Modeling and Analysis of an Age-Structured Malaria Model in the Sense of Atangana–Baleanu Fractional Operators

Abstract

In this paper, integer- and fractional-order models are discussed to investigate the dynamics of malaria in a human host with a varied age distribution. A system of differential equation model with five human state variables and two mosquito state variables was examined. Preschool-age (0–5) and young-age individuals make up our model’s division of the human population. We investigated the existence of an area in which the model is both mathematically and epidemiologically well posed. According to the findings of our mathematical research, the disease-free equilibrium exists whenever the fundamental reproduction number $R_0$ is smaller than one and is asymptotically stable. The disease-free equilibrium point is unstable when $R_0>1$. We showed that the endemic equilibrium point is unique for $R_0>1$. Also, the most influential control parameters of the spread of malaria were identified. Numerical simulations of both classical and fractional order were conducted, and we used ODE (45) for classical part and numerical technique developed by Toufik and Atangana for fractional order. The infected population will grow because of the high biting frequency of the mosquito and the high likelihood of transmission from the infected mosquito to the susceptible human. $R=1.622$, which is more than one, indicating that the mosquito vector keeps on growing. This supports the stability of the endemic equilibrium point theorem, which states that the disease becomes endemic when $R=1$. The susceptible human population will decrease because of the presence of the infective mosquito, which has a high biting frequency for the first couple of days. Since the infective mosquito bit the susceptible human, the susceptible human became infected and went to the infected human compartments. Then, the susceptible population will decrease and the infested human population will increase. After a certain amount of time, it becomes zero due to the growth of protected classes. In this case, a disease-free equilibrium point exists and is stable. This condition exists because $R_0=2.827\times {10}^{-5}$ is less than 1. This supports the theorem that the stability of the disease-free equilibrium point is obtained when $R_0<1$. Depending on equation, we have shown that the possibility of some endemic equilibria exists when $R_0<1$, that is, it undergoes backward bifurcation, even when the disease-free equilibrium is locally stable, and the result means that the society may misunderstand the level of malaria prevalence in the community.