Epidemic spreading

Abstract

We present an analysis of six deterministic models for epidemic spreading. The evolution of the number of individuals of each class is given by ordinary differential equations of the first order in time, which are set up by using the laws of mass action providing the rates of the several processes that define each model. The epidemic spreading is characterized by the frequency of new cases, which is the number of individuals that are becoming infected per unit time. It is also characterized by the basic reproduction number, which we show to be related to the largest eigenvalue of the stability matrix associated with the disease-free solution of the evolution equations. We also emphasize the analogy between the outbreak of an epidemic with a critical phase transition. When the density of the population reaches a critical value the spreading sets in, a result that was advanced by Kermack and McKendrick in their study of a model in which the recovered individuals acquire permanent immunization, which is one of the models analyzed here.

Complexidade na Física e seu Ensino: Apresentação da Edição Especial

“…Embora o modelo não seja capaz de fazer previsões quantitativas para epidemias reais, ele exibe transições de fases dinâmicas. Para complementação do atual estudo, recomenda-se a leitura do artigo de Tânia Tomé e Mário de Oliveira, publicado nesta revista, em qué e feita uma análise de seis modelos determinísticos de propagação epidêmica [30].…”

Simulation of deterministic compartmental models for infectious diseases dynamics

“…The reader can find mathematical details in Ref. [14]. We show the dynamical behaviours predict by some mathematical models, such as SI, SIS, SIR, SIRS, SEIR, and SEIRS.…”

“…A model * kiarosz@gmail.com was proposed by Ross [13] in 1916 for malaria. In the Ross model, known as SIS model [14], susceptible become infected and infected recover without immunity. Kermack and McKendrick [15] introduced a model, known as SIR model, in which the removed can be recovered, immune, or dead.…”

“…This law originated in practice and theory of chemical reaction kinetics [30]. Tomé and Oliveira [14] presented models for epidemic spreading and showed the analogy between the spreading of a disease with a critical phase transition. They also analysed the epidemic curve, which is a graphical representation of the number of identified cases over a period of time.…”

Effect of immunization through vaccination on the SIS epidemic spreading model

“…The model represented by the reaction equation ( 3) is known as the susceptible-infected-susceptible (SIS) model and the equations ( 4) or (1) describe the deterministic version of this model [6][7][8][9]. The model represented by the first reaction equation alone is known as the susceptible-infected (SI) model and the equation ( 2) describes the deterministic version of the model.…”