain, and the transmission coefficient is a nonnegative function of the total population . Susceptible are infected with drug-sensitive strain and drug-resistant strain entering classes and , respectively. For simplicity, we only account for treatment of sensitive strain, we assume that chemoprophylaxis of individuals in reduces their reactivation at a constant rate and that the initiation of therapeutics immediately removes individuals in from active status and places them into exposed class at a constant rate . The time before individuals in who does not received effective chemoprophylaxis become infectious is assumed to satisfy an exponential distribution, with mean waiting time . Thus, individuals leave the class to the class at a constant rate . The time before individuals in become infectious is also assumed to satisfy an exponential distribution, with mean waiting time . Thus, individuals leave the class to the class at a constant rate . Individuals sick with drug-sensitive strain receive treatment at rate and a proportion respond to treatment and move into exposed class , in the remaining proportion , inappropriate treatment results in the development of drug-resistant strain and the individuals move into class . Individuals in the infectious class and suffer additional disease-induced death at rate and , where . Then the following two strains tuberculosis model is formulated where , , , , and are assumed to be positive and . It is natural to assume that the transmission coefficient satisfies the following conditions:
Obviously, we can see that , corresponds to the standard incidence rate, that , corresponds to the mass action incidence rate, and that , corresponds to the saturating contact rate, where
2.1. Positivity and Boundedness
System (2.2) describes human population and therefore it is necessary to prove that all the variables , , , , are nonnegative for all time. Solutions of system (2.2) with positive initial data remains positive for all time and are bounded.
Theorem 2.1. If , , , , , the solutions , , , , of system (2.2) are positive for . For system (2.2), the region is positively invariant and all solutions starting in approach, enter, or stay in , where
Proof. Under the given initial conditions, it is easy to prove that the solutions of the system (2.2) are positive; if not, we assume a contradiction: that there exists a first time such that
there exists a
there exists a
there exists a
or there exists a
In the first case, we have
which is a contradiction meaning that , . In the second case, we have
which is a contradiction meaning that , . In the third case, we have
which is a contradiction meaning that , . In the fourth case, we have
which is a contradiction meaning that , . In the fifth case, we have
which is a contradiction meaning that , .
Thus, in all cases, , , , , remain positive for .
Let be any solution with nonnegative initial condition, adding all equations in system (2.2) gives It follows that where represents initial values of the total population. Thus , as . Therefore all feasible solutions of system (2.2) enter the region Hence, is positively invariant and it is sufficient to consider solutions of system (2.2) in . Existence, uniqueness and continuation results for system (2.2) hold in this region. It can be shown that is bounded and all the solutions starting in approach, enter or stay in .
3. Model Analysis
There are one disease-free equilibrium and two possible endemic equilibria for system (2.2), the endemic equilibria include boundary equilibrium (when only the drug-resistant TB-strain is present) and the interior equilibrium (when both strains exist).
3.1. Global Stability of the Disease-Free Equilibrium
The disease-free-equilibrium is given as The basic reproduction number is defined as the number of secondary infections produced by a single infectious individual during the entire infectious period. In our case the reproduction number defines the number of secondary TB infections produced by a single active TB individual during the entire infectious period. Mathematically it is defined as the spectral radius of the next generation matrix . To determine the stability of the disease-free steady state , we use the results in van den Driessche and Watmough . Reorder the components of as , , , , . Set The infected compartments are , , , and . Thus The dominant eigenvalues of are given by where and are reproduction numbers for drug-sensitive TB strain only and drug-resistant TB strain only, respectively. It implies that the spectral radius of the matrix is If , and , then . By Theorem 3.2 in van den Driessche and Watmough , we know that the disease-free steady state is locally asymptotically stable, is unstable if or . The following theorems provide the global stability of the disease-free equilibrium.
To conduct an analytical analysis of global asymptotical behaviors of the disease-free equilibrium point, first of all, we assume that there is no disease-induced death rate, that is, , . Consequently, the total population size satisfies the equation and as . Using results from Castillo-Chavez and Thieme  and Mischaikow et al. , we can obtain analytical results by considering the following limiting system of (2.2) in which the total population is assumed to be constant : Notice that the equation is eliminated from (3.6) and the variable is replaced by . By introducing the fractions we obtain the equivalent limiting system from (3.6) Obviously for all time .
For a bounded real-valued function on , we define
Lemma 3.1 (see ). Let be bounded and continuously differentiable. Then there are sequences with the following properties: for .
Theorem 3.2. In the absence of disease-induced death rate, the disease-free equilibrium of system (2.2) is globally asymptotically stable if , .
Proof. By Lemma 3.1 and the , equations in (3.8) we have
Using the equation in (3.8) and choosing such that we get It is shown that since , as , we have that . The inequalities in (3.12) also imply that .
Using the equation in (3.8) and choosing such that we get It is shown that since , as , we have that . The inequalities in (3.12) also imply that .
Hence Therefore, the disease-free equilibrium is globally asymptotically stable.
Next, we consider that there is disease-induced death rate, which means that , . Then, we can get the following theorem.
Theorem 3.3. For , , the disease-free equilibrium of system (2.2) is globally asymptotically stable if , .
Proof. Let us consider the following Lyapunov function: Its time derivative along the solutions of system (2.2) satisfies Now, using (2.3), one has , , then Since and , we have Thus, if , . It is obvious that if and only if , . Then, the largest invariant set of system (2.2) on the set , is the disease-free equilibrium . Therefore, it follows from LaSalle's invariance principle [31, 32] that is globally stable if , .
3.2. Existence and Uniqueness of the Drug-Resistant-TB-Strain-Only Equilibrium
Here, we present a result concerning the existence and uniqueness of the drug-resistant-TB-strain-only equilibrium for the system (2.2). To do this, we shall make use of the basic reproduction ratio and .
Let be boundary equilibrium of system (2.2). Then, the boundary equilibrium can be obtained by setting the right-hand side of each differential equations in system (2.2) equal to zero with the exception of second and third equation, giving adding the above three equations, we have
Substituting , , in the third equations of (3.23) yields where
Clearly, is a fixed point of (3.23), which corresponds to the disease-free equilibrium . Since , we get
It appears that and when . The existence of fixed point follows from the intermediate value-theorem. Now, is monotone increasing, so that has only one positive root in the interval .
Thus, we have established the following result
Lemma 3.4. When, the system (2.2) has a unique drug-resistant-TB-strain-only equilibrium with , , and all nonnegative.
3.3. Global Stability of the Drug-Resistant-TB-Strain-Only Equilibrium
Theorem 3.5. If , the drug-resistant-TB-strain-only equilibrium exists and is locally asymptotically stable.
Proof. Reorder the equilibrium as . Similarly as in the proof of locally stability of the disease-free steady state , we have where The spectral radius of is given by Since , and , then Thus As , we have , which implies that the drug-resistant-TB-strain-only equilibrium is locally asymptotically stable by Theorem 3.2 in van den Driessche and Watmough .
The following theorem provides the global stability of drug-resistant-TB-strain-only equilibrium.
Theorem 3.6. If there is no disease-induced death rate, the drug-resistant-TB-strain-only equilibrium is global asymptotically stable when .
Proof. If , from Theorem 3.5, is locally asymptotically stable. In the following, we only need to prove that the is a global attractor.
There is no disease-induced death rate, which means , using a similar argument as in the proof of Theorem 3.2, we can show that if , Then the equivalent limiting system of (3.6) is
Let be the vector field defined by system (3.35). Then for the Dulac function , there holds
From Dulac's criterion, we can see equivalent limiting system (3.35) does not have a limit cycle. Therefore, the local stability of implies the global stability in Int . This completes the proof of Theorem 3.6.
3.4. Existence and Uniqueness of the Interior Equilibrium
First of all, we assume that(W1), (W2),
then, existence, uniqueness and global stability of interior equilibrium can be obtained. Let be interior equilibrium of system (2.2). Then, the interior equilibrium (steady state with , ) can be obtained by setting the right-hand side of each of the five differential equations in system (2.2) equal to zero, giving adding the above five equations, we have
Substituting (3.40) into , satisfies the following equation: where Since , , if , then It means that has only one positive root in the interval if is monotone increasing and , which is equivalent to where Let When it shows that , and is monotone increasing, by the intermediate value-theorem, has only one positive root in the interval , which signify that is unique. Obviously, . In order to make certain that , , , , we obtain that , , , if and only if Due to , introducing (3.40) into (3.48), we acquire where and .
Now, we can get the following conclusion.
3.5. Global Stability of the Interior Equilibrium
Herein, we study the global stability of the interior equilibrium of system (2.2). For simplicity, we only consider the special case when the transmission coefficients of drug-sensitive TB-strain and drug-resistant TB-strain are equal, that is, , then, we get the following result.
Proof. Following , we consider the Lyapunov function: where and are positive constants to be determined later. Differentiating this function with respect to time along the solutions of system (2.2) yields Considering (3.37), we can deduce that Then, (3.53) becomes Now, using (3.37), we have Then, (3.55) can be rewritten as follows: Now, let and , then we get where We can choose For , we have which is less than or equal to zero by the arithmetic-geometric mean inequality, with the following equality if and only if