Abstract
We extend the deterministic model for the dynamics of toxoplasmosis proposed by Arenas et al. in 2010, by separating vaccinated and recovered classes. The model exhibits two equilibrium points, the disease-free and endemic steady states. These points are both locally and globally stable asymptotically when the threshold parameter Rv is less than and greater than unity, respectively. The sensitivity analysis of the model parameters reveals that the vaccination parameter $\pi$ is more sensitive to changes than any other parameter. Indeed, as expected the numerical simulations reveal that the higher the vaccination rate of susceptible individuals the smaller the value of the threshold Rv (i.e., increase in $\pi$ results in the decrease in Rv , leading to the eradication of toxoplasmosis in cats population.
Highlights
We extend the deterministic model for the dynamics of toxoplasmosis proposed by Arenas et al in 2010, by separating vaccinated and recovered classes
T. gondii is characterized as an intracellular parasite that lives in the host cell by regulating vital processes to acquire nutrients, guaranteeing its survival and evading the host immune system [12]
Ferreira et al [11], studied the oplasmosis include sulfonamides against murine toxoplasmo- dynamical behaviours of both deterministic and spatial models sis, combined therapy with sulfonamides and pyrimethamine as of toxoplasmosis disease in cat and human populations. They the standard therapy for toxoplasmosis in humans, spiramycin showed that the deterministic model exhibits a trans-critical biduring pregnancy to reduce transmission of the parasite from furcation and the system has no periodic orbits inside the posmother to fetus, clindamycin especially for patients allergic to itive octant
Summary
The DFE, P0 of model (2) is locally asymptotically stable on Ω if Rv < 1 and is unstable whenever Rv > 1. All the four eigenvalues of J(P0) have negative real parts and so, the DFE, P0 of the system (2) is locally asymptotically stable whenever Rv < 1. Global Stability of the DFE Using Lyapunov Principle method in conjunction with LaSalle’s. Invariant Principles [19], we establish the global asymptotic stability of the DFE as presented in the following theorem. If Rv 1, the DFE, P0 of model (2) is globally asymptotically stable on Ω. It follows from LaSalle’s Invariant Principles [19] that P0 is globally asymptotically stabile when Rv 1. Pe is the unique endemic equilibrium point of system (2) which exist if and only if Rv > 1. 21
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