Inference of an Epidemic Driven by Random Transmission Rate: Explaining the Dynamics of COVID-19 in Bogotá

The epidemic disease model is often used to predict an epidemic's characteristics and help plan an effective control strategy. Motivated by recent COVID-19 outbreaks, we develop and mathematically analyze a simple stochastic model that considers random perturbations in transmission rates and shows its validity using data from the Colombian city of Bogota. The stochastic epidemic model is stratified in the Susceptible-Exposed-Infectious-Recovered, \textit{SEIR}, type compartmental model with the randomness depicting the impact of stochastic variations due to external factors such as changes in environmental conditions and human behaviors that may impact the dynamics of an infectious disease. The analysis resulted in derivation of approximate distribution of \textit{eventual extinction of infection state}, \textit{persistence of infection in the mean}, and \textit{the quasi-stationary infectious state}). Finally, we illustrate inferences and model parameters using reported COVID-19 epidemic data from the Colombian city of Bogotá. The outbreak in Bogota is divided into six distinct periods, with transmission rates high during the initiation of the epidemic and even much higher during the last period, potentially due to the rapid spread of the Omicron variant in Colombia still has a low vaccination rate. Mathematics Subject Classification: 92D30