At present, a large number of people worldwide have been infected by coronavirus 2019 (COVID-19). When the outbreak of the COVID-19 pandemic begins in a country, its impact is disastrous to both the country and its neighbors. In early 2020, the spread of COVID-19 was associated with global aviation. More recently, COVID-19 infections due to illegal or undocumented immigration have played a significant role in spreading the disease in Southeast Asia countries. Therefore, the spread of COVID-19 of all countries’ border should be curbed. Many countries closed their borders to all nations, causing an unprecedented decline in global travel, especially cross-border travel. This restriction affects social and economic trade-offs. Therefore, immigration policies are essential to control the COVID-19 pandemic. To understand and simulate the spread of the disease under different immigration conditions, we developed a novel mathematical model called the Legal immigration and Undocumented immigration from natural borders for Susceptible-Infected-Hospitalized and Recovered people (LUSIHR). The purpose of the model was to simulate the number of infected people under various policies, including uncontrolled, fully controlled, and partially controlled countries. The infection rate was parameterized using the collected data from the Department of Disease Control, Ministry of Public Health, Thailand. We demonstrated that the model possesses nonnegative solutions for favorable initial conditions. The analysis of numerical experiments showed that we could control the virus spread and maintain the number of infected people by increasing the control rate of undocumented immigration across the unprotected natural borders. Next, the obtained parameters were used to visualize the effect of the control rate on immigration at the natural border. Overall, the model was well-suited to explaining and building the simulation. The parameters were used to simulate the trends in the number of people infected from COVID-19.

COVID-19 is a respiratory disease that can spread rapidly. Controlling the spread through vaccination is one of the measures for activating immunization that helps to reduce the number of infected people. Different types of vaccines are effective in preventing and alleviating the symptoms of the disease in different ways. In this study, a mathematical model, SVIHR, was developed to assess the behavior of disease transmission in Thailand by considering the vaccine efficacy of different vaccine types and the vaccination rate. The equilibrium points were investigated and the basic reproduction number R0 was calculated using a next-generation matrix to determine the stability of the equilibrium. We found that the disease-free equilibrium point was asymptotically stable if, and only if, R0<1, and the endemic equilibrium was asymptotically stable if, and only if, R0>1. The simulation results and the estimation of the parameters applied to the actual data in Thailand are reported. The sensitivity of parameters related to the basic reproduction number was compared with estimates of the effectiveness of pandemic controls. The simulations of different vaccine efficacies for different vaccine types were compared and the average mixing of vaccine types was reported to assess the vaccination policies. Finally, the trade-off between the vaccine efficacy and the vaccination rate was investigated, resulting in the essentiality of vaccine efficacy to restrict the spread of COVID-19.