Optimal control and cost-effectiveness analysis of anthrax epidemic model

Anthrax is an acute infectious zoonotic disease caused by the spore-forming Bacillus anthracis, a sporulating, nonmotile, gram-positive rod-shaped bacterium. The disease primarily affects domestic and wild herbivores and continues to threaten human health globally. Aside from causing severe and lethal infections in human and animal populations, an outbreak of anthrax may lead to large economic losses to livestock owners. In this research article, we proposed a nonlinear differential equation model for studying the transmission dynamics of anthrax in animals and humans with optimal control analysis taking into account a vaccination programme aimed at reducing the susceptibility of the animal population. The model system exhibits two distinctive equilibrium points, these are the anthrax-free equilibrium point and anthrax endemic equilibrium point. The anthrax-free equilibrium point is proved to be locally and globally asymptotically stable whenever the basic reproduction number is less than one (R0<1) and the endemic equilibrium point is locally asymptotically stable whenever the basic reproduction number is greater than unity (R0>1). The sensitivity analysis revealed that the factors that have higher influence on the spread of anthrax are the recruitment rate of animals Λa, pathogen shedding rate ξa, infection rate of susceptible animals βa, vaccination rate ρa, practicing of proper environmental hygiene θ and recovery rate in animals ϕa. The numerical simulation results reveal that managing the disease without optimal control is costly. It further demonstrates that the combined effect of vaccination, treatment of infected animals and practicing of proper environmental hygiene, comparatively has the ability to reduce the number of infected animals and humans as well as pathogens in the environment within a short period of time. The cost-effectiveness analysis also suggests that the combination of vaccination of animals, treatment of infected animals, and practicing of environmental hygiene is the most cost-effective strategy for combating anthrax epidemic.