Global dynamics of a diffusion-advection cholera model with vibrio age structure

For a multigroup cholera model with indirect transmission, the infection for a susceptible person is almost invariably transmitted by drinking contaminated water in which pathogens,V. cholerae, are present. The basic reproduction numberℛ0is identified and global dynamics are completely determined byℛ0. It shows thatℛ0is a globally threshold parameter in the sense that if it is less than one, the disease-free equilibrium is globally asymptotically stable; whereas if it is larger than one, there is a unique endemic equilibrium which is global asymptotically stable. For the proof of global stability with the disease-free equilibrium, we use the comparison principle; and for the endemic equilibrium we use the classical method of Lyapunov function and the graph-theoretic approach.

The spread and control of influenza are inevitably influenced by the age heterogeneity of the population. This paper investigates an age-structured SVEIQHR influenza model with physiological age structure, which includes the self-healing ability of latent individuals and the choice of home care or going to the hospital for infected individuals. First, the existence and uniqueness of the solution are proved using the operator semigroup theory. The expression of the basic reproduction number [Formula: see text] is derived through the characteristic equation. Then, we prove that the disease-free steady state is globally asymptotically stable if [Formula: see text]. There exists a unique endemic steady state, which is locally asymptotically stable when [Formula: see text]. Additionally, in order to minimize the number of infections at the lowest cost, we explored the optimal control measures for influenza by selecting strengthened vaccination and antiviral treatment as control variables. Finally, numerical simulations are conducted based on actual data. Our results show that it is essential to consider self-healing and home care for the prevention and control of influenza. Furthermore, vaccination and antiviral treatment can effectively prevent influenza infection and reduce the spread of influenza in the population.

Cholera is a common infectious disease caused by Vibrio cholerae, which has different infectivity. In this paper, we propose a cholera model with hyperinfectious and hypoinfectious vibrios, in which both human-to-human and environment-to-human transmissions are considered. By analyzing the characteristic equations, the local stability of disease-free and endemic equilibria is established. By using Lyapunov functionals and LaSalle's invariance principle, it is verified that the global threshold dynamics of the model can be completely determined by the basic reproduction number. Numerical simulations are carried out to illustrate the corresponding theoretical results and describe the cholera outbreak in Haiti. The study of optimal control helps us seek cost-effective solutions of time-dependent control strategies against cholera outbreaks, which shows that control strategies, such as vaccination and sanitation, should be taken at the very beginning of the outbreak and become less necessary after a certain period.