Mathematical modeling of echinococcosis in humans, dogs and sheep

Abstract

In this study, mathematical models for the dynamics of cystic echinococcosis transmission in populations of dogs, sheep, and people are developed and analyzed. The predator-prey interaction in these populations is rst considered and analyzed. The primary objective of taking this model into account is to determine su cient conditions to ensure the existence of stable equilibrium point which represent coexistence of the three populations. A mathematical model for the dynamics of cystic echinococcosis transmission in the absence of controls is then formulated and analyzed. Analytically, the basic reproduction number R0 and equilibrium points are determined. To examine the dynamics of the disease, stability analysis of the disease free equilibrium and endemic equilibrium is carried out. The results show that the disease-free equilibrium is globally asymptotically stable if R0 1, and unstable otherwise. It is further demonstrated that the endemic equilibrium is asymptotically stable if R0 > 1. To support analytic results numerical simulations are carried out. Sensitivity analyses of the critical parameters are performed. In the result it is shown that the transmission rate of echinococcus' eggs from the environment to sheep ( es) is the most in uential parameters in the dynamics of cystic echinococcosis. To this e ect, a model for the spread of cystic echinococcosis under interventions that involve vaccination of sheep and cleaning or disinfection of the environment is formulated and studied. The disease-free and endemic equilibrium points of the model are calculated. The control reproduction number Rc for the deterministic model is derived, and the global dynamics are established by the values of Rc. The disease-free equilibrium is globally asymptotically stable if and only if the control reproduction number Rc 1, and the disease will be wiped out of the populations. For Rc > 1, using Volterra-Lyapunov stable matrices, it is proven that the endemic equilibrium is globally asymptotically stable, and the disease persists. Sensitivity analyses on the control reproduction number Rc is carried out. It is revealed that the transmission rate from sheep to dog ( sd) is the most in uential parameter in the dynamics of cystic echinococcosis. To illustrate the analytical results and establish the long term behavior of the disease numerical simulations are performed. The impact of control strategies is investigated. It is shown that, whenever vaccination of sheep is carried out solely or in combination with cleaning or disinfection of the environment, transmission of cystic echinococcosis can be controlled. However, with cleaning or disinfecting of the environment alone, the disease persists in the populations. Furthermore, an optimal control approach is applied to a model of cystic echinococcosis in the populations of sheep, dog and human. The main objective is to reduce or eliminate the disease from the three populations while minimizing the intervention implementation costs. We used Pontryagin's Minimum Principle to solve the optimal control problem. Numerical simulations of the time evolution of infected sheep, dog and human populations are provided to illustrate the e ects of optimal and constant controls. It is noticed that optimal control strategy is better than the small amount constant controls in reducing the prevalence of the disease in the populations. While time independent control(s) is(are) administered at maximum amount, it is also noticed that the optimal control strategy is e ective as the time-dependent controls. We also calculate the Incremental Cost Effectiveness Ratio(ICER) to investigate the cost effectiveness of these strategies. Our results show that the most cost-effective strategy for cystic echinoccosis control is the combination of vaccination of sheep and cleaning or disinfection of the environment.