Mathematical Analysis of a General Two-Patch Model of Tuberculosis Disease with Lost Sight Individuals

and Applied Analysis 3


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][6][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 2 + 6-dimensions system is investigated using Lyapunov-LaSalle functions and quadratic forms.The function  = ∑  =1   (  −  *  ln   ) has a long story in epidemiology [15][16][17][18][19][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,

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  ( = 1, 2) is divided into +3 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 Λ  .The rate constant for nondisease related death is   ; thus 1/  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  = 1, 2 and  = 1, . . ., , 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 (1 −   )  of infected individuals who do not receive effective chemoprophylaxis becomes infectious with a rate constant   , so that 1/  represents the average latent period.Thus, individuals leave the class   to   at rate   (1 −   ).After receiving an effective therapy, infectious individuals can spontaneously recover from the disease with a rate constant   , entering the infected class   .A fraction   (1 −   )  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.

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 𝐼𝑅 2𝑛+6
+ 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 () ≥ 0 and   () ≥ 0, the matrix It is well known that linear Metzler system lets the nonnegative orthant invariant [23].Moreover ()/ ≤ Λ +    and   is a Metzler matrix.These prove the positive invariance of the nonnegative orthant  2+6 + by (1).

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,  0 , 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],  0 is a simply normalized bifurcation (transcritical) parameter for epidemiological models, such that  0 implies that the endemic steady state is stable (i.e., the infection persists), and  0 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  2 = ( The disease-free equilibrium can also be denoted by ( * , 0) since it is the solution of equation Λ +    * = 0 of the compact system (5).

Lemma 4.
Using the same method as in [17], the basic reproduction ratio of (1) , 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 ⟨ 2  / * ⟩ =  *  , is We can observe that, since  0 is the largest eigenvalue of the next generation matrix, where  denotes the spectral radius.
Remark 6.This lemma shows that if  0 ≤ 1, 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  0 ≤ 1.

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  1 function defined on an open set of   containing the closure Ω of a positively invariant set Ω such that the equilibrium  0 is in Ω.The following lemma holds.
Lemma 7. We assume that system (11) is point dissipative [1] on Ω.In other words there exists a compact set  ⊂ Ω such that, for any  ∈ Ω, there exists a time ( 0 ) > 0 such that, for any time  > ( 0 ), the trajectory with initial condition  0 is in the interior of .If there exists a  1 function  ≥ 0 defined on Ω, then (1) ()/ = ⟨()/()⟩ ≤ 0, for all  ∈ Ω; (2) the greatest invariant set  contained in  = { ∈ Ω, ()/ = 0} is contained in a positively invariant set on which the restriction of system (11) is globally and asymptotically stable on  at  0 .
Then,  0 is a globally and asymptotically stable equilibrium of system (11) on Ω.
Theorem 8.When  0 ≤ 1 (this implies  1 0 ≤ 1 and  2 0 ≤ 1, where   0 is the basic reproduction number for the patch ), the disease-free equilibrium (DFE), when it is unique, is globally and asymptotically stable in Γ  , since it is the unique equilibrium.This implies the global asymptotic stability of the DFE on the nonnegative orthant  2+6 + ; 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  0 ≤ 1.In this case the DFE cannot be globally asymptotically stable.

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.
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 (−  ) −1 gives From ( 16), Abstract and Applied Analysis 7 where From the second equation of ( 17), Using the expression of ⟨ 2 /⟩ in (16) gives Then where If ⟨ 1 /⟩ = 0, we obtain the disease-free equilibrium again.If not, the following equation holds: which can also be written as  1) for different initial conditions when  1 = 0.002 and  2 = 0.003 (such that  1 0 = 0.3691 and  2 0 = 0.4980).We can observe the global stability of DFE when  0 < 1.All other parameters are as in Table 1.

Let us set
(1) The limit when  → 0 is finite and is given by the monotony of the function  also depends on these parameters.
Proof.When an endemic equilibrium (, ) for system (5) exists,  is solution of () = 1. (41) Since the function  is not strictly monotone, the number of solutions of () = 1 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.

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 2 + 6 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.

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.

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   0 ,  = 1, 2, and  0 .The influence of parameter   0 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  0 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.

Figure 3 :
Figure3: Trajectories of latently infected individuals of (1) for different initial conditions when  1 = 0.002 and  2 = 0.003 (such that  1 0 = 0.3691 and  2 0 = 0.4980).The latent classes disappear at the end and only susceptible classes persist when  0 < 1.All other parameters are as in Table1.

Figure 4 :
Figure 4: Trajectories of infectious individuals in care center (a) and lost sight individuals (b) of (1) for different initial conditions when  1 = 0.002 and  2 = 0.003 (such that  1 0 = 0.3691 and  2 0 = 0.4980).The infectious and lost sight classes disappear at the end and only susceptible classes persist when  0 < 1.All other parameters are as in Table1.

Figure 7 :
Figure 7: Trajectories of latently infected individuals of (1) for different initial conditions when  1 = 2.082 and  2 = 2.173 (such that  1 0 = 3.1492 and  2 0 = 4.1085).The endemic equilibrium is locally and asymptotically stable when  0 > 1.In this case, the DFE is unstable.All other parameters are as in Table1.

Figure 8 :Figure 9 :
Figure 8: (a, b) Trajectories of infectious individuals in care center of (1) for different initial conditions.The endemic equilibrium exists and is stable when  0 > 1; the DFE is unstable in this case.

𝑟 21 ) 𝑘 21 𝐸 21 + 𝑏 12 𝐸 12 − (𝜇 2 + 𝑏 22 + (1 − 𝑟 22 ) 𝑘 22 ) 𝐸 22 ,
Figure 1: Flow chart of a two-patch model of tuberculosis with lost sight individuals in sub-Saharan Africa, with migrations at rate   between susceptible individuals of two populations, at rate   between latently infected individuals of two populations, at rate   between infectious individuals in care centre, and at rate   for infectious lost sight individuals.An individual of a given population (patch) has contacts only with individuals of the same population.