Dinâmica de epidemias: efeitos do atraso e das interações entre agentes

Abstract

The study of epidemic propagation has generated a series of researches proposing different models that tries to represent possible scenarios so that we could have more control over the propagation process. The aim of these studies is the ability not only of reproducing past epidemic evolutions but to forecast and, if possible, avoid new epidemic bursts or eradicate endemic states. Most of the research n this area are based on the Susceptible-Infective-Removed model, or SIR model, proposed by Kermack and McKendrick. Nevertheless, similar approaches using mathematical models or computational simulations have presented distinct behavior due to the fact that the first assumes a constant removal rate, which represents an exponential distribution of the infectious period in the population, while simulations in general uses a fixed infectious period identical to each individual in the population. In view of that, we have used a mathematical model which is a generalization of the SIR model using delayed equations, so that we can insert those time distributions explicitly into the equations. With this model, we show that the mathematical model can reproduce the average behavior of the time evolution given by simulations for several distributions and also the standard SIR model when we use an exponential distribution. We have done comparisons for model with and without vital dynamics and we show that the model with time delay can always reproduce the mean behavior of the simulations for all distributions tested. In addition, we can see that the time evolution of the epidemic spread is highly dependent on the infectious period distribution. When adding vital dynamics this dependence is also present in the epidemic threshold and endemic state. Given that this model allows us to use the population distribution of the infectious period, we applied it to study treatment policies where the effect of such treatment reduces this period and we show that is possible to determine the minimum fraction of the population that must be treated in order to prevent an epidemic burst or an endemic state. Although the SIR model is very useful to model a great variety of diseases, it can only be applied to those where the infected individuals acquire permanent immunity or die with the disease. Nevertheless, there is a huge class of diseases where the acquired immunity is only temporary, so that the individuals become susceptible again after a given period. Such diseases have a typical evolution represented by the Susceptible-Infective-Removed-Susceptible model, or SIRS model. They usually have an endemic state with cyclic epidemic bursts. The existence of such oscillations in epidemics has been, since a very long time, a challenge for the formulation of epidemiological models. If they result from external and seasonal forces or if they emerge from the intrinsic dynamics of the disease is an open question. It is known that fixed time delays destabilize the stationary states of the standard SIRS model, given rise to sustained oscillations for certain values of the epidemiological parameters of the model. In this work, starting from the standard SIRS model, we study a generalization of the terms relative to the infectious and immunity periods. We present oscillation diagrams (for the amplitude and period of oscillations) in terms of the parameters of the model, which shows how the shape of those characteristic time distributions (infectious and immunity) influence the oscillations. The model formulation is made with integro-differential equations with delay analyzed by numerical integration and linearization of the system. We also present a simulation of this model highlighting where it agrees with the results of the deterministic model and, when it diverges, explaining why it diverges. Along with those mean field models, we have also built an agent based model to study the impact of agent mobility in the disease propagation of a Susceptible-Infective-Susceptible (SIS) dynamic. In this model, by defining the basic reproduction rate in terms of the relevant parameters, we show that the endemic state has the same dependency on it as the mean field models, but the epidemic threshold is the same as the one obtained by the implementation of the SIS model in a bidimensinal lattice. Another important result of this approach is the fact that, given the agents density, it is possible to obtain oscillatory endemic states, a common result in real diseases but absent in the mean field models, being present only when the contact between agents is defined by a network.