In this paper, we formulate a mathematical model of vector-borne disease dynamics. The model is constructed by considering two models : a baseline model of vector population dynamics due to Lutambi et al. that takes into account the development of the aquatic stages and the female mosquitoes gonotrophic cycle and an SI-SIR model describing the interaction between mosquitoes and human hosts. We briefly study the baseline model of vectors dynamics and, for the transmission model, we explicitly compute the equilibrium points, and by using the method of Van den Driesshe and J. Watmough, we derive the basic reproduction number ℛ0. Otherwise, thanks to Lyapunov’s principle, Routh-Hurwitz criteria and a favorable result due to Vidyasagar, we establish the local and global stability results of the equilibrium points. Furthermore, we establish an interesting relationship between the mosquito reproduction number ℛ
v
and the basic reproduction number ℛ0. It then follows that aquatic stages and behavior of adult mosquitoes have a significant impact on disease transmission dynamics. Finally, some numerical simulations are carried out to support the theoretical findings of the study.
Malaria is a major public health issue in many parts of the world, and the anopheles mosquitoes which drive transmission are key targets for interventions. Consequently, a best understanding of mosquito populations dynamics is necessary in the fight against the disease. Hence, in this paper we propose a delayed mathematical model of the life cycle of anopheles mosquitoes by using delayed-logistic population growth. The model is formulated by inserting the time delay into the logistic population growth rate, that accounts for the period of growth from eggs to the last aquatic stage, which is pupae. Depending on the system parameters, we establish a threshold for survival and extinction of the anopheles mosquitoes population. Moreover, by choosing the time delay as a bifurcation parameter, we prove that the system loses its stability and a Hopf bifurcation occurs when time delay passes through some critical values. Finally, we perform some numerical simulations and the results are well in keeping with the analytical analysis.
This paper presents a co-infection mathematical model of COVID-19 and TB to study their synergistic dynamics. We first investigated the single infection models of each disease and then the co-infection dynamics of the two diseases. Indeed, we calculated the basic reproduction number of each model, and then we studied the existence and the stability of the equilibrium points. We subsequently proved that the TB only infection model and the co-infection model exhibit backward and forward bifurcations. In addition, we performed a sensitivity analysis of the basic reproduction numbers to determine which parameters influence them the most. Furthermore, we applied Pontryagin’s maximum principle to our co-infection model to assess the impact of the use of an imperfect vaccine for COVID-19, taken as an optimal control strategy. Finally, we presented the results of the numerical simulations to support the theoretical findings.
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