Dynamics of Epidemiological Models

Abstract

We study the SIS and SIRI epidemic models discussing different approaches to compute the thresholds that determine the appearance of an epidemic disease. The stochastic SIS model is a well known mathematical model, studied in several contexts. Here, we present recursively derivations of the dynamic equations for all the moments and we derive the stationary states of the state variables using the moment closure method. We observe that the steady states give a good approximation of the quasi-stationary states of the SIS model. We present the relation between the SIS stochastic model and the contact process introducing creation and annihilation operators. For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we present the phase transition lines using the mean field and the pair approximation for the moments. We use a scaling argument that allow us to determine analytically an explicit formula for the phase transition lines in pair approximation.

The effect of an acidified, ionized copper sulphate solution on digital dermatitis in dairy cows

“…Other treatments are not always effective and Thomsen et al (2008) evaluated three commercial hoof-care products, based on glutaraldehyde, organic acids or quaternary ammonium compounds, over an 8 week period and found no preventive and no curative effect in comparison with a negative control. This lack of an effect may have been partly due to the length of the study, because it might take several months before an effect is visible (Pinto et al, 2010). In our study the positive effect of Digiderm+ was the finding of significantly better prevention of disease, not a higher cure rate (Table 3).…”

Stationarity in moment closure and quasi-stationarity of the SIS model

“…In Fig. 3 b), we consider other infection rate values and we observe that the distance jhIi à m;b À hIi QS;b j decreases with m up to a certain mth moment closure approximation that can even occur before the break down of the stable equilibria for the value of the infection rate b under consideration (see simulations for other transition rates in Pinto et al [18,19]). Similar approximations are observed for the higher moments of infected individuals hI n i à ; n P 2.…”

Understanding COVID-19 Epidemics: A Multi-Scale Modeling Approach

“…Disease propagation is an inherently stochastic phenomenon and there are a number of reasons why one should use stochastic models to capture the disease transmission process. However, the mean-field approximation, where the dynamics of the mean quantities are approximated by neglecting correlations, is often used as a good approximation to get a first understanding of the behavior of stochastic systems in certain parameter regions [40,50,52].…”