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ln this paper, we extend the classical SEIRS model within the framework of the fractional model in the sense of Caputo–Fabrizio derivative modified with an auxiliary parameter. This model describes the transmission of the influenza virus and shows the influence of the disease resistance and the form of the non-linearity of the incidence rate on the spread of the infectious disease. The existence and uniqueness of the variables of the given model are examined using the fixed point theorems. A detailed analysis of the stability of the disease-free equilibrium point was presented. Finally, using the three-step fractional Adams–Bashforth scheme, the profile of each state variable is depicted for different values of the fractional-order $$\alpha $$ and compared with the first-order derivative curves.

We consider a third-order three-point boundary value problem. We introduce a generalized polynomial growth condition to obtain the existence of a nontrivial solution by using Leray-Schauder nonlinear alternative, then we give an example to illustrate our results.