Mathematical Modeling of Toxoplasmosis in Cats with Two Time Delays under Environmental Effects

In this paper, we construct a more realistic mathematical model to study toxoplasmosis dynamics. The model considers two discrete time delays. The first delay is related to the latent phase, which is the time lag between when a susceptible cat has effective contact with an oocyst and when it begins to produce oocysts. The second discrete time delay is the time that elapses from when the oocysts become present in the environment to when they are able to infect. The main aim in this paper is to find the conditions under which the toxoplasmosis can disappear from the cat population and to study whether the time delays can affect the qualitative properties of the model. Thus, we investigate the impact of the combination of two discrete time delays on the toxoplasmosis dynamics. Using dynamical systems theory, we are able to find the basic reproduction number R-0(d) that determines the global long-term dynamics of the toxoplasmosis. We prove that, if R-0(d) < 1, the toxoplasmosis will be eradicated and that the toxoplasmosis-free equilibrium is globally stable. We design a Lyapunov function in order to prove the global stability of the toxoplasmosis-free equilibrium. We also prove that, if the threshold parameter R-0(d) is greater than one, then there is only one toxoplasmosis-endemic equilibrium point, but the stability of this point is not theoretically proven. However, we obtained partial theoretical results and performed numerical simulations that suggest that, if R-0(d) > 1, then the toxoplasmosis-endemic equilibrium point is globally stable. In addition, other numerical simulations were performed in order to help to support the theoretical stability results.