On the analysis of a multi-regions discrete SIR epidemic model: an optimal control approach

Research on epidemic spread model based on cold chain input

In recent years, the new type of coronary pneumonia (COVID-19) has become a highly contagious disease worldwide, posing a serious threat to the public health. This paper is based on the SEIR model of the new coronavirus pneumonia, considering the impact of cold chain input and re-positive on the spread of the virus in the COVID-19. In the process of model design, the food cold chain and re-positive are used as parameters, and its stability is analyzed and simulated. The experimental results show that taking into account the cold chain input and re-positive can effectively simulate the spread of the epidemic. The research results have important research value and practical significance for the prevention and control of the COVID-19 and the prediction of important time nodes.

Stochastic optimization for vaccine and testing kit allocation for the COVID-19 pandemic

We present a formal mathematical modeling framework for a multi-agent sequential decision problem during an epidemic. The problem is formulated as a collaboration between a vaccination agent and learning agent to allocate stockpiles of vaccines and tests to a set of zones under various types of uncertainty. The model is able to capture passive information processes and maintain beliefs over the uncertain state of the world. We designed a parameterized direct lookahead approximation which is robust and scalable under different scenarios, resource scarcity, and beliefs about the environment. We design a test allocation policy designed to capture the value of information and demonstrate that it outperforms other learning policies when there is an extreme shortage of resources (information is scarce). We simulate the model with two scenarios including a resource allocation problem to each state in the United States and another for the nursing homes in Nevada. The US example demonstrates the scalability of the model and the nursing home example demonstrates the robustness under extreme resource shortages.

Long-term spatial and population-structured planning of non-pharmaceutical interventions to epidemic outbreaks

In this paper, we consider the problem of planning non-pharmaceutical interventions to control the spread of infectious diseases. We propose a new model derived from classical compartmental models; however, we model spatial and population-structure heterogeneity of population mixing. The resulting model is a large-scale non-linear and non-convex optimisation problem. In order to solve it, we apply a special variant of covariance matrix adaptation evolution strategy. We show that results obtained for three different objectives are better than natural heuristics and, moreover, that the introduction of an individual's mobility to the model is significant for the quality of the decisions. We apply our approach to a six-compartmental model with detailed Poland and COVID-19 disease data. The obtained results are non-trivialand sometimes unexpected; therefore, we believe that our model could be applied to support policy-makers in fighting diseases at the long-term decision-making level.

Optimal control of the spatial allocation of COVID-19 vaccines: Italy as a case study

While campaigns of vaccination against SARS-CoV-2 are underway across the world, communities face the challenge of a fair and effective distribution of a limited supply of doses. Current vaccine allocation strategies are based on criteria such as age or risk. In the light of strong spatial heterogeneities in disease history and transmission, we explore spatial allocation strategies as a complement to existing approaches. Given the practical constraints and complex epidemiological dynamics, designing effective vaccination strategies at a country scale is an intricate task. We propose a novel optimal control framework to derive the best possible vaccine allocation for given disease transmission projections and constraints on vaccine supply and distribution logistics. As a proof-of-concept, we couple our framework with an existing spatially explicit compartmental COVID-19 model tailored to the Italian geographic and epidemiological context. We optimize the vaccine allocation on scenarios of unfolding disease transmission across the 107 provinces of Italy, from January to April 2021. For each scenario, the optimal solution significantly outperforms alternative strategies that prioritize provinces based on incidence, population distribution, or prevalence of susceptibles. Our results suggest that the complex interplay between the mobility network and the spatial heterogeneities implies highly non-trivial prioritization strategies for effective vaccination campaigns. Our work demonstrates the potential of optimal control for complex and heterogeneous epidemiological landscapes at country, and possibly global, scales.

Controlling epidemic diseases based only on social distancing level: General case

The COVID-19 outbreak is an epidemic disease caused by the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). When a new virus emerges, generally, little is known about it, and no vaccines or other pharmaceutical interventions are available. In the case of a person-to-person transmission virus with no vaccines or other pharmaceutical interventions, the only way to control the virus outbreak is by keeping a sustained physical distancing between the individuals. However, to adjust the level of the physical distancing accurately can be so complicated. Any level above the necessary can compromise the economic activity, and any level below can collapse the health care system. This work proposes a controller to keep the number of hospitalized individuals below a limit, and a new group-structured model to describe the COVID-19 outbreak. The proposed controller is robust to the uncertainties in the parameters of the model and keeps the number of infected individuals controlled only by adjusting the social distancing level. Numerical simulations, to show the behavior of the proposed controller and model, are done.

An epidemiology-based model for the operational allocation of COVID-19 vaccines: A case study of Thailand

This paper addresses a framework for the operational allocation and administration of COVID-19 vaccines in Thailand, based on both COVID-19 transmission dynamics and other vital operational restrictions that might affect the effectiveness of vaccination strategies in the early stage of vaccine rollout. In this framework, the SIQRV model is first developed and later combined with the COVID-19 Vaccine Allocation Problem (CVAP) to determine the optimal allocation/administration strategies that minimize total weighted strain on the whole healthcare system. According to Thailand's second pandemic wave data (17^th January 2021, to 15^th February 2021), we find that the epicenter-based strategy is surprisingly the worst allocation strategy, due largely to the negligence of provincial demographics, vaccine efficacy, and overall transmission dynamics that lead to higher number of infectious individuals. We also find that early vaccination seems to significantly contribute to the reduction in the number of infectious individuals, whose effects tend to increase with more vaccine supply. With these insights, healthcare policy-makers should therefore focus not only on the procurement of COVID-19 vaccines at strategic levels but also on the allocation and administration of such vaccines at operational levels for the best of their limited vaccine supply.

Comprehensive compartmental model and calibration algorithm for the study of clinical implications of the population-level spread of COVID-19: a study protocol

INTRODUCTION

The complex dynamics of the coronavirus disease 2019 (COVID-19) pandemic has made obtaining reliable long-term forecasts of the disease progression difficult. Simple mechanistic models with deterministic parameters are useful for short-term predictions but have ultimately been unsuccessful in extrapolating the trajectory of the pandemic because of unmodelled dynamics and the unrealistic level of certainty that is assumed in the predictions.

METHODS AND ANALYSIS

We propose a 22-compartment epidemiological model that includes compartments not previously considered concurrently, to account for the effects of vaccination, asymptomatic individuals, inadequate access to hospital care, post-acute COVID-19 and recovery with long-term health complications. Additionally, new connections between compartments introduce new dynamics to the system and provide a framework to study the sensitivity of model outputs to several concurrent effects, including temporary immunity, vaccination rate and vaccine effectiveness. Subject to data availability for a given region, we discuss a means by which population demographics (age, comorbidity, socioeconomic status, sex and geographical location) and clinically relevant information (different variants, different vaccines) can be incorporated within the 22-compartment framework. Considering a probabilistic interpretation of the parameters allows the model's predictions to reflect the current state of uncertainty about the model parameters and model states. We propose the use of a sparse Bayesian learning algorithm for parameter calibration and model selection. This methodology considers a combination of prescribed parameter prior distributions for parameters that are known to be essential to the modelled dynamics and automatic relevance determination priors for parameters whose relevance is questionable. This is useful as it helps prevent overfitting the available epidemiological data when calibrating the parameters of the proposed model. Population-level administrative health data will serve as partial observations of the model states.

ETHICS AND DISSEMINATION

Approved by Carleton University's Research Ethics Board-B (clearance ID: 114596). Results will be made available through future publication.

Introduction to Group-Structured-Epidemic Model

The spread of an infectious disease in a population is a random process when considering a small group of individuals. However, to a great group of individuals, the use of deterministic behavior is better. Based on these facts, in the literature, there were proposed stochastic and deterministic epidemic models. This work proposes a mixed compartmental epidemic model that allows stratifying the population into groups, considers demographic and environmental variability, presents an approximation to stochastic effects, and contemplates the network effects. The proposed model has a compact form to assist in the synthesis of the control law and parameters estimation strategies. Its objective is to overcome the difficulties encountered when used purely deterministic or purely stochastic models. In the end, to detail and verify the functioning of the proposed model, we present a set of flowcharts and simulations.

Boundary optimal control of time-space SIR model with nonlinear Robin boundary condition

A boundary optimal control problem arising in time-space SIR epidemic models is treated. In this work we aim with the control of the flux of infected individuals crossing part of boundary. On the other side of the domain, we suppose a nonlinear boundary condition of third kind: nonlinear Robin boundary condition, this condition models immersing individual crossing this part of the boundary of the domain of study. We give the existence and uniqueness of the solution of both state and optimal control problem ending some numerical tests throughout a simple example.

Optimal control of alcoholism spreading through awareness over multiplex network

This paper proposes the SISRS epidemic model to represent alcohol addiction among people. The spreading of alcohol addiction is controlled by creating awareness among the people and also by treating them to overcome it. Multiplex network is used to study the dynamics of addiction. Alcoholism spreads over the physical contact layer and follows the SISRS process whereas human awareness spreads over the virtual contact layer and follows the UAU process. Based on the Microscopic Markov Chain Approach competing dynamics of spreading of alcohol addiction and human awareness diffusion are studied. Necessary conditions for the existence of an alcohol-free population are found. An optimal control problem using a suitable cost index is formulated to reduce the alcohol addicts and the optimal control strategy using Pontryagin’s Minimum Principle is determined. Numerical results are developed to find the effect of various parameters and to analyze the effects of different control strategies. The results obtained from this model are closer to the data collected in the National Survey of Drug Use and Health (NSDUH) from 2002 to 2018.

Analysis of the Tradeoff Between Health and Economic Impacts of the Covid-19 Epidemic

Various measures have been taken in different countries to mitigate the Covid-19 epidemic. But, throughout the world, many citizens don't understand well how these measures are taken and even question the decisions taken by their government. Should the measures be more (or less) restrictive? Are they taken for a too long (or too short) period of time? To provide some quantitative elements of response to these questions, we consider the well-known SEIR model for the Covid-19 epidemic propagation and propose a pragmatic model of the government decision-making operation. Although simple and obviously improvable, the proposed model allows us to study the tradeoff between health and economic aspects in a pragmatic and insightful way. Assuming a given number of phases for the epidemic (namely, 4 in this paper) and a desired tradeoff between health and economic aspects, it is then possible to determine the optimal duration of each phase and the optimal severity level (i.e., the target transmission rate) for each of them. The numerical analysis is performed for the case of France but the adopted approach can be applied to any country. One of the takeaway messages of this analysis is that being able to implement the optimal 4-phase epidemic management strategy in France would have led to 1.05 million of infected people and a GDP loss of 231 billions € instead of 6.88 millions of infected and a loss of 241 billions €. This indicates that, seen from the proposed model perspective, the effectively implemented epidemic management strategy is good economically, whereas substantial improvements might have been obtained in terms of health impact. Our analysis indicates that the lockdown/severe phase should have been more severe but shorter, and the adjustment phase occurred earlier. Due to the natural tendency of people to deviate from the official rules, updating measures every month over the whole epidemic episode seems to be more appropriate.

A multi-age mathematical modeling of the dynamics of population diabetics with effect of lifestyle using optimal control

Diabetes is a disease which caused by socio-environmental and / or genetic factors. The negative effect of socio-environmental or lifestyle leads a susceptible individual to become a diabetic. On the one hand, social interaction wields a great deal of influence over lifestyle. On the other hand, genetic factors are the main cause of the birth diabetes genetic disorder. Considering these above mentioned factors. In the present paper, we study a discrete age continuous mathematical model that describes the dynamics of diabetics. We highlight the negative impact of socio-environmental on diabetic patients according to age groups. We also suggest an optimal strategy to implement the best campaigns of rising awareness that aims at protecting diabetic patients from the negative impact of a lifestyle that leads them to complications. In addition to psychological treatment and follow-up of diabetic patients with complications, an awareness campaign will also be carried out for people with potential diabetes that aims at educating them about the dangerous of diabetes and its complications. Pontryagin's maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method. The numerical simulation is carried out using MATLAB.

A novel control set-valued approach with application to epidemic models

This work is considered in the framework of studies dedicated to the control problems, especially in epidemiology where the scientist are concerned to develop effective control strategies to minimize the number of infected individuals. In this paper, we set this problem as an asymptotic target control problem under mixed state-control constraints, for a general class of ordinary differential equations that model the temporal evolution of disease spread. The set of initial data, from which the number of infected people decrease to zero, is generated by a new type of Lyapunov functions defined in the sense of viability theory. The associated controls are provided via selections of adequately designed feedback map. The existence of such selections is improved by using Micheal selection theorem. Finally, an application to the SIRS epidemic model, with numerical simulations, is given to show the efficiency of our approach. To the best of our knowledge, our work is the first one that used a set-valued approach based on the viability theory to deal with an epidemic control problem.

Diffusion-reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study

The outbreak of COVID-19 in 2020 has led to a surge in interest in the research of the mathematical modeling of epidemics. Many of the introduced models are so-called compartmental models, in which the total quantities characterizing a certain system may be decomposed into two (or more) species that are distributed into two (or more) homogeneous units called compartments. We propose herein a formulation of compartmental models based on partial differential equations (PDEs) based on concepts familiar to continuum mechanics, interpreting such models in terms of fundamental equations of balance and compatibility, joined by a constitutive relation. We believe that such an interpretation may be useful to aid understanding and interdisciplinary collaboration. We then proceed to focus on a compartmental PDE model of COVID-19 within the newly-introduced framework, beginning with a detailed derivation and explanation. We then analyze the model mathematically, presenting several results concerning its stability and sensitivity to different parameters. We conclude with a series of numerical simulations to support our findings.

A multi-region discrete time mathematical modeling of the dynamics of Covid-19 virus propagation using optimal control

We study in this work a discrete mathematical model that describes the dynamics of transmission of the Corona virus between humans on the one hand and animals on the other hand in a region or in different regions. Also, we propose an optimal strategy to implement the optimal campaigns through the use of awareness campaigns in region j that aims at protecting individuals from being infected by the virus, security campaigns and health measures to prevent the movement of individuals from one region to another, encouraging the individuals to join quarantine centers and the disposal of infected animals. The aim is to maximize the number of individuals subjected to quarantine and trying to reduce the number of the infected individuals and the infected animals. Pontryagin's maximum principle in discrete time is used to characterize the optimal controls and the optimality system is solved by an iterative method. The numerical simulation is carried out using Matlab. The Incremental Cost-Effectiveness Ratio was calculated to investigate the cost-effectiveness of all possible combinations of the four control measures. Using cost-effectiveness analysis, we show that control of protecting susceptible individuals, preventing their contact with the infected individuals and encouraging the exposed individuals to join quarantine centers provides the most cost-effective strategy to control the disease.

Role of Media and Effects of Infodemics and Escapes in the Spatial Spread of Epidemics: A Stochastic Multi-Region Model with Optimal Control Approach

Mass vaccination campaigns play major roles in the war against epidemics. Such prevention strategies cannot always reach their goals significantly without the help of media and awareness campaigns used to prevent contacts between susceptible and infected people. Feelings of fear, infodemics, and misconception could lead to some fluctuations of such policies. In addition to the vaccination strategy, the movement restriction approach is essential because of the factor of mobility or travel. However, anti-epidemic border measures may also be disturbed if some infected travelers manage to escape and infiltrate into a safer region. In this paper, we aim to study infection dynamics related to the spatial spread of an epidemic in interconnected regions in the presence of random perturbations caused by the three above-mentioned reasons. Therefore, we devise a stochastic multi-region epidemic model in which contacts between susceptible and infected populations, vaccination-based and movement restriction optimal control approaches are all assumed to be unpredictable, and then, we discuss the effectiveness of such policies. In order to reach our goal, we employ a stochastic maximum principle version for noised systems, state and prove the sufficient and necessary conditions of optimality, and finally provide the numerical results obtained using a stochastic progressive-regressive schemes method.

Optimal Control and Computational Method for the Resolution of Isoperimetric Problem in a Discrete-Time SIRS System

We consider a discrete-time susceptible-infected-removed-susceptible “again” (SIRS) epidemic model, and we introduce an optimal control function to seek the best control policy for preventing the spread of an infection to the susceptible population. In addition, we define a new compartment, which models the dynamics of the number of controlled individuals and who are supposed not to be able to reach a long-term immunity due to the limited effect of control. Furthermore, we treat the resolution of this optimal control problem when there is a restriction on the number of susceptible people who have been controlled along the time of the control strategy. Further, we provide sufficient and necessary conditions for the existence of the sought optimal control, whose characterization is also given in accordance with an isoperimetric constraint. Finally, we present the numerical results obtained, using a computational method, which combines the secant method with discrete progressive-regressive schemes for the resolution of the discrete two-point boundary value problem.