Co-infection model for Covid-19 and Rubella with vaccination treatment: stability and threshold

Abstract

This study aimed to explore a co-infection transmission model between Covid-19 and Rubella that involves administering vaccinations for both diseases. These two diseases not only have the same characteristics, but also has the same pattern in terms of the causes of disease, spread, clinical manifestations, and vaccines as prevention efforts. This model provided answers to the question of whether one of these diseases or both will disappear from the human population through several steps in the mathematical modelling analysis, which consists of: 1) critical point analysis, 2) stability analysis, 3) next generation matrix, and 4) threshold value analysis. This research resulted in four critical points, which is a critical point of disease-free, Rubella critical point, Covid-19 critical point, and the critical point for both diseases. Based on the next generation matrix and the disease-free critical point, two basic reproduction numbers were generated, namely  for Rubella and  for Covid-19. The first condition, R₀₁ less than 1 and R₀₂ less than 1, the disease-free critical point will be stable such that Rubella and Covid-19 will disappear from the human population. The second condition, R₀₁ greater than 1 and R₀₂ less than 1, the disease-free critical point become unstable, which means that Rubella-infected people will be found in the population. The third condition, R₀₁ less than 1 and R₀₂ greater than 1, Covid-19 will be found in the human population.