Age-Infection Model and Control of Marek Disease

Abstract

We formulated three compartmental model of Marek Disease model. We first determined the basic Reproduction number and the existence of Steady (Equilibrium) states (disease-free and endemic). Conditions for the local stability of the disease-free and endemic steady states were determined. Further, the Global stability of the disease-free equilibrium (DFE) and endemic equilibrium were proved using Lyponav method. We went further to carry out the sensitivity analysis or parametric dependence on R0 and later formulated the optimal control problem. We finally looked at numerical Results on poultry productivity in the presence of Marek disease and we drew five graphs to demonstrate this. The first figure shows the effect of both vaccination (v) and biosecurity measures (u) on the latently infected birds. The population of infected birds increases speedily and then remains stable without the application of any control measure, with the controls, the population increases to about 145 and then begins to reduce from day 8 till it drops to 50 on day 20 and then remains stable. With this strategy, only bird vaccination (v) is applied to control the system while the other control is set to zero. In the second figure, the effect of bird vaccination and its’ positive impact is revealed, though there is an increase to about 160 before a decrease occurs. From the third figure, as the control (u) ranges from 0.2 to 0.9, we see that the bird population still has a high level of latently infected birds. This result from figure shows that the bird population is not free from the disease, hence, the biosecurity control strategy is not effective without vaccination of susceptible birds and hence it is not preferable as the only control measure for marek disease. The numerical result in the fourth figure shows that as the latently infected bird population increases without control, with vaccination it decreases as more susceptible birds are vaccinated. From the fifth figure we observe, that as the control parameter increases, the total deaths by infection reduces, also as the age of the infection increases to the maximum age of infection which is 6 months (thatis, T=24 weeks), the number of deaths increases to 30 in a day. Hence, control measures should be applied at the early ages of infection in order to avoid high mortality rate during the outbreak of the disease.