Optimal vaccination and treatment policies with constrained inequalities to study limited vaccination resources for a multistrain reaction–diffusion SEIR model of COVID-19

The availability of an effective vaccine is a major challenge in COVID-19 vaccination campaigns. In this article, we propose a comprehensive strategy incorporating distributed controls via vaccination and treatment while considering the limitations of available resources expressed as inequalities constrained. Our approach utilizes a spatiotemporal model that includes multiple strains of the virus and considers spatial heterogeneity, providing a more accurate representation of disease transmission. By using a generalized two-strain ODE model, we effectively capture the evolution of COVID-19, including various strains and variants. To validate our model, we establish the positivity and boundedness of the system’s solution. Through the exploration of vaccination and treatment controls. Our study aims to reduce the spread of the disease between strains and identify optimal values that minimize the number of infected individuals and associated costs. This work provides useful insights into effective strategies for controlling the COVID-19 epidemic, even with limited vaccination resources. Furthermore, we demonstrate the existence of controls and describe their characterizations in terms of state and adjoint functions. The penalty method is used to treat the inequalities constrained. More importantly, our numerical simulation shows that adequate vaccination resources and simultaneous treatment can effectively reduce and even eliminate the spread of COVID-19.