The outburst of the CoVid-19 pandemic has raised several questions leading to a complex system in terms of modeling. Indeed, the modeling of the epidemic, at the level of a country, needs considering each of the different sources of contamination as well as the public health authorities' strategy, in a specific way. With this in mind, in the present paper, we develop a mathematical model of the CoVid-19 epidemic in Senegal. In the model, the population is subdivided into five compartments: susceptible, infected but asymptomatic, symptomatic, quarantined, and recovered immune people. In addition, due to its important impact on the propagation of the disease, we add one more variable: the number of infected objects. %\vspace{1mm} Therefore, the model corresponds to a system of six non-linear ordinary differential equations we submit to an analytical study to prove the relevancy of the model, simulate the evolution of the epidemic, and retrieve epidemiological parameters, namely the infection rate and the basic reproduction number, ${\cal R}_0$. Based on the senegalese territory CoVid-19 data, we simulate various scenarios as for the evolution of the epidemic in the country, in order to predict the peak and its magnitude with regard to the application of barrier measures. We also explore the option of collective immunity with special protection for vulnerable people. In doing so, non-available parameters are identified using some mathematical identification technics.

A general mathematical model for schistosomiasis is formulated that incorporates the miracidia and cercariae dynamics, since parasites play an important role in the transmission dynamics of schistosomiasis. This model is an extension of the study undertaken in Diaby etc[?] concerning the evolution of a schistosomiasis infection. Meanwhile, optimal control theory is applied to the proposed model. In the first part of our analysis we describe and propose a complete mathematical analysis of a new mathematical model for schistosomiasis infection with fixed control for both drug and biological treatment. It also includes a net inflow of competitor snails into the aquatic region at the rate u per unit of time as control term. Schistosomiasis is associated with water resource development such as dams and irrigation schemes, where the snail is the intermediate host of the parasite breeds. The snail intermediate host breeds in slow-flowing or stagnant water. We establish a deterministic model to explore the role of biological control strategy. We derive the basic reproduction number R₀ and establish that the global dynamics are completely determined by the values of R₀. It is shown that the disease can be eradicated when R₀ ≤ 1. In the case where R₀ > 1, we prove the existence, uniqueness and global asymptotic stability of an endemic equilibrium. We also show how the control u can be chosen in order to eradicate the disease. In the second part, we take the controls as time dependent and obtain the optimal control strategy to minimize both infected humans and snails populations. All the analytical results are verified by simulations works. Some important conclusions are given at the end of the paper.