Analysis of a Multiscale Model for CholeraMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We analyze the asymptotic behavior of a multiscale cholera model where the population is structured by the pathogen load inside the individuals. A threshold for the pathogen load divides the population in susceptible and infected people. The pathogen load is increased by booster events (representing the uptake of pathogens from the environment) which describe a stochastic process. The immune response of the body degrades the pathogens time-continuously. The booster events depend on the pathogen in the environment which we assume to be constant due to time-scale arguments. On the population level, this results in a transport equation with non-local terms. We derive a non-negative stationary solution for the pathogen load of infected people. Furthermore, we prove the existence of a stable pathogen distribution for the susceptible people following the standard approach for fragmentation-aggregation equations from Gyllenberg and Heijmans. Combining the stationary solution for infected people and the pseudo-stationary solution for susceptible people weighted by the amount of infected respectively susceptible people depending on time, we obtain an ODE system (SI-model), which we hope will approximate the original dynamics well.