Dynamic analysis and optimal control of leptospirosis based on Caputo fractional derivative

Caputo fractional derivative solves the fractional initial value problem in Riemann-Liouville (R-L) fractional calculus. The definition of a Caputo-type derivative is in the same form as the definition of an integral differential equation, including the restriction of the value of the integral derivative to the value of the unknown function at the endpoint $t=a$. Therefore, this paper introduced the Caputo fractional derivative (CFD) to establish the transmission model of leptospirosis. First, to ensure that the model had a particular significance, we proved the dynamic properties of the model, such as nonnegative, boundedness, and stability of the equilibrium point. Second, according to the existence mode and genetic characteristics of pathogenic bacteria of leptospirosis, and from the perspective of score optimal control, we put forward measures such as wearing protective clothing, hospitalization, and cleaning the environment to prevent and control the spread of the disease. According to the proposed control measures, a control model of leptospirosis was established, and a forward-backward scanning algorithm (FB algorithm) was introduced to optimize the control function. Three different disease control strategies were proposed. Finally, the numerical simulation of different fractional orders used the fde12 (based on Adams–Bashforth–Moulton scheme) solver. The three optimized strategies, A, B, and C, were compared and analyzed. The results showed that the optimized control strategy could shorten the transmission time of the disease by about 80 days. Therefore, the above methods contributed to the study of leptospirosis and the World Health Organization.