# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks and Heterogeneous Media

June 2022 , Volume 17 , Issue 3

Special issue on advanced mathematical methodologies to contrast COVID-19 pandemic

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2022, 17(3): i-ii doi: 10.3934/nhm.2022020 +[Abstract](212) +[HTML](88) +[PDF](68.77KB)
Abstract:
2022, 17(3): 293-310 doi: 10.3934/nhm.2022008 +[Abstract](347) +[HTML](273) +[PDF](667.3KB)
Abstract:

This paper focuses on the multiscale modeling of the COVID-19 pandemic and presents further developments of the model [7] with the aim of showing how relaxations of the confinement rules can generate sequential waves. Subsequently, the dynamics of mutations into new variants can be modeled. Simulations are developed also to support the decision making of crisis managers.

2022, 17(3): 311-331 doi: 10.3934/nhm.2022009 +[Abstract](325) +[HTML](95) +[PDF](1130.64KB)
Abstract:

Deterministic compartmental models for infectious diseases give the mean behaviour of stochastic agent-based models. These models work well for counterfactual studies in which a fully mixed large-scale population is relevant. However, with finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. In this article, we consider the classical stochastic Susceptible-Infected-Recovered (SIR) model, and derive a martingale formulation consisting of a deterministic and a stochastic component. The deterministic part coincides with the classical deterministic SIR model and we provide an upper bound for the stochastic part. Through analysis of the stochastic component depending on varying population size, we provide a theoretical explanation of finite size effects. Our theory is supported by quantitative and direct numerical simulations of theoretical infinitesimal variance. Case studies of coronavirus disease 2019 (COVID-19) transmission in smaller populations illustrate that the theory provides an envelope of possible outcomes that includes the field data.

2022, 17(3): 333-357 doi: 10.3934/nhm.2022010 +[Abstract](287) +[HTML](101) +[PDF](1898.49KB)
Abstract:

In this paper, we investigate the well-posedness and dynamics of a class of hybrid models, obtained by coupling a system of ordinary differential equations and an agent-based model. These hybrid models intend to integrate the microscopic dynamics of individual behaviors into the macroscopic evolution of various population dynamics models, and can be applied to a great number of complex problems arising in economics, sociology, geography and epidemiology. Here, in particular, we apply our general framework to the current COVID-19 pandemic. We establish, at a theoretical level, sufficient conditions which lead to particular solutions exhibiting irregular oscillations and interpret those particular solutions as pandemic waves. We perform numerical simulations of a set of relevant scenarios which show how the microscopic processes impact the macroscopic dynamics.

2022, 17(3): 359-384 doi: 10.3934/nhm.2022011 +[Abstract](273) +[HTML](70) +[PDF](2154.01KB)
Abstract:

The ongoing COVID-19 pandemic highlights the essential role of mathematical models in understanding the spread of the virus along with a quantifiable and science-based prediction of the impact of various mitigation measures. Numerous types of models have been employed with various levels of success. This leads to the question of what kind of a mathematical model is most appropriate for a given situation. We consider two widely used types of models: equation-based models (such as standard compartmental epidemiological models) and agent-based models. We assess their performance by modeling the spread of COVID-19 on the Hawaiian island of Oahu under different scenarios. We show that when it comes to information crucial to decision making, both models produce very similar results. At the same time, the two types of models exhibit very different characteristics when considering their computational and conceptual complexity. Consequently, we conclude that choosing the model should be mostly guided by available computational and human resources.

2022, 17(3): 385-400 doi: 10.3934/nhm.2022012 +[Abstract](285) +[HTML](108) +[PDF](764.24KB)
Abstract:

We present a new epidemic model highlighting the roles of the immunization time and concurrent use of different vaccines in a vaccination campaign. To this aim, we introduce new intra-compartmental dynamics, a procedure that can be extended to various other situations, as detailed through specific case studies considered herein, where the dynamics within compartments are present and influence the whole evolution.

2022, 17(3): 401-425 doi: 10.3934/nhm.2022013 +[Abstract](290) +[HTML](108) +[PDF](1931.03KB)
Abstract:

Uncertainty in data is certainly one of the main problems in epidemiology, as shown by the recent COVID-19 pandemic. The need for efficient methods capable of quantifying uncertainty in the mathematical model is essential in order to produce realistic scenarios of the spread of infection. In this paper, we introduce a bi-fidelity approach to quantify uncertainty in spatially dependent epidemic models. The approach is based on evaluating a high-fidelity model on a small number of samples properly selected from a large number of evaluations of a low-fidelity model. In particular, we will consider the class of multiscale transport models recently introduced in [13,7] as the high-fidelity reference and use simple two-velocity discrete models for low-fidelity evaluations. Both models share the same diffusive behavior and are solved with ad-hoc asymptotic-preserving numerical discretizations. A series of numerical experiments confirm the validity of the approach.

2022, 17(3): 427-442 doi: 10.3934/nhm.2022015 +[Abstract](255) +[HTML](83) +[PDF](402.63KB)
Abstract:

Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible \begin{document}$S$\end{document} and removed \begin{document}$R$\end{document} populations by ODEs and the infected \begin{document}$I$\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for \begin{document}$S$\end{document} and \begin{document}$R$\end{document} contains terms that are related to the measure \begin{document}$I$\end{document}. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases.

2022, 17(3): 443-466 doi: 10.3934/nhm.2022016 +[Abstract](282) +[HTML](83) +[PDF](2781.75KB)
Abstract:

During the Covid-19 pandemic a key role is played by vaccination to combat the virus. There are many possible policies for prioritizing vaccines, and different criteria for optimization: minimize death, time to herd immunity, functioning of the health system. Using an age-structured population compartmental finite-dimensional optimal control model, our results suggest that the eldest to youngest vaccination policy is optimal to minimize deaths. Our model includes the possible infection of vaccinated populations. We apply our model to real-life data from the US Census for New Jersey and Florida, which have a significantly different population structure. We also provide various estimates of the number of lives saved by optimizing the vaccine schedule and compared to no vaccination.

2022, 17(3): 467-494 doi: 10.3934/nhm.2022017 +[Abstract](265) +[HTML](107) +[PDF](1813.16KB)
Abstract:

In classical epidemic models, a neglected aspect is the heterogeneity of disease transmission and progression linked to the viral load of each infected individual. Here, we investigate the interplay between the evolution of individuals' viral load and the epidemic dynamics from a theoretical point of view. We propose a stochastic particle model describing the infection transmission and the individual physiological course of the disease. Agents have a double microscopic state: a discrete label, that denotes the epidemiological compartment to which they belong and switches in consequence of a Markovian process, and a microscopic trait, measuring their viral load, that changes in consequence of binary interactions or interactions with a background. Specifically, we consider Susceptible–Infected–Removed–like dynamics where infectious individuals may be isolated and the isolation rate may depend on the viral load–sensitivity and frequency of tests. We derive kinetic evolution equations for the distribution functions of the viral load of the individuals in each compartment, whence, via upscaling procedures, we obtain macroscopic equations for the densities and viral load momentum. We perform then a qualitative analysis of the ensuing macroscopic model. Finally, we present numerical tests in the case of both constant and viral load–dependent isolation control.

2021 Impact Factor: 1.41
5 Year Impact Factor: 1.296
2021 CiteScore: 2.2