This paper considers the Susceptible-Infected-Vaccinated-Recovered (SIRV) deterministic model with a non linear force of infection and treatment, where individual humans that are vaccinated losses their vaccination after some time and become vulnerable to infections. The basic reproduction number R 0 obtained from the model system is an epidemic threshold that determines if a disease will continue to ravage the human population or not. The model state equations considered in this paper possess two steady-state solutions such that if R 0 < 1, the infection-absent steady-state solutions are locally and globally asymptotically stable. Also, if R 0 > 1, a unique infection-persistent steady-state solutions are established, which is also locally and globally asymptotically stable. Thus, it leads to the persistence of infections in the human host population. Finally, numerical simulations were carried out to validate our theoretical results.

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In this paper, a new conjugate gradient (CG) parameter is proposed through the convex combination of the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP) CG update parameters such that the conjugacy condition of Dai-Liao is satisfied. The computational efficiency of the PRP method and the convergence profile of the FR method motivated the choice of these two CG methods. The corresponding CG algorithm satisfies the sufficient descent property and was shown to be globally convergent under the strong Wolfe line search procedure. Numerical tests on selected benchmark test functions show that the algorithm is efficient and very competitive in comparison with some existing classical methods.