Stability Analysis of a Deterministic Epidemic Model in Metapopulation Setting

We present in this article an epidemic model with saturated in metapopulation setting. We develop the mathematical modelling of HIV transmission among adults in Metapopulation setting. We discussed the positivity of the system. We calculated the reproduction number, If 0 1 j R ≤ for 1,2,3,4 j = , then each infectious individual in Sub-Population j infects on average less than one other person and the disease is likely to die out. Otherwise, if 0 1 j R > for 1,2,3,4 j = , then each infectious individual in Sub-Population j infects on average more than one other person; the infection could therefore establish itself in the population and become endemic. An epidemic model, where the presence or absence of an epidemic wave is characterized by the value of 0 j R both ideas of the inner equilibrium point of stability properties are discussed.


Introduction
Numerous mathematical models have been developed in order to understand disease transmissions and behavior of epidemics.One of the earliest of these models as discussed by Kermack [1], by considering the total population into three classes, namely, susceptible (S) individuals, infected (I) individuals, and recovered (R) individuals which is known to us as SIR epidemic model.This SIR or SI epidemic model is very significant in todays analysis of diseases.SIR Model: The SIR model labels these three compartments S = number susceptible, I = number infectious, and R = number recovered.This is a good and simple model for many infectious diseases.Birth→S→I→R→Death and The SI model labels these two compartments S = number susceptible and I = number infectious.This is a good and simple model for many infectious diseases.Birth→S→I→Death.
In the mathematical epidemiology area an key concept is associated to the basic reproduction number ( 0 R ).This is defined as the second expected number produced from just a one individual in a susceptible population.For any infectious disease, one of the most key concerns is its capacity to invade a population, as studied by various authors [2].This can be expressed by a threshold parameter: if the disease free equilibrium is locally asymptotically stable, then the disease cannot invade the population and 0 1 R < , whereas if the number of infected individuals grows, the disease can invade the population and 0 1 R > , as studied by various authors [3].

Compartmental Model and Differential Equations
In this section, we approached this study by using SI deterministic model.In our model system the recruitment into the susceptible human population is only by births ( λ ).The size of the human population is decreased by natural deaths ( µ ), infected and awareness/education.Uneducated and educated infected female youth move to the classes YUF S , YEF S respectively at the rate From the proposed schematics of the compartment model shown (see Figure 1), we extracted a metapopulation model for HIV dynamics among the youth coupled with awareness/education i.e., we extended the single patch disease model to include multiple patches.A schematic of the Metapopulation Model (see Figure 2) for HIV transmission in the youths coupled with awareness/education in each patch i,

Positivity and Boundedness
The theory of ordinary differential equations requires that, for every set of initial conditions ( )

S t S t S t S t I t I t I t I t T t
for every 0 t ≥ such that the solution will remain in Ω with probability of one.
2) The solution ( Evidently, the coefficients of system (2.0) are locally Lipschitz continuous.
Hence, for any given initial condition ( )

S t S t S t S t I t I t I t I t T t
for every , where T is the final time.Here, it can be deduced that Hence, by comparison theorem; ( ) ( ) . It can therefore be verified that the solution ( )

Calculation of the Basic Reproduction Number
The basic reproduction number ( 0 R ) is defined as an infections originating from an infected individual that invades a population originally of susceptible individuals.0 R is used to predict whether the epidemic will spread or die out.
In the next part, we will analyze the dynamics of , YUF i I so as to be able to obtain 0 R .Here, the functions (F) and (V) denote the matrix of the infection rates and the matrix of the transition rates respectively.
Let us thus look at the following system of differential equations (The reduce model from 2.0). ( The above system can be represented in matrix form as .

I fI vI = +
where f is the matrix of the infection rates and v is the matrix of the transition rates.
The spectral radius of the Metzler Matrix, ( ) , is defined as the largest eigenvalue of the Metzler Matrix [4].Thus: , , ,

Stability of the Disease Free Equilibrium Stability
Consider the differential equation ( ) liaponouv stable and quasi-asymptotically stable [5].

Local Asymptotic Stability
A point x * is an equilibrium point of the system if ( ) x * is locally stable if all solutions which start near x * (meaning that the initial conditions are in a neighborhood of x * ) remain near x * for all time.The equilibrium point x * is said to be locally asymptotically stable if x * is locally stable and, furthermore, all solutions starting near x * tend towards x * as t → ∞ [5].

Global Asymptotic Stability
The system ( ) is globally asymptotically stable if for every trajectory ( ) x t , we have ( ) as t → ∞ (implies x * is the unique equilibrium point) [5].

Liapunov stability
An important technique in stability theory for differential equations is one known as the direct method of Liapunov.A Liapunov function is constructed to prove stability or asymptotic stability of an equilibrium in a given region.1) If ( ) We note that in case 1 the function V is a Liapunov function and in case (2) V is a strict Liapunov function.

Global Stability of the Disease-Free Equilibrium
In this section, we prove that 0 E is actually globally asymptotically stable when 0 1 j R ≤ .Therefore, the model (2.0) demonstrates global threshold dynamics.We shall achieve our goal by constructing an appropriate Lyapunov functional.

Conclusion
In this study, we approached using deterministic model.We developed a mathematical model of HIV transmission among adults in Meta-population setting in Ethiopia.Our model captures the disease induced deaths in transmission as HIV is known to cause deaths in transmission.Mathematical analysis was done and it was established that in the absence of the disease a disease free equilibrium will always exist if 0 1 j R ≤ for 1, 2,3, 4 j = . We also established that the endemic equilibrium exists in the presence of the disease that is when 0 1 j R > for 1, 2,3, 4 j = , with the infectious population greater than zero.Reducing the infection in the vector population reduces 0 j R for 1, 2,3, 4 j = , greatly.Thus the best methods of controlling HIV transmission is to target the Infected uneducated female youth, Infected educated female youth, Infected uneducated male youth, Infected educated male youth.0 j R is a threshold that completely determines the global dynamics of disease transmission.

1 β, YEM I respectively at the rate 2 β
whereas uneducated and educated infected males youth move to the classes YUM I resulting in an increase in the youth infectious classes.The infectious classes are all decreased by natural deaths ( µ ) and disease induced deaths ( δ , 1 δ ).YEF S and YEM I is decreased further as a result of the infected educated, II vertical transmission and tested youths going through the Antiretroviral therapy thus moving to the treatment class Y T at the rate α .We assume that once a person becomes infected with HIV they do not fully recover as there is no immunity to HIV and that only the educated and tested persons qualify the antiretroviral therapy.

Figure 1 .
Figure 1.Proposed schematics of the compartmental model.

Figure 2 .
Figure 2. Schematics of the metapopulation model.

T
employing the same intuition used in proving proposition 3.1, we see that system (2.0) with non-negative initial conditions ≥ has a non-negative solution defined in R and the set

1 j
3,4 j = , then each infectious individual in Sub-Population j infects on average less than one other person and the disease is likely to die out Otherwise, If 0 infectious individual in Sub-Population j infects on average more than one other person; the infection could therefore establish itself in the population and become endemic.An SIR epidemic model, where the presence or absence of an epidemic wave is characterized by the value of 0 j R .

Definition 5 . 1 . 1 C
A positive-definite function V in an open neighborhood of the origin is said to be a Liapunov function for the autonomous differential system, function in a neighborhood U of the origin.
. It can therefore be ascertained that the function is a Lyapunov function for system , , , , , so that the infection does not persist in the metapopulation and under this condition the endemic equilibrium point does not exist.The DFE is unstable for P. K. Desalegn et al.