Memory Effects in Population Dynamics : Spread of Infectious Disease as a Case Study

Abstract

Abstract. Modification of behaviour in response to changes in the environment or ambient conditions, based on memory, is typical of the human and, possibly, many animal species.One obvious example of such adaptivity is, for instance, switching to a safer behaviour when in danger, from either a predator or an infectious disease. In human society such switching to safe behaviour is particularly apparent during epidemics. Mathematically, such changes of behaviour in response to changes in the ambient conditions can be described by models involving switching. In most cases, this switching is assumed to depend on the system state, and thus it disregards the history and, therefore, memory. Memory can be introduced into a mathematical model using a phenomenon known as hysteresis. We illustrate this idea using a simple SIR compartmental model that is applicable in epidemiology. Our goal is to show why and how hysteresis can arise in such a model, and how it may be applied to describe a variety of memory effects. Our other objective is to introduce a unified paradigm for mathematical modelling with memory effects in epidemiology and ecology. Our approach treats changing behaviour as an irreversible flow related to large ensembles of elementary exchange operations that recently has been successfully applied in a number of other areas, such as terrestrial hydrology, and macroeconomics. For the purposes of illustrating these ideas in an application to biology, we consider a rather simple case study and develop a model from first principles. We accompany the model with extensive numerical simulations which exhibit interesting qualitative effects.

Um estudo sobre a memória epidemiológica: modelo SIRC fracionário

“…A dinâmica das populações entre os compartimentos fica mais bem apresentada no diagrama abaixo, figura (1): (PIMENOV et al, 2012). Esta memória imunológicaé uma capacidade do sistema imune adaptativo responsável por propiciar respostas mais eficientes e rápidasà ocorrência de exposiçãoà antígenos encontrados previamente, seja na forma de exposição natural, ou na forma de vacinas.…”

Asymptotics of sign-changing patterns in hysteretic systems with diffusive thresholds

“…The resulting model is a reaction-diffusion system including, as reaction terms, discontinuous hysteresis relay operators and the integral of those. This integral can be interpreted as the Preisach operator [8,13,31,32,34,35,43,44,46,50] with a time dependent density (the density is a component of the solution describing the varying distribution of bacteria). In [23], we have shown that fitness, competition and diffusion can act together to select a nontrivial distribution of phenotypes (states) over the population of thresholds.…”

The Memory Effect on Fractional Calculus: An Application in the Spread of Covid-19

“…In [13], a system of fractional differential equations is used to study the effect of memory on epidemic evolution. In [10], the hysteresis effect is used to model biological phenomena that activate defense mechanisms of certain living beings. For example, in a situation of spreading of an infectious disease for human beings, depending on type of disease, previous experiences can reveal that social distance or additional hygiene habits are defensive behaviors to prevent its propagation.…”