一类新的由比例纯跳Lévy噪声驱动的随机时空SIR型模型的适定性结果
Well-posedness results for a new class of stochastic spatio-temporal SIR-type models driven by proportional pure-jump Lévy noise
AI总结：
本文首次将纯跳Lévy噪声引入随机时空SIR型流行病模型，利用半群理论、截断技术和Banach不动点定理证明温和解的存在唯一性、连续依赖性及几乎必然正性，并通过数值模拟展示其相对于高斯噪声的影响。
摘要：
本文首次尝试将流行病时空动力学中的大规模不连续变化纳入建模。具体而言，我们提出了一类扩展的流行病模型，该模型由耦合的随机半线性偏微分方程控制，并由纯跳Lévy噪声驱动。基于所考虑的发病率函数类型，借助半群理论、截断技术和Banach不动点定理，我们证明了温和解的存在性和路径唯一性，且该解连续依赖于初始数据。此外，通过基于预解算子的正则化技术，我们得到温和解可以由一列适当收敛的强解逼近。利用这一结果，对于正的初始状态，我们推导出所得解的几乎必然正性。最后，我们给出了若干数值模拟的结果，以展示所考虑的随机噪声类型相比于以往文献中使用的高斯噪声的影响。我们所建立的结果为进一步研究与新提出的流行病模型类相关的其他问题奠定了基础，例如渐近行为分析、最优控制以及识别问题，这些问题主要依赖于生物学上可行解的存在性和唯一性。
英文摘要：
This paper provides a first attempt to incorporate the massive discontinuous changes in the spatio-temporal dynamics of epidemics. Namely, we propose an extended class of epidemic models, governed by coupled stochastic semilinear partial differential equations, driven by pure-jump Lévy noise. Based on the considered type of incidence functions, by virtue of semi-group theory, a truncation technique and Banach fixed point theorem, we prove the existence and pathwise uniqueness of mild solutions, depending continuously on the initial datum. Moreover, by means of a regularization technique, based on the resolvent operator, we acquire that mild solutions can be approximated by a suitable converging sequence of strong solutions. With this result at hand, for positive initial states, we derive the almost-sure positiveness of the obtained solutions. Finally, we present the outcome of several numerical simulations, in order to exhibit the effect of the considered type of stochastic noise, in comparison to Gaussian noise, which has been used in the previous literature. Our established results lay the ground-work for investigating other problems associated with the new proposed class of epidemic models, such as asymptotic behavior analyses, optimal control as well as identification problems, which primarily rely on the existence and uniqueness of biologically feasible solutions.
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