Simulation of Local Stability Analysis of An Epidemic Model with Constant Removal Rate of the Infective Between 2017-2019

In this thesis we consider an epidemic model with a constant removal rate of infective individuals is proposed to understand the effect of limited resources for treatment of infective on the disease spread. It is found that it is unnecessary to take such a large treatment capacity that endemic equilibria disappear to eradicate the disease. It is shown that the outcome of disease spread may depend on the position of the initial states for certain range of parameters. It is also shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, subcritical Hopf bifurcation. Keywords: Epidemic model, nonlinear incidence rate, basic reproduction number, local and global stability DOI : 10.7176/JHMN/61-01 Publication date : April 30 th 2019

If we assume that individuals are mixed randomly then each potential transmission may be from an infected population to a susceptible population which results in a new infected population. Or a transmission may occur from an infected population to another infected population in which case nothing happening since the person is already infected. Or the potential transmission may occur from an infected person to a recovered or immune person. In this case again nothing changes. This description of the SIR model was made more mathematical by a formulated differential equation for the proportion of individuals in each class.

The Model Equation
Based on the above assumption the dynamics of the disease can be express using the system of ordinary differential equation , where all parameter are positive and h(I) is the removal rate of infective individuals due to the treatment of infective. The definition of h(I) implies that a constant removal rate for the infective is used until the disease disappears. We suppose that the treated infective become recovered when they are treated in treatment sites. Suppose that 0 ( ) 0 0 r for I h I for I       (2) where r > 0 is a constant and it represents the capacity of treatment for infective. This means that we use a constant removal rate for the infective until the disease disappears.
Parameters Parameters definition A Recruitment rate(New born or immigrants) Μ Natural death rate Β The contact rate, defined to be the average number of effective contacts with other(susceptible) individuals per infective per unit time Γ The rate at which an infectious individual recovered per unit time Table 2.

Parameters and their definitions
Variables Definition S(t) The number of susceptible individuals at time, t I(t) The number of infected individuals at time, t R(t) The number of recovered individuals at time, t Table 2.2 Variables and definitions of populations used as variable

2.MODEL ANALYSIS
This section deals with the study of stability properties of different equilibriums point of the model. Stability analysis is crucial in this study since we would be able to know whether disease free equilibrium point and endemic equilibrium point would be stable so that the disease would persist or not.

Positivity and Boundedness of Solutions
We can show from system (1) that the state variables are non-negative and the solutions remain positive for all time t  0. Here the parameters in the system are assumed to be positive. We also show that the feasible solutions are bounded in a region: , I is a function of t Integrating both sides of the inequalities, we get Solving for S (t) we get S(t) = C0 e -(μ+ I) t . But at t = 0, we have S(t) ≥ S(0)e -(μ+βI)t , since (μ+βI) >0 Hence, S(t)  0. From the second equation of system (1) we have Integrating both sides of the inequalities, we get ln(I(t))  -(μ + ) t + C. Then I(t)  C0e -(μ+ ) t . But at t = 0, we have I(t)  I(0)e -(μ+ )t  0, since (μ+γ) > 0 Hence, I(t)  0. From last equation in system (1) The solution of the linear differential equation then becomes  (1) are contained and remain in the region Г for all time 0  t Proof: Suppose that (S(t), I(t), R(t)) is a solution of system (1). Then since N(t) = S(t)+I(t)+R(t) is the population at any given time t, it is non-negative, that is N(t)  0 for all t  0. Thus, the lower bound for S(t), I(t) and R(t) is 0. To find the upper bound of the system consider the following equation N = S + I + R and N′ = (S + I + R)′ Then (S + I + R)′ = A -μS -μ I -μ R = A -μ (S + I + R) = A -μ N The solution of the linear differential equation then becomes Using this result together with Lemma1, we have that 0 ≤ N ≤ A/μ which implies that N and all other variable (S, I and R) are bounded and all the solutions starting in Г stay in Г with respect to the system. Therefore, one can show that the removal rate has significant effects on the dynamics of the system. Since the first two equations in system (1) are independent of the variable R it suffices to consider the following reduced model: It is assumed that all the parameters are positive constants.

Equilibria and Stability Analysis
In this section we investigate the existence for the system described by equation. System (3) has always a disease-free equilibrium point and unique endemic equilibrium point. The equilibrium points of the system are obtained by solving the following equations simultaneously We detail each of these equilibriums points as follows based on equation (4)

The Disease Free Equilibrium point
The equilibrium state in the absence of infection is known as the disease free equilibrium point. The disease free equilibrium point is obtained when I = 0. Hence the disease free equilibrium point of system (4) is given by E0 = (  A , 0).

The Endemic Equilibrium point
The equilibrium state with the presence of infection (i.e. I ≠ 0) is known as endemic equilibrium point or non-zero equilibrium point. The endemic equilibrium point of system (4) is the solution of the system of equations A -μ S -SI = 0 SI -(μ + )I -r = 0. Therefore, the endemic equilibrium point is given by Since the discriminate (form the second equation) ((A -r) -μ(μ + )) 2 -4r μ (μ + ) is non-negative, then at least a positive solution exists.

The Basic Reproduction Number (R0)
The basic reproduction number of an infectious disease is one of the fundamental concepts in mathematical epidemiology. It is defined as the average number of secondary infections caused by an infectious individual during his or her entire period of infectiousness. The basic reproduction number is an important non-dimensional quantity in epidemiology as it sets the threshold in the study of a disease both for predicting its outbreak and for evaluating its control strategies. The condition R0 < 1 means that every infectious individual will cause less than one secondary infection and hence the disease will die out and when R0 >1 every infectious individual will cause more than one secondary infection and hence the disease will invade the population. A large value of R0 may indicate the possibility of a major epidemic. The properties and complexities of R0 depend on the number of infective and intervention strategies.
Thus, the basic reproduction number R0 often takes as the threshold quantity that determines whether or not an infectious disease will spread through a population. Since Since the basic reproductive number R0 is given by (5) can be rewritten as Then equation (6) has no positive solution.
Case2: If R0 > 1 and   . 3. two interior equilibrium points if condition (case 3) is satisfied. The two endemic equilibrium points of system (3) are given by E1 = (S1, I1) and E2 = (S2, I2) Where We know that quarantine is an important method to decrease the spread of disease. In classical epidemic models, the treatment rate is assumed to be proportional to the number of infective. In fact, this assumption is irrational because every community should have a suitable capacity for treatment. If it is too large, the community pays for unnecessary cost. If it is too small, the community has the risk of the outbreak of a disease. This means that unnecessary to increase removal rate r to make the disease disappear. The removal rate can be small, so the resources for treatment are saved.
Therefore, the positive solution of system (3) is bounded. Note that the non-negative I-axis repels positive solutions of system (3) and that there is no equilibrium on the non-negative S-axis. If R0  1 or condition (case 1) holds, the susceptible population may be any constant in [0, ∞) , which depends on initial conditions. However, the infective go to extinction in a finite time, implying the disappearance of disease. Theorem 1: [25] The equilibrium point E1 = (S1, I1) is saddle whenever it exists and E2 = (S2, I2) is a center. Proof: We begin by analyzing the stability of these two equilibria. The Jacobean matrix of (3) is given by Now at (S1, I1), the Jacobean matrix becomes It follows that (S1, I1) is saddle point. Since this implies that an eigen-value remains real positive and the other real negative for increasing r.
The Jacobean matrix of system (3) at the equilibrium point (S2, I2) is We can immediately conclude that the endemic equilibrium 1 with low number of infected individuals is always a saddle, and that the endemic equilibrium 2 with high number of infected individuals is a node or focus.  (11) and unstable otherwise.
Now we find the condition under which the tr(J2) = 0.
From condition (13) we have that tr(J2) = 0 is equivalent to It follows from the definition of I2 that tr(J2) = 0 is equivalent to Taking squares on both sides of (15) and simplifying the resulting equation, we obtain Then by using quadratic formula we get (19) Since (16) is equivalent to βA-3μ 2 -μγ-3μ 3 /γ < βr By definition of D1 and D2 we have

Local Stability Analysis Local Stability of Diseases -Free Equilibrium point
In this section we are going to discuss about the local stability of the disease free equilibrium point. In the absence of the infectious diseases the model has unique diseases free equilibrium at E0.The stability analysis of the diseasefree equilibrium E0 determines the thresholds (reproductive number) of the epidemic. The Jacobean matrix of system (1) at the disease free equilibrium point is given by Then the characteristic equation of the Jacobean matrix is given by  2 + (2μ +  -A/ μ) + μ (μ +) -A = 0 And hence the corresponding eigen-values are 1 = -μ and 2 = -(μ +) + A/ μ. In the following theorem established the local stability of the disease free equilibrium point. Theorem 3: [27, 30],The disease free equilibrium E0 is locally asymptotically stable if R0 < 1, otherwise unstable. Proof: The eigen-values of the characteristics equation λ1 < 0 and λ2 < 0 if R0 < 1 then the diseases free equilibrium E0 is locally asymptotically stable.

Local Stability of Endemic Equilibrium point
In this section, we discuss the local stability of endemic equilibrium of system (3) Therefore, tr(J2) < 0 and det(J2) > 0. This implies that the eigen-value of Jacobean matrix J2 has negative real part, and hence E2 is locally asymptotically stable.

Bifurcation Analysis
The main purpose of this section is to get an insight into how the dynamics of the system changes depending on the system parameters. If a parameter is allowed to vary, the dynamics of the system may change. The stability of an equilibrium may change from an equilibrium point may appear or disappear a periodic solution may appear or disappear as the values of parameter varies. An equilibrium point may become unstable and a periodic solution may appear or a new stable equilibrium point may appear making the previous equilibrium point unstable. The change in the qualitative behavior of solution as a control parameter is varied is known as a bifurcation and the parameter values at which bifurcation occur are called bifurcation points.
An epidemic models, the reproduction number works as the threshold quantity for the stability of the diseasefree equilibrium. The usual situation is that for R0 < 1 the diseases-free equilibrium point is the only equilibrium and it is asymptotically stable, but it loses its stability as R0 increases through 1, where a stable endemic equilibrium emerges, which depends continuously on R0. Such a transition of stability between the disease-free equilibrium point and the endemic equilibrium point is called forward bifurcation.
However, it is possible to have a very different situation at R0 =1, as there might exist positive equilibria also for values of R0 less than 1. In this case we say that the system undergoes a backward bifurcation at R0 =1, when for values of R0 in an interval to the left of 1, multiple positive equilibria coexist,( such as Disease free equilibrium and endemic equilbruim) typically one unstable and the other is stable.
When forward bifurcation occurs, the condition R0 < 1 is usually a necessary and sufficient condition for disease eradication, whereas it is no longer sufficient when a backward bifurcation occurs. In fact, the backward bifurcation scenario involves the existence of the trans-critical bifurcation at R0 = 1and of a saddle-node bifurcation at R0=R0 SN <1(where R0 SN stand for saddle-node bifurcation) In particularly the backward bifurcation may be qualitatively described as follows. In the neighborhood of 1, for R0 < 1, the stability of disease free equilibrium point is exists with two endemic equilibria: a smaller equilibrium (i.e., with a smaller number of infective individuals) which is unstable and a larger one (i.e., with a larger number of infective individuals) is stable. These two endemic equilibria disappear by saddle-node bifurcation when the basic reproductive number R0 is decreased below the critical value R0 = R0 SN > 1 For R0 > 1, there are only two equilibria: the disease free equilibrium point, which is unstable, and the larger endemic equilibrium, which is stable.As a consequence, in the backward bifurcation scenario, if R0 is nearly below unity, then the disease control strongly depends on the initial sizes of the various sub-populations. On the contrary, reducing R0 below the saddle-node bifurcation value R0 SN , may result in disease eradication, which is guaranteed provided that the disease free equilibrium is globally asymptotically stable. Hence, determining the sub-threshold R0 SN may have a crucial importance in terms of disease control.
The linear stability analysis shows that diseases free equilibrium is locally asymptotically stable if βA < μ(μ+γ) and an unstable if βA > μ(μ+γ), while the endemic equilibrium points is locally asymptotically stable if βA > μ(μ+γ) and an unstable if βA < μ (μ +γ). Furthermore, at βA = μ (μ +γ, so there is a trans-critical bifurcation. There is the threshold condition R0 = 1 is equivalent to the threshold condition describes at ΒA = μ (μ +γ) When the disease free equilibrium point and the endemic equilibrium point exist and exchange stability, a unique stable endemic equilibrium point arises from the bifurcation point R0 and increases as R0 increases (β increases by fixing A and μ). Thus, it shows that infectious free equilibrium exists for all R0, while endemic infections only exist for R0 > 1.
Let us verify that the existence of a Hopf bifurcation in (3) and determine its direction.  (22) then there is a family of unstable limit cycles if r is less than and close to A0 (that is subcritical Hopf bifurcation occurs when r passes through the critical value A0). Proof: Let r = A0.Then the tr(J2) = 0. It follows from (16) that From the above equations one can see that tr(J2) = 0 and det(J2) > 0. Thus from the above conditions Perform coordinate transformation by x = S -S2 and y = I -I2 then system (3) will become We know that tr(J2) = 0 and hence we get And we have the following: I  S  I  I  S  I  S  I  I  S   I  S  I  I  S  I  S   I  S  I  I  S  I  S Then after some algebraic calculations we obtain 2 2 2 Then the conclusion of this theorem follows from L. Perko [12] As an example, we fix A = 8, μ = 0.1,  = 1,  = 1. Then we obtain We know that there is an unstable limit cycle when r is less than and near A0 from theorem 4.5.1, which is shown that there is an unstable limit cycle when r decrease from 5.2023
For the initial value, the approximated solutions S (t), I (t), and R (t) are displayed in the figure (6.1-6.4) given below. This decrease may be possibly because of the high rate of recovery due to mass vaccination, since individual become permanently immune upon recovery. The contact rate also has large impact on the spread of a disease through a population. The higher the rates of contact, the more rapid the spread of the disease, it is also observed that as the contact rate decreases, the fraction of individuals infected decreases at a faster rate as would be expected logically. And I(t) decreasing (I'(t)< 0) when . This show that the disease is dies out. The condition      ) 0 ( S is sufficient and necessary to start an outbreak, otherwise the number of infected individuals is decreasing from the very beginning. This graph also demonstrates that the contact rate has large impact on the spread of the disease through population. If the contact rate is observed to be high then the rate of infection of the disease will also be high as would be expected logically. However, there exists another parameter to consider, as more individuals are infected with the disease and I(t) grows, some individuals are also leaving the infected class by being cured and then join the Recovered class.

CONCLUSION
In this thesis, by combining qualitative and bifurcation analyses we have studied the global behavior of an epidemic model with a constant removal rate of the infective individuals to understand the effect of the treatment capacity on the disease transmission. We have shown that there are two possibilities for the outcome of the disease transmission. First, if R0 < 1 there is a disease -free equilibrium which is asymptotically stable and the infections dies-out. Second, if R0 > 1 the usual situation is there is an endemic equilibrium which is asymptotically stable and the infections persist. If the endemic equilibrium is unstable the instability commonly arises from a Hopf bifurcation and the infection still persists.
More precisely, as R0 is increase through 1 there is an exchange of stability between the disease -free equilibrium and the endemic equilibrium (which is negative as well as unstable and the biologically meaningless if R0 < 1. There is bifurcation or change in equilibrium behavior at R0 = 1 but the equilibrium infective population size depends on continuously on R0 such a transition is a transcritional bifurcation. In this case, reducing the basic reproductive number R0 below one may fail to control the disease.
Generally the model has shown success in attempting to predict the causes of diseases transmission within a population. The model strongly indicated that the spread of a disease largely depend on the contact rates with infected individuals within a population