Mathematical analysis of the dynamics of a fractional-order tuberculosis epidemic in a patchy environment under the influence of re-infection

In human societies, memory has a significant impact on the procedure of progression and control of epidemics. Therefore, fractional differential equations govern an epidemic model, including the memory effect. Since infectious diseases can spread from one place to another, human movement impacts the prevalence of such diseases. In this study, we present a two-patch model for tuberculosis with fractional order and exogenous re-infection, in which susceptible individuals can move without any obstacle between the patches. The present work establishes the effect of population movement and the backward bifurcation phenomenon on the prevalence of the tuberculosis epidemic with fractional-order derivatives. Furthermore, to show that the proposed model has backward bifurcation, we provide some conditions between the model's parameters, so in this case, the model of tuberculosis has numerous boundary points. We show the tuberculosis model can exhibit backward bifurcation neither in case of exogenous re-infection, with some conditions, nor in the situation that the model is vacant from any re-infection. In contrast, the disease-free equilibrium of the model is globally asymptotically stable when R0 & LE;1$$ {R}_0\le 1 $$. Furthermore, we investigate the fractional-order derivative & alpha;$$ \alpha $$ on the spread of infection. The results show that & alpha;$$ \alpha $$ can perform the function of adventure or expertise of people about the disease background.