Dynamics of an Epidemic Model under the Influence of Environmental Stress

Abstract

We have considered a compartmental epidemiological model with infectious disease to observe the influence of environmental stress on disease transmission. The proposed model is well-defined as the population at each compartment remains positive and bounded with time. Dynamical behaviour of the model is observed by the stability and bifurcation analysis at the equilibrium points. Also, numerical simulation supports the theoretical proofs and the result shows that the system undergoes a forward bifurcation around the disease-free equilibrium. Our results indicate that with the increase of environmental pollution, the overall infected population increases. Also, the disease transmission rate among the susceptible and stressed population from asymptomatically infected individuals plays a crucial role to make a system endemic. A corresponding optimal control problem has also been proposed to control the disease prevalence as well as to minimize the cost by choosing the vaccination policy before being infected and treatment policy to the infected as control variables. Numerical figures indicate that the vaccination provided to susceptible needs some time to reduce the disease transmission but the vaccination provided to stressed individuals works immediately after implementation. The treatment policy for symptomatically infected individuals works with a higher rate at an earlier stage but the intensity decreases with time. Simultaneous implementation of all control interventions is more useful to reduce the size of overall infective individuals and also to minimize the economic burden. Hence, this research clearly expresses the impact of environmental pollution (specifically the influence of environmental stress) on the disease transmission in the population.

Mathematical model to assess the impact of contact rate and environment factor on transmission dynamics of rabies in humans and dogs

“…If , it means that every infectious individual will cause more than one secondary infection, leading to the persistence of rabies in the entire population. In order to determine , the next generation matrix, as applied by [28] , [29] , [30] , and the Jacobian Matrix are used, such that where is the new infections in the compartment i while and are the transfer terms in and out of the compartment i , respectively. From equation (5) , we define and by The Jacobian Matrices F and V at the disease free equilibrium point are given by equation (25) : …”