Human mobility has played a critical role in the spread of COVID-19. The understanding of mobility helps in getting information on the acceleration or control of the spread of disease. The COVID-19 virus has been spreading among several locations despite all the best efforts related to its isolation. To comprehend this, a multi-patch mathematical model of COVID-19 is proposed and analysed in this work, where-in limited medical resources, quarantining, and inhibitory behaviour of healthy individuals are incorporated into the model. Furthermore, as an example, the impact of mobility in a three-patch model is studied considering the three worst-hit states of India, i.e. Kerala, Maharashtra and Tamil Nadu, as three patches. Key parameters and the basic reproduction number are estimated from the available data. Through results and analyses, it is seen that Kerala has a higher effective contact rate and has the highest prevalence. Moreover, if Kerala is isolated from Maharashtra or Tamil Nadu, the number of active cases will increase in Kerala but reduce in the other two states. Our findings indicate that the number of active cases will decrease in the high prevalence state and increase in the lower prevalence states if the emigration rate is higher than the immigration rate in the high prevalence state. Overall, proper travel restrictions are to be implemented to reduce or control the spread of disease from the high-prevalence state to other states with lower prevalence rates.

When a disease spreads in a population, individuals tend to change their behavior due to the presence of information about disease prevalence. Therefore, the infection rate is affected and incidence term in the model should be appropriately modified. In addition, a limitation of medical resources has its impact on the dynamics of the disease. In this work, we propose and analyze an Susceptible-Exposed-Infected-Recovered (SEIR) model, which accounts for the information-induced non-monotonic incidence function and saturated treatment function. The model analysis is carried out, and it is found that when R0 is below one, the disease may or may not die out due to the saturated treatment (i.e., a backward bifurcation may exist and cause multi-stability). Further, we note that in this case, disease eradication is possible if medical resources are available for all. When R0 exceeds one, there is a possibility of the existence of multiple endemic equilibria. These multiple equilibria give rise to rich and complex dynamics by showing various bifurcations and oscillations (via Hopf bifurcation). A global asymptotic stability of a unique endemic equilibrium (when it exists) is established under certain conditions. An impact of information is shown and also a sensitivity analysis of model parameters is performed. Various cases are considered numerically to provide the insight of model behavior mathematically and epidemiologically. We found that the model shows hysteresis. Our study underlines that a limitation of medical resources may cause bi(multi)-stability in the model system. Also, information plays a significant role and gives rise to a rich and complex dynamical behavior of the model.

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