This paper proposes and analyses a new fractional-order SIR type epidemic model with a saturated treatment function. The detailed dynamics of the corresponding system, including the equilibrium points and their existence and uniqueness, uniformboundedness, and stability of the solutions are studied. The threshold parameter, basic reproduction number of the system which determines the disease dynamics is derived, and the condition of occurrence of backward bifurcation is also determined. Some numerical works are conducted to validate our analytical results for the commensurate fractional-order system. Hopf bifurcations for the fractional-order system are studied by taking the order of the fractional differential as a bifurcation parameter.

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Presently, monkeypox virus infection has spread worldwide in the ongoing outbreak that began in the UK. To study the transmission dynamics of monkeypox, we formulate here a seven-compartmental (five compartments for the human population and two compartments for animals or rodents) fractional-order mathematical model. The existence and uniqueness of the solution of the proposed fractional order model are examined here. The basic reproduction number for humans ($\mathfrak{R}_0^h$) and animals ($\mathfrak{R}_0^a$) are obtained through the next-generation matrix approach. Depending on the values of $\mathfrak{R}_0^h$ and $\mathfrak{R}_0^a$, we observed that the fractional order model has three equilibria, namely, monkeypox-free equilibrium, animal-free endemic equilibrium, and endemic equilibrium. Also, the stability of all equilibria is checked in this present article. We found that the model goes through transcritical bifurcation at $\mathfrak{R}_0^a=1$ for any values of $\mathfrak{R}_0^h$ and at $\mathfrak{R}_0^h=1$ for $\mathfrak{R}_0^a<1$. Best of our knowledge, this is the first work where the fractional order optimal control for monkeypox is formulated and solved considering vaccination and treatment controls. Several feasible parameter values are used in the simulations to visualize and verify the findings, from which the results show that fractional order is more appropriate. Finally, parameters involved in the expression of $\mathfrak{R}_0^h$ and $\mathfrak{R}_0^a$ are scaled using the sensitivity index approach.

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