Departamento de Física, Instituto Federal da Bahia-40110-150 Salvador, Brazil.
Instituto de Física, Universidade Federal da Bahia-40210-340 Salvador, Brazil.
Phys Rev E. 2017 Jun;95(6-1):062135. doi: 10.1103/PhysRevE.95.062135. Epub 2017 Jun 29.
The use of stochastic models to study the dynamics of infectious diseases is an important tool to understand the epidemiological process. For several directly transmitted diseases, reinfection is a relevant process, which can be expressed by endogenous reactivation of the pathogen or by exogenous reinfection due to direct contact with an infected individual (with smaller reinfection rate σβ than infection rate β). In this paper, we examine the stochastic susceptible, infected, recovered, infected (SIRI) model simulating the endogenous reactivation by a spontaneous reaction, while exogenous reinfection by a catalytic reaction. Analyzing the mean-field approximations of a site and pairs of sites, and Monte Carlo (MC) simulations for the particular case of exogenous reinfection, we obtained continuous phase transitions involving endemic, epidemic, and no transmission phases for the simple approach; the approach of pairs is better to describe the phase transition from endemic phase (susceptible, infected, susceptible (SIS)-like model) to epidemic phase (susceptible, infected, and removed or recovered (SIR)-like model) considering the comparison with MC results; the reinfection increases the peaks of outbreaks until the system reaches endemic phase. For the particular case of endogenous reactivation, the approach of pairs leads to a continuous phase transition from endemic phase (SIS-like model) to no transmission phase. Finally, there is no phase transition when both effects are taken into account. We hope the results of this study can be generalized for the susceptible, exposed, infected, and removed or recovered (SEIR_{I}^{E}) model, for which the state exposed (infected but not infectious), describing more realistically transmitted diseases such as tuberculosis. In future work, we also intend to investigate the effect of network topology on phase transitions when the SIRI model describes both transmitted diseases (σ<1) and social contagions (σ>1).
使用随机模型来研究传染病的动力学是理解流行病学过程的重要工具。对于一些直接传播的疾病，再次感染是一个相关的过程，可以通过病原体的内源性再激活或由于与感染个体的直接接触而导致的外源性再感染来表示（再感染率 σβ 小于感染率 β）。在本文中，我们检查了模拟内源性再激活的随机易感者、感染者、恢复者、再次感染者（SIRI）模型，这种再激活是由自发反应引起的，而外源性再感染是由催化反应引起的。通过对位点和双位点的平均场近似以及对特定外源性再感染情况的蒙特卡罗（MC）模拟，我们得到了涉及地方性、流行和无传播阶段的连续相变；对于简单方法，双位点方法更适合描述从地方性阶段（类似于易感者、感染者、易感者（SIS）的模型）到流行阶段（类似于易感者、感染者和移除或恢复（SIR）的模型）的相变，考虑到与 MC 结果的比较；再感染会增加疫情爆发的峰值，直到系统达到地方性阶段。对于内源性再激活的特定情况，双位点方法导致从地方性阶段（类似于 SIS 的模型）到无传播阶段的连续相变。最后，当同时考虑这两种效应时，没有相变。我们希望这项研究的结果可以推广到易感者、暴露者、感染者和移除或恢复（SEIR_{I}^{E}）模型，对于描述更真实的传染病，如结核病，暴露者（已感染但无传染性）的状态更能反映出传染病的特征。在未来的工作中，我们还打算研究当 SIRI 模型描述传染病（σ<1）和社会传播（σ>1）时，网络拓扑对相变的影响。
