Dynamics of a mathematical model of virus spreading incorporating the effect of a vaccine

TL;DR

This paper develops and analyzes a mathematical model of virus spread that includes vaccination and environmental infection, revealing conditions for backward bifurcations and comparing model predictions with COVID-19 data from Turkey.

Contribution

It introduces a novel epidemiological model incorporating imperfect vaccination and environmental infection, analyzing bifurcation phenomena and validating with real COVID-19 data.

Findings

Backward bifurcations can occur depending on the infection force model.

Multiple endemic steady states may exist under certain conditions.

Model predictions align with COVID-19 data from Turkey.

Abstract

The COVID-19 pandemic led to widespread interest in epidemiological models. In this context the role of vaccination in influencing the spreading of the disease is of particular interest. There has also been a lot of debate on the role of non-pharmaceutical interventions such as the disinfection of surfaces. We investigate a mathematical model for the spread of a disease which includes both imperfect vaccination and infection due to virus in the environment. The latter is studied with the help of two phenomenological models for the force of infection. In one of these models we find that backward bifurcations take place so that for some parameter values an endemic steady state exists although the basic reproduction ratio $R_0$ is less than one. We also prove that in that case there can exist more than one endemic steady state. In the other model all generic transcritical bifurcations are forward bifurcations so that these effects cannot occur. Thus we see that the occurrence of backward bifurcations, which can be important for disease control strategies, is dependent on the details of the function describing the force of infection. By means of simulations the predictions of this model are compared with data for COVID-19 from Turkey. A sensitivity analysis is also carried out.