Analysis of a non-integer order compartmental model for cholera and COVID-19 incorporating human and environmental transmissions

Fractional differential operators have increasingly gained wider applications in epidemiological modelling due to their ability to capture memory effect in their definitions; an attribute which lacks in the concept of classical integer derivatives. In this paper, employing the Caputo fractional operator with singular kernel, the co-dynamical model for cholera and COVID-19 diseases is proposed and analyzed, incorporating both direct and indirect transmission routes for cholera. The necessary conditions for existence of the unique solution of the proposed model are studied. Using the results from fixed point theory, Ulam-Hyers stability analysis of the system is performed. The model is fitted to real data from Pakistan and the optimized order of the fractional derivative for which the system fits well to data is obtained. Other numerical assessments of the model are also executed. Phase portraits of the infected classes with different initial conditions and various order of the fractional derivative are obtained in the cases when the reproduction number R-0<1 and R-0>1 . It is observed that the trajectories for the infected compartments tend towards the infection-free steady state when R-0<1 and the endemic steady state when R-0>1 irrespective of the initial conditions and the order of the fractional derivative. Increment in the COVID-19 vaccine efficacy, keeping the vaccination rate fixed at delta(omega)(1)=0.8 , resulted in a decline in the COVID-19 disease. Also, increasing the COVID-19 vaccination rate, keeping the vaccine efficacy for COVID-19 fixed at phi(omega)(c)=0.85 , led to a decline in the COVID-19 associated reproduction number. The simulations also pointed out the impact of COVID-19 and cholera vaccinations, direct and indirect transmissions of cholera.