Yulin Zhao

Emerging disease dynamics in a model coupling within-host and between-host systems.

Epidemiological models and immunological models have been studied largely independently. However, the two processes (between- and within-host interactions) occur jointly and models that couple the two processes may generate new biological insights. Particularly, the threshold conditions for disease control may be dramatically different when compared with those generated from the epidemiological or immunological models separately. An example is considered in this paper for an environmentally driven infectious disease such as Toxoplasma gondii. The model explicitly couples the within-host and between-host dynamics. The within-host sub-system is linked to a contaminated environment E via an additional term g(E) to account for the increase in the parasite load V within a host due to the continuous ingestion of parasites from the contaminated environment. The parasite load V can also affect the rate of environmental contamination, which directly contributes to the infection rate of hosts for the between-host sub-system. When the two sub-systems are considered in isolation, the dynamics are standard and simple. That is, either the infection-free equilibrium is stable or a unique positive equilibrium is stable depending on the relevant reproduction number being less or greater than 1. However, when the two sub-systems are explicitly coupled, the full system exhibits more complex dynamics including backward bifurcations; that is, multiple positive equilibria exist with one of which being stable even if the reproduction number is less than 1. The biological implications of such bifurcations are illustrated using an example concerning the spread and control of toxoplasmosis.

Fast and slow dynamics of malaria model with relapse.

Two mathematical models of malaria with relapse are studied. When the vector population size is constant, complete analyses of the dynamics are conducted. The geometric singular perturbation theory is used to analyze the full dynamics. On the critical manifold, from next generation matrix method, we obtain the basic reproduction number. The global stability of disease-free equilibrium and the uniformly persistence of malaria have also been analyzed. While the vector population size is variable, the basic reproduction number and the stability of disease-free as well as the malaria-infected equilibrium have been obtained in a similar way. Some numerical simulations are also given.