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

• Studied a model coupling within- and between-host disease dynamics.
• Applied the model results to toxoplasmosis.
• The coupled model can generate new disease dynamics which are absent without coupling.
• The appearance of a backward bifurcation makes it challenging for disease control.

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.