Bravo de la Parra Rafael, Sanz-Lorenzo Luis
U.D. Matemáticas, Universidad de Alcalá, Alcalá de Henares, Spain.
Dpto. Matemática Aplicada, ETSI Industriales, Universidad Politécnica de Madrid, Madrid, Spain.
Adv Differ Equ. 2021;2021(1):478. doi: 10.1186/s13662-021-03633-0. Epub 2021 Oct 30.
The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviors, or others. To include a sufficiently general disease model, we first build up from first principles a discrete-time susceptible-exposed-infectious-recovered-susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number . Then, we propose a general full model that includes sequentially the two processes at different time scales and proceed to its analysis through a reduced model. The basic reproduction number of the reduced system gives a good approximation of of the full model since it serves at analyzing its asymptotic behavior. As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches.
这项工作的主要目的是提出一类通用的双时间尺度离散时间流行病模型。在所提出的框架中，疾病动态被认为作用于比另一个不同过程更慢的时间尺度上，后一过程可以代表空间位置之间的移动、个体活动或行为的变化等。为了纳入一个足够通用的疾病模型，我们首先从基本原理构建一个离散时间易感-暴露-感染-康复-易感（SEIRS）模型，并借助其基本再生数来刻画疾病的根除或地方性流行情况。然后，我们提出一个通用的完整模型，该模型依次包含不同时间尺度上的两个过程，并通过一个简化模型对其进行分析。简化系统的基本再生数给出了完整模型基本再生数的良好近似，因为它有助于分析其渐近行为。作为所提出的通用框架的一个例证，结果表明存在这样的条件，即对于一个在集合种群中考虑孤立斑块的局部地方性疾病，通过在斑块之间建立适当的移动可以在全球范围内将其根除。
