Modelling the role of cross-immunity between two different strains of leishmania

We develop a mathematical model to understand the role of cross-immunity between two different strains of leishmania  . Our results show that R0R0, the basic reproduction number of the model decreases as the cross-immunity increases; moreover, if the cross-immunity between the two different strains is perfect, then the disease-free equilibrium is globally asymptotically stable. Results also show that in the case of partial cross-immunity the model undergoes backward bifurcation, where R0<1R0<1 is not enough for eradication of the disease. We find that if the force of infection of both strains in not equal to zero, then both strains will coexist without competition. Numerical results show that at the endemic equilibrium the number of infected humans with one strain is higher than the number of infected humans with the same strain after they recovered from the infection of the other one, which shows that the cross-immunity reduces the number of infections.