Juher David, Kiss Istvan Z, Saldaña Joan
Departament d׳Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, Catalonia, Spain.
School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK.
J Theor Biol. 2015 Jan 21;365:457-68. doi: 10.1016/j.jtbi.2014.10.013. Epub 2014 Oct 23.
The existence of a die-out threshold (different from the classic disease-invasion one) defining a region of slow extinction of an epidemic has been proved elsewhere for susceptible-aware-infectious-susceptible models without awareness decay, through bifurcation analysis. By means of an equivalent mean-field model defined on regular random networks, we interpret the dynamics of the system in this region and prove that the existence of bifurcation for this second epidemic threshold crucially depends on the absence of awareness decay. We show that the continuum of equilibria that characterizes the slow die-out dynamics collapses into a unique equilibrium when a constant rate of awareness decay is assumed, no matter how small, and that the resulting bifurcation from the disease-free equilibrium is equivalent to that of standard epidemic models. We illustrate these findings with continuous-time stochastic simulations on regular random networks with different degrees. Finally, the behaviour of solutions with and without decay in awareness is compared around the second epidemic threshold for a small rate of awareness decay.
对于无意识衰减的易感-知晓-感染-易感模型，通过分岔分析，已在其他地方证明了存在一个灭绝阈值（不同于经典的疾病入侵阈值），该阈值定义了流行病缓慢灭绝的区域。借助在规则随机网络上定义的等效平均场模型，我们解释了该区域内系统的动态，并证明了这个第二流行阈值的分岔存在性关键取决于无意识衰减。我们表明，当假设存在恒定的意识衰减率时，无论该衰减率多小，表征缓慢灭绝动态的连续平衡点都会坍缩为一个唯一的平衡点，并且由此从无病平衡点产生的分岔等同于标准流行病模型的分岔。我们在具有不同度数的规则随机网络上通过连续时间随机模拟来说明这些发现。最后，针对较小的意识衰减率，比较了在第二流行阈值附近有意识衰减和无意识衰减情况下解的行为。
