Bifurcation and optimal control analysis of delayed models for huanglongbing

In this paper, a delayed differential model of citrus Huanglongbing infection is analyzed, in which the latencies of the citrus tree and Asian citrus psyllid are considered as two time delay factors. We compute the equilibrium points and the basic reproductive numbers with and without time delays, i.e. [Formula: see text] and [Formula: see text], and then show that [Formula: see text] completely determines the local stability of the disease-free equilibrium. Moreover, the conditions for the existence of transcritical bifurcation are derived from Sotomayor’s Theorem. The stability of the endemic equilibrium and the existence of Hopf bifurcation are investigated in four cases: (1) [Formula: see text], (2) [Formula: see text], (3) [Formula: see text] and (4) [Formula: see text]. Optimal control theory is then applied to the model to study two time-dependent treatment efforts and minimize the infection in citrus and psyllids, while keeping the implementation cost at a minimum. Numerical simulations of the overall systems are implemented in MatLab for demonstration of the theoretical results.

Dynamics of a dengue epidemic model with class-age structure

We introduce the class-age-dependent rates of the infected and vaccinated class in the compartmental model of dengue transmission. An age-structured host-vector interaction model incorporating vaccination effects is formulated and analyzed for the spread of dengue. Moreover, the basic reproduction number is derived, which serves as a threshold value determining the stability of the equilibrium points. By constructing suitable Lyapunov functional, the global asymptotic stability of the equilibria of the model is established in terms of the basic reproduction number. In particular, the disease-free equilibrium of the model is globally asymptotically stable if the basic reproduction number is less than one, while the disease persists and the unique endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than one. The analysis of our model indicates that our model is realistic to give a hint to control the transmission of dengue. Furthermore, it follows from the formulation of the infection-free equilibrium of susceptible humans [Formula: see text] and the basic reproduction number [Formula: see text] that both of them are decreasing with respect to the vaccination parameter [Formula: see text], which indicates that appropriate vaccinating program may contribute to prevent the transmission of Dengue disease.

Optimal control of a malaria model with asymptomatic class and superinfection

In this paper, we introduce a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations. The model assumes that asymptomatic individuals can get re-infected and move to the symptomatic class. In the case of an incomplete treatment, symptomatic individuals move to the asymptomatic class. If successfully treated, the symptomatic individuals recover and move to the susceptible class. The basic reproduction number, R₀, is computed using the next generation approach. The system has a disease-free equilibrium (DFE) which is locally asymptomatically stable when R₀<1, and may have up to four endemic equilibria. The model exhibits backward bifurcation generated by two mechanisms; standard incidence and superinfection. If the model does not allow for superinfection or deaths due to the disease, then DFE is globally stable which suggests that backward bifurcation is no longer possible. Simulations suggest that total prevalence of malaria is the highest if all individuals show symptoms upon infection, but then undergoes an incomplete treatment and the lowest when all the individuals first move to the symptomatic class then treated successfully. Total prevalence is average if more individuals upon infection move to the asymptomatic class. We study optimal control strategies applied to bed-net use and treatment as main tools for reducing the total number of symptomatic and asymptomatic individuals. Simulations suggest that the optimal control strategies are very dynamic. Although they always lead to decrease in the symptomatic infectious individuals, they may lead to increase in the number of asymptomatic infectious individuals. This last scenario occurs if a large portion of newly infected individuals move to the symptomatic class but many of them do not complete treatment or if they all complete treatment but the superinfection rate of asymptomatic individuals is average.