A mathematical model for the control of carrier-dependent infectious diseases with direct transmission and time delay

In this paper, a non-linear delay mathematical model for the control of carrier-dependent infectious diseases through insecticides is proposed and analyzed. In the modeling process, it is assumed that disease spreads due to direct contact between susceptibles and infectives as well as through carriers (indirect contact). Further, it is assumed that insecticides are used to kill carriers and the rate of introduction of insecticides is proportional to the density of carriers with some time lag. The model analysis suggests that as delay in using insecticides exceeds some critical value, the system loses its stability and Hopf-bifurcation occurs. The direction, stability and period of the bifurcating periodic solutions arising through Hopf-bifurcation are also analyzed using normal form concept and center manifold theory. Numerical simulation is carried out to confirm the obtained analytical results.