A mathematical model of an SIR epidemic model with constant recruitment and two control variables using control terms and a deterministic system of differential equation is presented and analyzed mathematically and numerically. We intend to control the susceptible and infected individuals with educational campaign and treatment strategies. We analyzed the model by non-dimensionalizing the system of equations of our SIR epidemic model and derived our basic reproduction number.We aim to minimize the total number of infective individuals and the cost associated with the use of educational campaign and treatment on [0, T ]. We used Pontryagin's maximum principle to characterize the optimal levels of the two controls. The resulting optimality system is solved numerically. The results show that the optimal combination of treatment and educational campaign strategy required to achieve the set objective will depend on the relative cost of each of the control measures. The results from our simulation is discussed.

SUMMARYWe study the stability of a delay susceptible-infective-recovered epidemic model with time delay. The model is formulated under the assumption that all individuals are susceptible, and we analyse the global stability via two methods-by Lyapunov functionals, and-in terms of the variance of the variables. The main theorem shows that the endemic equilibrium is stable. If the basic reproduction number R 0 is less than unity, by LaSalle invariance principle, the disease-free equilibrium E s is globally stable and the disease always dies out. By applying the integral averaging theory, we also investigate the stability in variance of the model.

SUMMARYWe describe an SIR epidemic model with a discrete time lag, analyse the local stability of its equilibria as well as the effects of delay on the reproduction number and on the dynamical behaviour of the system. The model has two equilibria-a necessary condition for local asymptotic stability is given. The proofs are based on linearization and the application of Lyapunov functional approach. An upper bound of the critical time delay for which the model remains valid is derived. Numerical simulations are carried out to illustrate the effect of time delay which tends to reduce the epidemic threshold.

Abstract:We considered an MSEIR epidemic model as proposed by Hethcote (2000). Homotopy analysis method is used to solve the problem because of its advantages over methods.

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In [1], there was a typographical error in the entries of the off-diagonal elements of the matrix A(t), starting on the line before Equation (15). The purpose of this current note is to correct this mistake and propose a direction for future work. Choose W 1 , W 2 > 0 such that W 2 = W 1 e − h I * , and rewriting this equation with some little rearrangement,V(U(t)) now readṡwhere V(t, h) = col(U 1 (t),U 1 (t −h)), and A(t) =I * +S * −S * +(U 1 (t)+U 2 (t)) −S * +(U 1 (t)+U 2 (t)) I * +S * . Then, the determinant of A(t) is positive if and only if:=[(I * +2S * ) 2 −(−S * +(U 1 (t)+U 2 (t))) 2 ] > 0.That is, (I * +U 1 (t)+U 2 (t))(I * +2S * −(U 1 (t)+U 2 (t))) > 0, with solution satisfying the following inequality:Since is greater than zero, the matrix A is positive definite. For any > 0, considerOne can also see that direct computation of eigenvalues, say 1,2 of A(t) gives:1,2 (t) = I * +S * ±|S * −(U 1 (t)+U 2 (t))|. Since in 1 U 1 (t)+U 2 (t) S * , the minimum between the eigenvalues is min (t) = I * +U 1 (t)+U 2 (t), which is positive if U 1 (t)+ U 2 (t)<2S * . Hence, for any > 0, 1, ={(U 1 ,U 2 ) ∈ 1 : U 1 (t)+U 2 (t) > −2S * + }. It results that min (t) . Thus, = , provided that if (U 1 (0),U 2 (0)) ∈ 1, , then (U 1 (t),U 2 (t)) ∈ 1, for all t > 0. In future studies of similar problems, it would be interesting to prove global stability of the interior equilibrium by showing that solutions starting in 1 − 1, ultimately will enter 1, and remain there.