Sensitivity and stability analyses of COVID-19 model with waning immunity and relapse: a case study of Nigeria

Abstract

The Coronavirus Disease 2019 (COVID-19) experienced an exponential increase in cases between October 2023 and January 2024, with a growth factor of 6.53, following a period of gradual decline in previous years. Nigeria, a densely populated developing country in Africa with limited preparedness for epidemics, was caught unawares by this sudden re-emergence. This study investigates COVID-19 re-infections among individuals who had previously recovered or had been vaccinated, using Nigeria as case study. A deterministic compartmental model capturing the transmission, prevention, and control dynamics of COVID-19 was developed. The qualitative properties of the model, namely positivity and boundedness of its solutions, were established. Stability analysis of the model’s equilibria was conducted by deriving and analyzing the effective reproduction number, Re. The model was shown to exhibit a stable disease-free equilibrium when Re < 1 and a unique globally stable endemic equilibrium when Re > 1. Sensitivity analysis of Re was carried out by employing the forward index sensitivity approach. The results indicated that recruitment into the susceptible population and contact rate have unit sensitivity indices. Waning immunity of the vaccinated class was shown to have a direct relationship with Re, resulting in re-infections, whereas vaccination and recovery rates have an inverse relationship. Numerical simulations were conducted using the fourth-order Runge-Kutta method implemented via a MATLAB subroutine. Model parameters were estimated using data from the Nigeria Centre for Disease Control (NCDC) and weekly trends of the COVID-19 pandemic. Each parameter sensitive to Re was varied and their effect on the re-emergence of infection discussed. Results indicated that disease prevalence increases with higher rates of relapse, contact, and recruitment into the susceptible population and decreases with higher rates of treatment and vaccination