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The objective of this article is to formulate a mathematical model to make a vaccination strategy to reduce outbreaks of influenza A (H1N1) via a fractional model, taking into account the time it takes for the vaccine to be active. For this purpose, the SIR model is modified by using the Caputo fractional derivative, unifying the unit of time on both sides of each equation, and adding the control variable with a time delay (the vaccine variable). Meanwhile, the theory of optimal control is used to construct an algorithm that enables us to determine the optimal vaccination strategy. The forward and backward Euler method has been used to find the optimal solutions numerically. The numerical simulation is based on data from Morocco's experience with influenza A (H1N1).

Severe acute respiratory syndrome (SARS) is a very dangerous disease that affects the human respiratory system. In this article, we discuss the optimal control of this disease via a fractional SVEIR epidemic model together with two control variables (treatment and vaccination). For this purpose, we first design a fractional optimal control problem and then apply Pontryagin's minimal principle in a fractional version to find the optimal control. Also, the forward and backward fractional Euler methods (FEM) are used to solve the state and co-state equations, respectively. The results gave a new treatment and vaccine strategy for breaking dawn and preventing the spread of SARS.

In this paper, we studied the neutral stochastic functional differential equations with infinite delay (NSFDEwID for short).The existence and uniqueness of solutions to NSFDEwID at the state space  have been addressed under the local Lipschitz condition and and Linear growth condition and we introduced a prove of lemma 3.4 because of it is presented in many articles without prove.