Analysis of backward bifurcation induced by nonlinear vertical transmission in COVID-19 dynamics with optimal control

A mathematical model is designed to investigate transmission dynamics of COVID-19 in a setting where both horizontal and vertical transmission of COVID-19 occur concurrently. The model incorporated two nonlinear vertical transmission routes of COVID-19, namely transmission of the COVID-19 to the fetus and also due to exposure at birth. Theoretical analysis revealed that nonlinear vertical transmission induces backward bifurcation phenomenon where COVID-19 exist even when basic reproduction number is below unity. Other bifurcation structure exhibited by the model is the atypical hysteresis effect. Time profiles disclosed that saturating constants associated with nonlinear vertical transmission rates cause a delay in the COVID-19 peak. The nonlinear autonomous model was transformed to a nonlinear non-autonomous model through incorporation of time-dependent controls. Numerical findings showed that optimal vaccination of susceptible individuals and optimal management of infectious cohort have the most beneficial impact, while monitoring of recovered individuals with the aim of preventing loss of infection-acquired immunity has the least benefit. Further, combination of optimal vaccination of susceptible and optimal management of infectious individuals was shown to have the highest efficiency index, demonstrating the capacity of the combined control measures to avert the most COVID-19 cases in the population.