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.

In this paper, we propose a two-patch model with border control to investigate the effect of border control measures and local non-pharmacological interventions (NPIs) on the transmission of COVID-19. The basic reproduction number of the model is calculated, and the existence and stability of the boundary equilibria and the existence of the coexistence equilibrium of the model are obtained. Through numerical simulation, when there are no unquarantined virus carriers in the patch-2, it can be concluded that the reopening of the border with strict border control measures to allow people in patch-1 to move into patch-2 will not lead to disease outbreaks. Also, when there are unquarantined virus carriers in patch-2 (or lax border control causes people carrying the virus to flow into patch-2), the border control is more strict, and the slower the growth of number of new infectious in patch-2, but the strength of border control does not affect the final state of the disease, which is still dependent on local NPIs. Finally, when the border reopens during an outbreak of disease in patch-2, then a second outbreak will happen.

Among many epidemic prevention measures, isolation is an important method to control the spread of infectious disease. Scholars rarely study the impact of isolation on disease dissemination from a quantitative perspective. In this paper, we introduce an isolation ratio and establish the corresponding model. The basic reproductive number and its biological explanation are given. The stability conditions of the disease-free and endemic equilibria are obtained by analyzing its distribution of characteristic values. It is shown that the isolation ratio has an important influence on the basic reproductive number and the stability conditions. Taking the COVID-19 in Wuhan as an example, isolating more than 68% of the population can control the spread of the epidemic. This method can provide precise epidemic prevention strategies for government departments. Numerical simulations verify the effectiveness of the results.

A new modification of the SIR epidemic model incorporating vaccination is proposed in the present paper. The recent trend of vaccinating against COVID-19 pandemic reveals a strong control of infectious disease. On the other hand, it is observed in some countries that, the vaccine application offers less control over the spread of virus, since some portion of vaccinated people is not totally protected/immuned and viable to infection again after a while due to weak/loss immunity offered by the vaccine. This requires transition from vaccinated department to infected for COVID-19. This character of COVID-19 helps us reconsideration of the vaccinated department by letting some part of it being exposed to the infection again. Taking this into account, as a result of modification of the SIR model, the epidemiology is now governed with three main epidemic dimensionless parameters, having provided an initial fraction of infected individuals. The dimensionless model with these parameters is analyzed initially from the stability point of view. The effects of weak immunity are then illustrated numerically on some chosen parameter range. How some of the countries applying the COVID-19 vaccine programs affected by weak/loss immunity is eventually examined with the modified model. The rate of vaccination as well as the basic Reproduction number are found to affect the epidemic demography of the population subject to weak or loss of immunity. In the case of a high vaccination rate, the countries are not anticipated to be highly influenced by the weak immunity of low level, whereas weak immunity prolongs the contagious disease by appearance of secondary multiple peaks in the epidemic compartments with relatively small vaccination rates and basic Reproductive numbers.