Abstract
This paper deals with the mathematical analysis of a general class of epidemiological models with multiple infectious stages for the transmission dynamics of a communicable disease. We provide a theoretical study of the model. We derive the basic reproduction number
Appendix A: Proof of Theorem 3
Herein, we give the proof of Theorem 3 on the GAS of the endemic equilibrium
Consider the following Lyapunov function:
where ai (i = 1,2,3) and bk (k = 1,…,n) are defined as
The time derivative of U(S,E,Ik,L,R) satisfies
By considering model system (2) at the endemic equilibrium, one has
Introducing eq. (22) into eq. (21) yields
Now, let
The coefficients ai et bk are chosen such that the coefficients of
Now, multiplying the fifth equation of (25) by
On the other hand, from eq. (22), one has
Adding the above equation in the right-hand side of eq. (26) gives
Note that when the fifth equation of (25) is satisfied, then all equations of (25) are also satisfied. This is why in the sequel, we only consider the following system of equations:
Solving the above equations give the expressions of ai and bk defined as in eq. (20). Thus, after plugging the expressions of ai and bk given as in eq. (20) into eq. (24), one obtains
Using the fact that
Now, multiplying the second, third, fourth, and fifth equations of (29) by E*, L*, R*,
Using the expressions of
Now, let F1(u), F2(u), F3(u), and Gk(u) (k = 2,…,n) where
Adding eq. (34) to the right-hand side of (31) gives
Now, we shall choose the functions F1(u), F2(u), F3(u), F4(u), and Gk(u), k = 2,…,n which make
After plugging these expressions given as in eq. (36) into eq. (35), one finally obtains
Since the arithmetic mean exceeds the geometric mean, the following inequalities hold:
Now, let
The next step is to show that the functions Dk is nonpositive for all
□
Appendix B: Useful inequalities
In this appendix, we give inequalities which are necessary to demonstrate that the time derivative of the Lyapunov function (19) is nonpositive. A key tool is the arithmetic geometric means inequality, which we state here.
(Arithmetic geometric means inequality): Let
Furthermore, exact equality only occurs if
An immediate consequence of the arithmetic geometric means inequality follows.
Let
Furthermore, exact equality only occurs if
We also have the following result
Let
□
Proof
We have
□
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