A mathematical model of transmission cycle of CC-Hemorrhagic fever via fractal–fractional operators and numerical simulations

Nowadays, the rapid spread of various tick-borne viruses has caused various diseases in the animal population of livestock and poultry, in which the human population is not safe. Crimean-Congo hemorrhagic fever is one of the common diseases between animals and humans that causes many deaths of both populations in the large areas of the world every year. Identification and control methods of this epidemic have led virologists to study the dynamics and behavior of these viruses in different transmission cycles in recent years based on mathematical models. In this paper, we present an advanced mathematical model of transmission cycle of viruses of the Crimean-Congo hemorrhagic fever between livestock, ticks and humans in a fractal–fractional system of six initial value problems. In fact, we extend the standard integer-order model to a two-parametric six-compartmental fractal–fractional hybrid model with power-law type kernels. To study the existence of solution for such a system, we first use a special family of contractions titled ϕ−ψ-contractions and also in the next step, we use the Leray–Schauder fixed point theorem. The Banach contraction principle helps us to prove the uniqueness of solutions. We try to investigate the stability behaviors of the solutions in the context of the Ulam's criterion, and then use Lagrange polynomials to obtain a numerical algorithm to find the approximate solutions of the mathematical model of Crimean-Congo hemorrhagic fever. Finally, by changing the values of the fractal dimension and fractional order in a closed interval, we analyze the convergence and stability of the solutions graphically. We see that all solutions have stable behaviors and at smaller fractal dimensions, decay and growth rates in susceptible and infected groups are slower, and vice versa. The accurate results of fractal–fractional operators in mathematical modeling motivate us to use them in different models.