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In this paper, we consider an endemic Susceptible-Exposed-Infectious-Recovered model with continuous age-structure for exposed and infectious classes. This work is appropriate for those infectious diseases that have exposed period such as tuberculosis and herpes virus infection. Key result is the existence of an optimal control problem. To control the spread of the disease, we use an objective functional with a suitable control and present an optimal control problem.Finally, we present some simulations that support our theoretical findings and show the effectiveness of the model.

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In this article, we extend the concept of triple Laplace transform to the solution of fractional order partial differential equations by using Caputo fractional derivative. The concerned transform is applicable to solve many classes of partial differential equations with fractional order derivatives and integrals. As a consequence, fractional order telegraph equation in two dimensions is investigated in detail and the solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. The same problem is also solved by taking into account the Atangana-Baleanu fractional derivative. Numerical plots are provided for the comparison of Caputo and Atangana-Baleanu fractional derivatives.