(IME - 2020/2021 - 2 FASE)Text 3Mathematical Model

(IME - 2020/2021 - 2ª FASE)

Text 3

Mathematical Modeling of Epidemic Diseases; A Case Study of the COVID-19 Coronavirus

Reza Sameni

Abstract — The outbreak of the Coronavirus COVID-19 has taken the lives of several thousands worldwide and lockedout many countries and regions, with yet unpredictable global consequences. In this research we study the epidemic patterns of this virus, from a mathematical modeling perspective. The study is based on an extensions of the well-known susceptible-infected recovered (SIR) family of compartmental models. It is shown how social measures such as distancing, regional lockdowns, quarantine and global public health vigilance, influence the model parameters, which can eventually change the mortality rates and active contaminated cases over time, in the real world. As with all mathematical models, the predictive ability of the model is limited by the accuracy of the available data and to the so-called level of abstraction used for modeling the problem. In order to provide the broader audience of researchers a better understanding of spreading patterns of epidemic diseases, a short introduction on biological systems modeling is also presented and the Matlab source codes for the simulations are provided online.

I. INTRODUCTION

Since the outbreak of the Coronavirus COVID-19 in January 2020, the virus has affected most countries and taken the lives of several thousands of people worldwide. By March 2020, the World Health Organization (WHO) declared the situation a pandemic, the first of its kind in our generation. To date, many countries and regions have been locked-down and applied strict social distancing measures to stop the virus propagation. From a strategic and healthcare management perspective, the propagation pattern of the disease and the prediction of its spread over time is of great importance, to save lives and to minimize the social and economic consequences of the disease. Within the scientific community, the problem of interest has been studied in various communities including mathematical epidemiology, biological systems modeling, signal processing and control engineering.

In this study, epidemic outbreaks are studied from an interdisciplinary perspective, by using an extension of the susceptible-exposed-infected-recovered (SEIR) model, which is a mathematical compartmental model based on the average behavior of a population under study. The objective is to provide researchers a better understanding of the significance of mathematical modeling for epidemic diseases. It is shown by simulation, how social measures such as distancing, regional lockdowns and public health vigilance, can influence the model parameters, which in turns change the mortality rates and active contaminated cases over time.

It should be highlighted that mathematical models applied to real-world systems (social, biological, economical, etc.) are only valid under their assumptions and hypothesis. Therefore, this research— and similar ones— that address epidemic patterns, do not convey direct clinical information and dangers for the public, but should rather be used by healthcare strategists for better planning and decision making. Hence, the study of this work is only recommended for researchers familiar with the strength points and limitations of mathematical modeling of biological systems. The Matlab codes required for reproducing the results of this research are also online available in the Git repository of the project. In Section II, a brief introduction to mathematical modeling of biological systems is presented, to highlight the scope of the present study and to open perspectives for the interested researchers, who may be less familiar with the context. The proposed model for the outspread of the Coronavirus is presented in Section III. The article is concluded with some general remarks and future perspectives.

Adapted from: Mathematical Modeling of Epidemic Diseases; A Case Study of the COVID-19 Coronavirus. Avaliable at: <arxiv.org/abs/2003.11371> [Accessed 6th June 2020].

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Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China

B. Ivorra, M.R. Ferrández, M. Vela-Pérez and A.M. Ramosa

Abstract 

In this paper we develop a mathematical model for the spread of the coronavirus disease 2019 (COVID-19). It is a new θ-SEIHRD model (not a SIR, SEIR or other general purpose model), which takes into account the known special characteristics of this disease, as the existence of infectious undetected cases and the different sanitary and infectiousness conditions of hospitalized people. In particular, it includes a novel approach that considers the fraction θ of detected cases over the real total infected cases, which allows to study the importance of this ratio on the impact of COVID-19. The model is also able to estimate the needs of beds in hospitals. It is complex enough to capture the most important effects, but also simple enough to allow an affordable identification of its parameters, using the data that authorities report on this pandemic.

We study the particular case of China (including Chinese Mainland, Macao, Hong-Kong and Taiwan, as done by the World Health Organization in its reports on COVID-19), the country spreading the disease, and use its reported data to identify the model parameters, which can be of interest for estimating the spread of COVID-19 in other countries. We show a good agreement between the reported data and the estimations given by our model. We also study the behavior of the outputs returned by our model when considering incomplete reported data (by truncating them at some dates before and after the peak of daily reported cases). By comparing those results, we can estimate the error produced by the model when identifying the parameters at early stages of the pandemic. Finally, taking into account the advantages of the novelties introduced by our model, we study different scenarios to show how different values of the percentage of detected cases would have changed the global magnitude of COVID-19 in China, which can be of interest for policy makers.

Keywords: Mathematical model, θ-SEIHRD model, COVID-19, Coronavirus, SARS-CoV-2, Pandemic, Numerical simulation, Parameter estimation

1. Introduction

Modeling and simulation are important decision tools that can be useful to control human and animal diseases. However, since each disease exhibits its own particular biological characteristics, the models need to be adapted to each specific case in order to be able to tackle real situations.

Coronavirus disease 2019 (COVID-19) is an infectious disease emerging in China in December 2019 that has rapidly spread around China and many other countries. On 11 February 2020, the World Health Organization (WHO) renamed the epidemic disease caused by 2019-nCoV as strain severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)

This is a new virus and a completely new situation. On 30 January 2020, WHO declared it to be a Public Health Emergency of International Concern. As of 11 March 2020, the disease was confirmed in more than 118,000 cases reported globally in 114 countries, more than 90 percent of cases are in just four countries (two of those China and the Republic of Korea - have significantly declining epidemics) and WHO declared it to be a pandemic, the first one caused by a coronavirus. On 1 April 2020 there are 872,481 and 43,275 official reported cases and deaths, respectively, and there is no vaccine specifically designed for this virus, with proven effectiveness.

There are some mathematical models in the literature that try to describe the dynamics of the evolution of COVID-19. (...) Other works propose SEIR type models with little variations and some of them incorporate stochastic components. COVID-19 is a disease caused by a new virus, which is generating a worldwide emergency situation and needs a model taking into account its known specific characteristics.

Adaptec from: Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China. In: Elsevier Public Health Emergency Colletion, 2020. Avaliable at: <https://www.mcbi.nlm.nih.gov/pmc/articles/PMC7190554/> [Accessed 4th June 2020].

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