Complex dynamics of a fractional-order monkeypox transmission system with saturated recovery function

Every country is continuously experiencing the monkeypox disease that started in the United Kingdom in May 2022. A detailed understanding of the transmission mechanism is required for controlling the disease. Based on real needs, we have developed an eight-compartmental mathematical model using a system of fractional-order differential equations to understand the behavior of the disease. The fractional-order model is adopted to discuss the effect of memory in reducing monkeypox infection. The next-generation matrix method determines the basic reproduction number for human beings and beasts, which are \(\mathscr {R}_0^m\) and \({\mathscr {R}}_0^b\), respectively. Depending on the numerical values of \({\mathscr {R}}_0^m\) and \({\mathscr {R}}_0^b\), the feasibility and the nature of the equilibrium points are studied. It is found that the system exhibits two transcritical bifurcations; one occurs at \({\mathscr {R}}_0^b=1\) for any value of \({\mathscr {R}}_0^m\), and the other occurs at \({\mathscr {R}}_0^m=1\) when \({\mathscr {R}}_0^b<1\). The effectiveness of the parameters has been discussed with the help of global sensitivity analysis. Further, we have investigated the optimal control policies, considering vaccination and treatment as two dynamic control variables. The incremental cost-effectiveness ratio and the infected averted ratio are determined to assess the cost-effectiveness of all practical control strategies. From the fractional-order optimal control problem, we have experienced that simultaneous use of both treatment and vaccination controls gives better results than using any single control in reducing infected humans. The global sensitivity analysis shows that controlling certain system parameters can regulate monkeypox infection. Further, our cost-effectiveness analysis shows that treatment control is the most cost-effective method for the monkeypox virus.