Abstract
A two-patch model, , , is used to analyze the spread of tuberculosis, with an arbitrary number of latently infected compartments in each patch. A fraction of infectious individuals that begun their treatment will not return to the hospital for the examination of sputum. This fact usually occurs in sub-Saharan Africa, due to many reasons. The model incorporates migrations from one patch to another. The existence and uniqueness of the associated equilibria are discussed. A Lyapunov function is used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium is globally and asymptotically stable. When it is greater than one, there exists at least one endemic equilibrium. The local stability of endemic equilibria can be illustrated using numerical simulations. Numerical simulation results are provided to illustrate the theoretical results and analyze the influence of lost sight individuals.
1. Introduction
For a given system, the focus in qualitative mathematical epidemiology is the long-term dynamics. The simplest possible attractor is a globally and asymptotically stable equilibrium. Equilibrium can be shown to be globally and asymptotically stable, using Poincaré-Bendixson theory [1], Bendixson’s negative criterion [2, 3], or the generalized version of Dulac [4]. Another method of Li and Muldowney [5–7] for demonstrating global stability in dimensions has been developed more recently, with applications in three [8, 9] and four dimensions [10, 11]. For higher-dimensional systems, the theory of quadratic forms [12] or Lyapunov’s method can be used [13, 14]. Lyapunov’s method requires finding a function such that the flow always crosses the level sets from higher values of to lower values. When such a function can be found, then any isolated minimum of the function is a stable equilibrium of the flow. In this paper, the stability of a -dimensions system is investigated using Lyapunov-LaSalle functions and quadratic forms. The function has a long story in epidemiology [15–20]. Volterra himself originally discovered this function, although he did not use the vocabulary and the theory of Lyapunov functions. Since epidemic models are Lotka-Volterra-like models, the pertinence of this function is not surprising.
The issue of modeling tuberculosis motivates the model studied in this paper. It is an extension with two patches and latent classes of the SEIL model in [21], where denotes the lost sight individuals. These lost sight individuals usually occur in sub-Saharan Africa. For example, according to the Cameroonian National Program of Fight against Tuberculosis, about 10% of infectious individuals that begun their therapy treatment lost sight. Therefore, this fact cannot be neglected in the modeling of TB. By allowing for an arbitrary number of latently infected compartments, the model allows for the approximation of a wide class of distributions of latency durations. This is of particular importance for tuberculosis since latency may last for years or even decades.
In the model given here, there are migrations between people of two patches, who have the same epidemiological characteristics. We introduce a direct transfer from the class of susceptible individuals toward the compartment of infectious individuals in each patch. The reason is that there are some individuals with bad immune effector protection, due to many reasons. We also incorporate a transfer from the infectious class to the exposed class to take into account the fact that some infectious individuals apparently recover but actually harbor TB bacteria. Let us give now the outlines of the paper. In Section 2, the model is constructed; the variables and parameters of the model are explained. In Section 3, the mathematical properties of the model are given. We present the computation thresholds as basic reproduction numbers, which are bifurcation parameters , using the same method as in [22]. The equilibria of the model are computed: equilibrium without disease, endemic equilibria, or coexistence equilibria. The stability of each equilibrium is investigated, using the bifurcation parameters . In Section 4, numerical simulations are done to illustrate the results. In Section 5, we give the conclusion.
2. Model Construction
The model consisted of two patches. Each patch denotes a given population and the two populations are very much closed. Based on epidemiological status, the population of a given patch is divided into classes: susceptible , latently infected , infectious , and lost sight . All recruitments in a given patch are into the susceptible class and occur at a constant rate . The rate constant for nondisease related death is ; thus is the average lifetime. Infectious and lost sight classes of a patch have addition death rates due to the disease with rates constants and , respectively. Since we do not know if lost sight class is recovered, died, or is still infectious, we assume that a fraction of them is still infectious and can transmit disease to susceptible class. Transmission of Mycobacterium tuberculosis occurs following adequate contacts between susceptible and infectious or lost sight classes that continue to have disease. We assume that infected individuals are not infectious and thus are not capable of transmitting bacteria. We use the universal incidence expressions and to indicate successful transmission of M. tuberculosis due to nonlinear contacts dynamics in the population by infectious and lost sight classes, respectively. A fraction of the newly infected individuals is assumed to undergo fast progression directly to the infectious class, while the remainder are latently infected and enter the latent class. Once latently infected with M. tuberculosis, an individual will remain so for life unless reactivation occurs. To account for treatment, we define , with and , as the fraction of infected individuals receiving effective chemoprophylaxis, and as the rate of effective per capita therapy. We assume that chemoprophylaxis of latently infected individuals reduces their reactivation at rate . Thus, a fraction of infected individuals who do not receive effective chemoprophylaxis becomes infectious with a rate constant , so that represents the average latent period. Thus, individuals leave the class to at rate . After receiving an effective therapy, infectious individuals can spontaneously recover from the disease with a rate constant , entering the infected class . A fraction of infectious individuals that began their treatment will not return to the hospital for the examination of sputum. After some time, some of them will return to the hospital with the disease at a constant rate . This can be the situation in many African countries or refugees camps in Africa or elsewhere.
The transfer diagram of the model is given by Figure 1.
This yields the following set of differential equations: where and .
System (1) can also be written as where is the scalar usual product in , , , , , = , is the canonical basis of , the matrices and are Metzler stable [23], and and are Metzler stable and given, respectively, bywhere , , , and , .
Let us give another expression of system (1). System (2) can be written as where is now the usual scalar product in , , , , , , , , , , , and .
3. Mathematical Properties
3.1. Positivity of the Solutions
Since the variables considered here are nonnegative quantities, we have to be sure that their values are always nonnegative.
Theorem 1. The nonnegative orthant is positively invariant by (1). This means that every trajectory, which begins in the positive orthant, will stay inside.
Proof. System (4) can be written in the following form: Since and , the matrix is Metzler. It is well known that linear Metzler system lets the nonnegative orthant invariant [23]. Moreover and is a Metzler matrix. These prove the positive invariance of the nonnegative orthant by (1).
3.2. Boundedness of the Trajectories
Lemma 2. The simplex + , is a compact forward and absorbing set for (1).
Proof. From system (1), if denotes the entire population of (1), one has .
Then, the dynamics of at any time is given by
Thus, , where . It follows that .
In the simplex , (1) is mathematically well posed. The following lemma also holds.
Lemma 3. The simplex is a compact forward invariant set for (1).
3.3. Local Stability of the Disease-Free Equilibrium
Many epidemiological models have a threshold condition which can determine whether an infection will be eliminated from the population or become endemic. The basic reproduction number, , is defined as the average number of secondary infections produced by an infected individual in a completely susceptible population [16]. As discussed in [24, 25], is a simply normalized bifurcation (transcritical) parameter for epidemiological models, such that implies that the endemic steady state is stable (i.e., the infection persists), and implies that the uninfected steady state is stable (i.e., the infection can be eliminated from the population).
Equation (1) has a disease-free equilibrium given by which always exists in the nonnegative orthant . The explicit expressions of and are The disease-free equilibrium can also be denoted by since it is the solution of equation of the compact system (5).
Lemma 4. Using the same method as in [17], the basic reproduction ratio of (1) is , where denotes the spectral radius.
Proof. The expressions, which are coming from the other compartment, due to contamination, are given by the following matrix: The expressions, which are coming from the other compartment, due to reasons different from contamination, are given by . The next generation matrix, since , is We can observe that, since is the largest eigenvalue of the next generation matrix, where denotes the spectral radius.
Consequently, from Theorem 2 of [22], the following lemma holds.
Lemma 5. The disease-free equilibrium of (1) is locally and asymptotically stable whenever and unstable if .
Remark 6. This lemma shows that if , a small flow of infectious individuals will not generate large outbreaks of the disease. To control the disease independently of the initial total number of infectious individuals, a global asymptotic stability property has to be established for the DFE when .
3.4. Global Stability of the Disease-Free Equilibrium
The following result helps to determine the stability and is related to LaSalle’s principle [13]. Consider the differential equation where is a function defined on an open set of containing the closure of a positively invariant set such that the equilibrium is in . The following lemma holds.
Lemma 7. We assume that system (11) is point dissipative [1] on . In other words there exists a compact set such that, for any , there exists a time such that, for any time , the trajectory with initial condition is in the interior of . If there exists a function defined on , then(1), for all ;(2)the greatest invariant set contained in is contained in a positively invariant set on which the restriction of system (11) is globally and asymptotically stable on at .Then, is a globally and asymptotically stable equilibrium of system (11) on .
Theorem 8. When (this implies and , where is the basic reproduction number for the patch ), the disease-free equilibrium (DFE), when it is unique, is globally and asymptotically stable in , since it is the unique equilibrium. This implies the global asymptotic stability of the DFE on the nonnegative orthant ; that is, the disease dies out in both two populations.
Remark 9. At least one endemic equilibrium can exist and coexist with the disease-free equilibrium when . In this case the DFE cannot be globally asymptotically stable.
Proof. Following the same method as in [23], system (4) can be written in a pseudotriangular form as
where , , , = , and . The Jacobian matrix of system (4) at disease-free equilibrium is . The Jacobian matrix of system (4) at disease-free equilibrium is . The matrix is clearly a Metzler stable matrix and, using a result in [8], there exists a vector such that . The matrix is a Metzler and irreducible matrix, which is stable if . With this condition satisfied and using the same previous result in [22], the stability modulus of the matrix is in the spectrum of (Perron-Frobenius theorem, proof of Lemma 2) and there exists a vector such that , with . Let us show the global stability of the DFE.
System (12) is in pseudotriangular form since the matrix depends on . There are many results for the stability of triangular systems [21–25]. Using LaSalle’s principle [13] one can obtain attractivity of equilibrium. But, for nonlinear systems, attractivity does not generally implies stability. We can use now the result of Lemma 7. Let us consider the following candidate Lyapunov function:
The function is positive and it is time derivative along the trajectories of (4) which gives
Then, ensures that for all and that holds when . This proves the global asymptotic stability on [14]. This achieves the proof that the DFE is globally and asymptotically stable.
3.5. Existence of Endemic Equilibrium
Definition 10. An equilibrium for a multipatch model as (1) is called endemic equilibrium when the two populations coexist (the density of each compartment is different from zero) at this equilibrium.
Lemma 11. An endemic equilibrium of (1) can be determined, using an equation .
Proof. An endemic equilibrium of (1) is obtained by setting the right-hand side of (4) which equals zero, giving Multiplying the second equation of (15) by gives From (16), where From the second equation of (17), Using the expression of in (16) gives Then where and .
If , we obtain the disease-free equilibrium again. If not, the following equation holds: which can also be written as This expression of in (23) is positive when one of these conditions is satisfied:
Case 1. . This yields the condition
Case 2. and . This yields the condition In sum, exists when satisfies condition (25) or condition (26). Now, using the first equation of system (15) gives Then, Putting (28) into the first equation of (17) gives Putting (28) into the second equation of (17) gives Then, in the compact form, From (1), which can be written in semicompact form as where , , and .
At equilibrium, Using (16) in (34), where , , , and . Using (28) in (35) gives Using (31) in (36) gives Let us set
One can observe that it is useful to analyze how many times the function intersects the line in the plane . The monotony of the function depends on the parameters of system (1). Then, the following lemma and theorem hold.
Lemma 12. The function satisfies the following properties.(1)The limit when is finite and is given by (2)The limit when is infinite: .
Theorem 13. These properties hold for system (1).(1)There is no boundary equilibrium. This means that the disease cannot persist in one patch while disappearing in the other one.(2)If , there exists at least one solution of , depending on the monotony of the function , and therefore there exists at least one endemic equilibrium for system (5) which coexists with the disease-free equilibrium. This situation corresponds to a backward bifurcation.(3)If , there exists at least one solution of , depending on the monotony of the function , and therefore there exists at least one endemic equilibrium for system (5) which coexists with the disease-free equilibrium. There is no backward bifurcation in this case since the endemic equilibrium no longer exists (it disappears as the disease-free equilibrium).(4)If , at least one solution of can exist, depending on parameters of system (1), since the monotony of the function also depends on these parameters.(5)If , at least one solution of can exist, depending on the monotony of the function , and therefore there exists at least one endemic equilibrium for system (5), which coexists with the disease-free equilibrium.(6)If , at least one solution of can exist, depending on parameters of system (1), since the monotony of the function also depends on these parameters.(7)If , at least one solution of can exist, depending on parameters of system (1), since the monotony of the function also depends on these parameters.
Proof. When an endemic equilibrium for system (5) exists, is solution of . and are given by (31) as
and are given by (16) as , .
The expression of is given by (16) as
Since the function is not strictly monotone, the number of solutions of depends on the parameters of system (1). Therefore, there can be more than one endemic equilibrium.
Remark 14. In the second case of the theorem, we observe that at least one endemic equilibrium coexists with the disease-free equilibrium. Then, there can be a backward bifurcation in this case.
3.6. Stability of Endemic Equilibrium
The stability of endemic equilibrium is always a big challenge in epidemiology. The problem is more difficult here since we have equations. For multipatch and universal incidence law, results concerning the global stability of endemic equilibrium are limited. Next we will illustrate some results concerning our model.
4. Numerical Simulations
System (1) is simulated with parameter values presented in Table 1 using real data of the cities of Yaounde and Bafia [21].
Next we use the values in Table 1 to generate several curves (Figures 2, 3, 4, 5, 6, 7, 8, and 9) in order to illustrate our theoretical results.
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(b)
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5. Conclusion
The long-term dynamics of our system has been completely investigated. The model exhibits rich dynamics, depending on the values of the bifurcation parameters , , and . The influence of parameter is significant on the spread of tuberculosis since it quantifies the intensity of pathogens transmission. The stability of equilibria depends on these parameters. We have transcritical bifurcation parameters and backward bifurcation. When the basic reproduction number is less than unity, tuberculosis can be controlled in each population if the DFE is the unique equilibrium. It can be more difficult if the DFE coexists with at least one endemic equilibrium (this is the situation of backward bifurcation). In this case, the disease-free equilibrium can be locally and asymptotically stable, as well as the endemic equilibrium. When the basic reproduction number is greater than unity, tuberculosis is endemic and can be difficult to control in the population. The disease in our model cannot persist in one population while disappearing in the other one.
Disclosure
Part of this work was realized during the visit of Jean Jules Tewa at the UMI 209 UMMISCO Laboratory of University Cheikh Anta Diop, Dakar, Senegal.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.