Epidemic Influenza Model with Time Delay

In the previous chapter, two different types of vaccine were used for the two strain epidemic model; the results have demonstrated the significance of choosing the right vaccination. This chapter added delay to model (1), which is given in chapter 9. Here delay describes the incubation period. The model consists of four equilibrium points; disease-free equilibrium, endemic based to strain1, endemic with respect to strain2, and endemic with respect to both strains.

The global stability analysis of the equilibrium points was carried out through the use of Lyapunov functions. Two basic reproduction ratios r1and r2are found, and we have shown that if both are less than one, the disease dies out. If one of the ratios is less than one, an epidemic occurs with respect to the other. It was also shown that any strain with the highest basic reproduction ratio would automatically outperform the other strain, thereby eliminating it. Condition for the existence of endemic equilibria was also given.

Numerical simulations were carried out to support the analytic results and show the vaccine's effect for strain1 against strain 2 and the vaccine for strain 2 against strain 1. It is found that the population for infectives to strain 2 increases when the vaccine for strain1 is absent and vice versa.

Keywords: Basic Reproduction Ratios, Delay, Global Stability Analysis, Two Strain, Vaccine.