Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model

Improved uniform persistence for partially diffusive models of infectious diseases: cases of avian influenza and Ebola virus disease

Past works on partially diffusive models of diseases typically rely on a strong assumption regarding the initial data of their infection-related compartments in order to demonstrate uniform persistence in the case that the basic reproduction number $ \mathcal{R}_0 $ is above 1. Such a model for avian influenza was proposed, and its uniform persistence was proven for the case $ \mathcal{R}_0 > 1 $ when all of the infected bird population, recovered bird population and virus concentration in water do not initially vanish. Similarly, a work regarding a model of the Ebola virus disease required that the infected human population does not initially vanish to show an analogous result. We introduce a modification on the standard method of proving uniform persistence, extending both of these results by weakening their respective assumptions to requiring that only one (rather than all) infection-related compartment is initially non-vanishing. That is, we show that, given $ \mathcal{R}_0 > 1 $, if either the infected bird population or the viral concentration are initially nonzero anywhere in the case of avian influenza, or if any of the infected human population, viral concentration or population of deceased individuals who are under care are initially nonzero anywhere in the case of the Ebola virus disease, then their respective models predict uniform persistence. The difficulty which we overcome here is the lack of diffusion, and hence the inability to apply the minimum principle, in the equations of the avian influenza virus concentration in water and of the population of the individuals deceased due to the Ebola virus disease who are still in the process of caring.

Spatiotemporal modeling of first and second wave outbreak dynamics of COVID-19 in Germany

The COVID-19 pandemic has kept the world in suspense for the past year. In most federal countries such as Germany, locally varying conditions demand for state- or county-level decisions to adapt to the disease dynamics. However, this requires a deep understanding of the mesoscale outbreak dynamics between microscale agent models and macroscale global models. Here, we use a reparameterized SIQRD network model that accounts for local political decisions to predict the spatiotemporal evolution of the pandemic in Germany at county resolution. Our optimized model reproduces state-wise cumulative infections and deaths as reported by the Robert Koch Institute and predicts the development for individual counties at convincing accuracy during both waves in spring and fall of 2020. We demonstrate the dominating effect of local infection seeds and identify effective measures to attenuate the rapid spread. Our model has great potential to support decision makers on a state and community politics level to individually strategize their best way forward during the months to come.

Modelling the aqueous transport of an infectious pathogen in regional communities: application to the cholera outbreak in Haiti

A mathematical model is developed to describe the dynamics of the spread of a waterborne disease among communities located along a flowing waterway. The model is formulated as a system of reaction-diffusion-advection partial differential equations in this spatial setting. The compartments of the model consist of susceptible, infected, and recovered individuals in the communities along the waterway, together with a term representing the pathogen load in each community and a term representing the spatial concentration of pathogens flowing along the waterway. The model is applied to the cholera outbreak in Haiti in 2010.

Nonlinear two-point boundary value problems: applications to a cholera epidemic model

This paper is concerned primarily with constructive mathematical analysis of a general system of nonlinear two-point boundary value problem when an empirically constructed candidate for an approximate solution (quasi-solution) satisfies verifiable conditions. A local analysis in a neighbour- hood of a quasi-solution assures the existence and uniqueness of solutions and, at the same time, provides error bounds for approximate solutions. Applying this method to a cholera epidemic model, we obtain an analytical approximation of the steady-state solution with rigorous error bounds that also displays dependence on a parameter. In connection with this epidemic model, we also analyse the basic reproduction number, an important threshold quantity in the epidemiology context. Through a complex analytic approach, we determine the principal eigenvalue to be real and positive in a range of parameter values.