An Approach for the Global Stability of Mathematical Model of an Infectious Disease

: The global stability analysis for the mathematical model of an infectious disease is discussed here. The endemic equilibrium is shown to be globally stable by using a modiﬁcation of the Volterra–Lyapunov matrix method. The basis of the method is the combination of Lyapunov functions and the Volterra–Lyapunov matrices. By reducing the dimensions of the matrices and under some conditions, we can easily show the global stability of the endemic equilibrium. To prove the stability based on Volterra–Lyapunov matrices, we use matrices with the symmetry properties (symmetric positive deﬁnite). The results developed in this paper can be applied in more complex systems with nonlinear incidence rates. Numerical simulations are presented to illustrate the analytical results.


Introduction
Mathematical modeling has the best predictive analysis to accurately predict the prevalence of infectious diseases, and with the help of predicting the prevalence of infection, effective strategies for disease control can be determined [1][2][3][4]. The mathematical models used for infectious diseases have evolved rapidly in recent decades. One of the reasons for this progress and development is the improvement of researchers' ability to collect data [5][6][7][8][9][10][11].

The Mathematical Model
The mathematical model of the epidemic model with two controls was proposed by Kar and Jana [26]. In this model, we have four types of population, which are represented by susceptible S(t), infected I(t), recovered R(t) and vaccinated V(t).

Parameter Description a
The total recruitment u 1 The constant vaccination control σ Transmission rates from vaccinated to susceptible α The reciprocal of half-saturation λ The infection force parameter λSI 1+αI The saturated infection rate m Infected population rate that have recovered naturally ρ 1 The portion recovered (0 < ρ 1 < 1) u 2 The constant treatment control b The effectiveness of the treatment bu 2 The rate by which the infected populations recovered βR The part of the recovered class becomes susceptible ρ 2 Recovered sections that go to recovery class (0 < ρ 2 < 1) γ Death rate of infected people due to disease attack d The natural death rate

Equilibrium of the Model for Fixed Controls
Throughout this paper, we assume that the controls u 1 and u 2 are constant. System (1) has two possible nonnegative equilibria. The first one is E 0 (S 1 , 0, 0, V * ) where This equilibrium is the disease free equilibrium. The other equilibrium is E * = (S * , I * , R * , V * ), where )R 0 , The average rate of infection in susceptible individuals caused by a number of secondary infections is called the basic reproduction number R 0 . Now, we want to get the basic reproduction number of System (1). Let us introduce matrices F and V, as follows: Then by applying the next generation matrix method developed by van den Driessche and Watmough [31], the basic reproduction number R 0 is the spectral radius of the next generation operator

Proposition 1. The closed set
is positively invariant.
Proof. Let (S(t), I(t), R(t), V(t)) be any solution with positive initial conditions. We have, The time derivative of N(t) along the solution of (1) is Using theory of differential equations, we get and for t → ∞, we have Hence, Γ is positively invariant and it is sufficient to consider solutions of System (1) in it.

Global Stability of the Endemic Equilibrium
In this section, we are concerned with the global stability of (1) in a positively invariant set of Γ. To do this, we define the Lyapunov function as follows: where w 1 , w 2 , w 3 and w 4 are positive constants. Calculating the time derivative of L along the trajectories of System (1), we obtain: Then, we add the expression λS * I 1+αI into the first and second square bracket and then subtract it. As a result, we obtain To establish the global stability of the endemic equilibrium E * , we investigate that the matrix A defined in Equation (4) is Volterra-Lyapunov stable. Below we briefly review the following prerequisites: Here, we recall the basic definitions related to Volterra-Lyapunov stable matrices [26]. Suppose, A n×n is a real matrix.
(D1) All the eigenvalues of A have negative (positive) real parts if and only if there exists a matrix H > 0 (that is, mean H is symmetric positive definite) such that H A + A T H T < 0(> 0) [32]. [32,33] is Volterra-Lyapunov stable if and only if: [34,35]. Suppose the nonsingular D n×n = [d ij ], (n ≥ 2), M n×n = diag(m 1 , · · · , m n ) is a positive diagonal matrix and H = D −1 , such that: it is possible to choose m n > 0 such that MD + D T M T > 0. Note that, D denote the (n − 1) × (n − 1) matrix obtained from D by deleting its last row and last column. (4) is Volterra-Lyapunov stable.

Theorem 1. The matrix A defined in Equation
Proof. Clearly −A 44 > 0. Let us consider D = −Ã, denote the 3 × 3 matrix obtained from −A by deleting its last row and last column. From Equation (4), we obtain Based on (L2), we state and prove the following results. The first Lemma, proves that D = −Ã is diagonal stable and in the next Lemma, we show the H = −A −1 is diagonal stable. Therefore, all the conditions of (L2) are satisfied. Hence the matrix A is Volterra-Lyapunov stable. Lemma 1. The matrix D defined in Equation (5), is diagonal stable.
Proof. Let's now discuss the diagonal stability of D. It is guaranteed by the following steps: Step 1. It is obvious that D 33 > 0.
Step 3. Now, we must show that D −1 is diagonal stable. Let us consider the D −1 as following: where, ], ), ).
Now, we have D −1 as: Following some calculations, we obtain that It is easy to see, D −1 11 > 0 and D −1 22 > 0. Therefore, D −1 is diagonal stable.

Proof.
We can obtain the −A −1 as following: where, ), ), ), ). First, we show that det(−A) > 0: Also, we can show that det(H) > 0 (see the Appendix A). It remains to show that H −1 is diagonal stabe. Define The h 11 is writen as The h 22 is writen as It is easy to see det(H) > 0, see the Appendix B. Therefore, H −1 is diagonal stable.
Summarizing the above discussions, we have the following conclusions for the globally asymptotically stablity of the endemic equilibrium.
Proof. Lemmas 1 and 2 with the aid of Theorem 1, guarantee that the endemic equilibrium of the model System (1) is globally asymptotically stable.

Numerical Simulations and Discussion
In this section, we present some numerical simulations of System (1) using the basic reproduction number R 0 , to support the analytical results. Parameters were taken from [26].

Discussion
The authors in [27], applied the original method for proving the global stability of endemic equilibrium of the system of three-dimensional and four-dimensional. At first, they define D = −A and E = (−A) −1 , to discuss the Volterra-Lyapunov stability of A 3×3 . Hence, following the steps they concluded that A 3×3 is a Volterra-Lyapunov stable matrix:

2.
To prove that D is Volterra-Lyapunov stable, they performed another process. Defined where, Q 2×2 is positive. Finally, by some algebraic and matrix manipulations, showed that P 2×2 > 0.
To compare the results in this paper with the original method, the process of proving the stability of matrix A 4×4 is shown in Figure 7. According to our investigations on different systems, and as the authors mentioned in Section 6 [27], the implementation of the method for the higher dimensions systems (in the second step proposed by the authors) is very difficult and complex. Therefore, the use of the modified method, can reduce the complexity of the calculations.
Based on Figure 7, by decreasing the size of the matrix A 4×4 to A 3×3 , applying (L2), finally reducing to 2 × 2 matrix, and using (L1), it can easily be proved that A is a Volterra-Lyapunov stable matrix.

Conclusions
We have investigated the global stability of the endemic equilibrium point of an infectious disease model. In this paper, using the modified Volterra-Lyapunov matrices method, the stability of the model has been analyzed. The main advantage of this modification is its application to various systems of epidemics, infection diseases and even chaotic dynamical systems. This leads to better performance and reduces the complexity of the proofs. The numerical results verify the effectiveness of the proposed scheme. ).
Hence, it is clear to see det(H) > 0. The proof is then complete.
It is easy to see that a 21 a 13 a 32 − a 31 a 13 a 22 = 0, hence we show that a 11 a 22 a 33 − a 11 a 23 a 32 − a 21 a 12 a 33 + a 31 a 12 a 23 > 0.
Hence, it is clear to see det(H) > 0. The proof is then complete.