Mathematical Epidemiology Research Articles

Public healthcare system capacity during COVID‐19: A computational case study of SARS‐CoV‐2

AimCoronavirus Disease (COVID‐19) is spreading typically to the human population all over the world and the report suggests that scientists have been trying to map the pattern of the early transmission of Severe Acute Respiratory Syndrome Coronavirus 2 (SARS‐CoV‐2) since it has been reported as an epidemic. Our main aim is to show if the rise‐in‐cases proceeds in a gradual and staggered manner instead of soaring quickly then we can suppress the burden of the health system. In this new case study, we are attempting to show how to control the outbreak of the infectious disease COVID‐19 via mathematical modeling. We have examined that the method of flattening the curve of the coronavirus, which increases the recovery rate of the infected individuals and also helps to decrease the number of deaths. In this pandemic situation, the countries like Russia, India, the United States of America (USA), South Africa, and the United Kingdom (UK) are leading in front where the virus is spreading in an unprecedented way. From our point of view, we establish that if these countries are following the method of flattening the curve like China and South Korea then these countries can also overcome this pandemic situation.MethodWe propose a Susceptible, Infected, and Recovered (SIR) mathematical model of infectious disease with onset data of COVID‐19 in Wuhan and international cases, which has been propagated in Wuhan City to calculate the transmission rate of the infectious virus COVID‐19 until now. To understand the whole dynamics of the transmission rate of coronavirus, we portray time series diagrams such as growth rate diagram, flattening the pandemic curve diagram, infected and recovered rate diagram, prediction of the transmission of the disease from the available dataset in Wuhan, and internationally exported cases from Wuhan.ResultsWe have observed that the basic reproduction number in Wuhan declined from 2.2 (95% Confidence Interval [CI] 1.15‐4.77) to 1.05 (0.41‐2.39) and the mean incubation period was 5.2 days (95% [CI], 4.1‐7.0). Interestingly the mean value lies between 2 and 2.5 for COVID‐19. The doubling time of COVID‐19 was registered 7.4 days (95% CI, 5.3‐19) in the early stages and now the value decreases to −4.9 days. Similarly, we have observed the doubling time of the epidemic in South Korea decreased to −9.6 days. Currently, the doubling time of the epidemic in Russia, India, and the USA are 19.4 days, 16.4 days, and 41 days, respectively. We have investigated the growth rate of COVID‐19 and plotted the curve flattening diagram against time.ConclusionVia flattening the curve method, China and South Korea control the transmission of the fatal disease COVID‐19 in the human population. Our results show that these two countries initially sustained pandemics in a large portion of the human population in the form of virus outbreaks that basically prevented the virus from spreading further and created ways to prevent community transmission. The majority portion of the people are perfectly fine, who are quarantined strictly and never get sick, but the portion of people who have developed symptoms is quickly isolated further.

What Is the Estimated COVID-19 Reproduction Number and the Proportion of the Population That Needs to Be Immunized to Achieve Herd Immunity in Malaysia? A Mathematical Epidemiology Synthesis

We aimed to determine Malaysia’s COVID-19 reproduction number and herd immunity threshold through a mathematical epidemiology synthesis. Using time-series incidence data, the time-dependent reproduction number (Rt) was yielded over time during the COVID-19 containment measures in Malaysia. The value of Rt at the beginning of the epidemic and prior to any interventions in place was used to determine the proportion of the population that needs to be immunized to achieve herd immunity. Rt was strongly influenced by interventions being put in place. We established that at least 74% of the Malaysian population needs to be vaccinated to achieve herd immunity against COVID-19. This threshold estimate is somewhat influenced by the availability of an efficacious vaccine. A vaccine with 95% efficacy would approximately synthesize a herd immunity threshold of 78%. We conclude that Rt is a valid estimator to determine the effectiveness of control measures and a parameter of use to synthesize herd immunity thresholds in the current COVID-19 pandemic.

Simplicial SIRS epidemic models with nonlinear incidence rates.

Mathematical epidemiology that describes the complex dynamics on social networks has become increasingly popular. However, a few methods have tackled the problem of coupling network topology with complex incidence mechanisms. Here, we propose a simplicial susceptible-infected-recovered-susceptible (SIRS) model to investigate the epidemic spreading via combining the network higher-order structure with a nonlinear incidence rate. A network-based social system is reshaped to a simplicial complex, in which the spreading or infection occurs with nonlinear reinforcement characterized by the simplex dimensions. Compared with the previous simplicial susceptible-infected-susceptible (SIS) models, the proposed SIRS model can not only capture the discontinuous transition and the bistability of a complex system but also capture the periodic phenomenon of epidemic outbreaks. More significantly, the two thresholds associated with the bistable region and the critical value of the reinforcement factor are derived. We further analyze the stability of equilibrium points of the proposed model and obtain the condition of existence of the bistable states and limit cycles. This work expands the simplicial SIS models to SIRS models and sheds light on a novel perspective of combining the higher-order structure of complex systems with nonlinear incidence rates.

COVID-19 outbreaks follow narrow paths: A computational phase portrait approach based on nonlinear physics and synergetics

From the perspective of mathematical epidemiology, COVID-19 epidemics emerge due to instabilities in epidemiological systems. It is shown that the COVID-19 outbreaks follow highly specified paths in epidemiological state spaces. These paths are described by phase portraits that can be readily computed from epidemiological models defined in terms of nonlinear dynamical systems. The paths are predicted by order parameters and amplitude equations that are well known in nonlinear physics and synergetics to exist at instability points. The approach is illustrated for SIR, SEIR and SEIAR models and epidemic outbreaks in China, Italy and West Africa. Identifying such COVID-19 order parameters may help in forecasting COVID-19 epidemics and predicting the impacts of intervention measures.

Perinatology clinic in the coronavirus disease-2019 pandemic: what harms, often teaches

Background Compartmental models simplify the mathematical modeling of infectious diseases based on reported cases. In the absence of precautions, personal protective equipment, quarantine and social distancing, a Susceptible–Exposed–Infectious–Recovered (SuEIR) model with Unscented Kalman Filter for coronavirus disease-19 (COVID-19) Forecasts in Turkey has revealed 174 641 infected people on August 15, 2020, whilst the reported case was 12 216. Through numerical experiments, the effects of quarantine, social distancing, and COVID-19 testing on the dynamics of the outbreak varies. We herein present the documentation of the work in a perinatology clinic during COVID-19 pandemic to find the reflection in a pandemic hospital as even in the pandemic, pregnancy complications and fetal diagnosis/therapy are time-sensitive and cannot be delayed. During the prevention of the horizontal transmission to the health-care workers (HCWs), testing all pregnant women with nasopharyngeal/oropharyngeal swabs for severe acute respiratory syndrome coronavirus (SARS-COV-2) undergoing birth, ultrasound examinations, invasive procedures appear to be the gold standard so that appropriate precautions can be taken if the screen is positive. Though it is logical, it may be incompatible with a busy obstetric practise as a pending polymerase chain reaction (PCR) result should never delay any emergent procedure. Objective We aim to describe the development of COVID-19 disease of 408 HCW out of 1462 by the exposure to pregnant women while providing obstetric care in a single tertiary perinatology unit under strict clinical triage, recommended precautions and wearing personal protective equipment and compare the maternal and perinatal outcome with those of the preceding three months. Study design A prospective cohort study involving the pregnant women and the HCW with positive PCR for SARS-COV-2 were carried out to correlate with the horizontal transmission while documenting the perinatal work. Results 25 HCW, including nurses/midwives: 11, doctors: 7 and health technicians: 3 and support staff: 4 developed positive PCR for SARS-COV-2 while providing healthcare to 162 cases: mild-moderate (n = 146), severe (n = 12) and critical (n = 1) and asymptomatic (n = 3) in obstetric population. 22 out of 25 HCW were working in the perinatology unit. COVID-19 clinic was asymptomatic (n = 8), mild-moderate (13) or severe (n = 2) in HCW. However, “Exposed” group in the SuEIR model, both the pregnant women and the HCW that have already been infected and have not been tested, which have been also capable of infecting the “Susceptible” group could not be determined. Some of the HCW and the pregnant women in the “Exposed” group were tested and transferred to the “Infectious” group (which were reported to be PCR positive), while the rest of them who recovered, transitted to the so-called “Unreported Recovered” group. The ratio of the women with severe pre-eclampsia admitted to intensive care unit increased significantly during the lockdown (p = .01). Conclusions In a nonstop pandemic perinatology clinic, exposure to 162 PCR positive pregnant women may be correlated with a 5.4% (22/408) documented horizontal transmission in the frontline HCW despite clinical triage and personal protective equipment.

Adaptive mesh refinement and coarsening for diffusion-reaction epidemiological models.

The outbreak of COVID-19 in 2020 has led to a surge in the interest in the mathematical modeling of infectious diseases. Disease transmission may be modeled as compartmental models, in which the population under study is divided into compartments and has assumptions about the nature and time rate of transfer from one compartment to another. Usually, they are composed of a system of ordinary differential equations in time. A class of such models considers the Susceptible, Exposed, Infected, Recovered, and Deceased populations, the SEIRD model. However, these models do not always account for the movement of individuals from one region to another. In this work, we extend the formulation of SEIRD compartmental models to diffusion–reaction systems of partial differential equations to capture the continuous spatio-temporal dynamics of COVID-19. Since the virus spread is not only through diffusion, we introduce a source term to the equation system, representing exposed people who return from travel. We also add the possibility of anisotropic non-homogeneous diffusion. We implement the whole model in libMesh, an open finite element library that provides a framework for multiphysics, considering adaptive mesh refinement and coarsening. Therefore, the model can represent several spatial scales, adapting the resolution to the disease dynamics. We verify our model with standard SEIRD models and show several examples highlighting the present model’s new capabilities.

Optimal Perturbations of Systems with Delayed Independent Variables for Control of Dynamics of Infectious Diseases Based on Multicomponent Actions

In this paper, we apply optimal perturbations to control mathematical models of infectious diseases expressed as systems of nonlinear differential equations with delayed independent variables. We develop the method for calculation of perturbations of the initial state of a dynamical system with delayed independent variable producing maximal amplification in the given local norm taking into account weights of perturbation components. For the model of experimental virus infection, we construct optimal perturbation for two types of stationary states, with low or high viral load, corresponding to different variants of chronic virus infection flow.

A case study of Covid-19 epidemic in India via new generalised Caputo type fractional derivatives.

The first symptomatic infected individuals of coronavirus (Covid‐19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid‐19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor‐Corrector scheme. For the proposed model of Covid‐19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non‐integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's‐Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non‐linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non‐linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non‐integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.

Mathematical Analysis of Epidemic Disease Models and Application to COVID-19

Mathematical models for infectious diseases are considered. The exact solutions of SIR model are analyzed in terms of oscillatory motion in susceptible–infected (S–I) phase space. SIR model can not...

Facilitating Understanding, Modeling and Simulation of Infectious Disease Epidemics in the Age of COVID-19

Interest in the mathematical modeling of infectious diseases has increased due to the COVID-19 pandemic. However, many medical students do not have the required background in coding or mathematics to engage optimally in this approach. System dynamics is a methodology for implementing mathematical models as easy-to-understand stock-flow diagrams. Remarkably, creating stock-flow diagrams is the same process as creating the equivalent differential equations. Yet, its visual nature makes the process simple and intuitive. We demonstrate the simplicity of system dynamics by applying it to epidemic models including a model of COVID-19 mutation. We then discuss the ease with which far more complex models can be produced by implementing a model comprising eight differential equations of a Chikungunya epidemic from the literature. Finally, we discuss the learning environment in which the teaching of the epidemic modeling occurs. We advocate the widespread use of system dynamics to empower those who are engaged in infectious disease epidemiology, regardless of their mathematical background.

Calculating prescription rates and addiction probabilities for the four most commonly prescribed opioids and evaluating their impact on addiction using compartment modelling.

In 2016, more than 11 million Americans abused prescription opioids. The National Institute on Drug Abuse considers the opioid crisis a national addiction epidemic, as an increasing number of people are affected each year. Using the framework developed in mathematical modelling of infectious diseases, we create and analyse a compartmental opioid-abuse model consisting of a system of ordinary differential equations. Since $40\%$ of opioid overdoses are caused by prescription opioids, our model includes prescription compartments for the four most commonly prescribed opioids, as well as for the susceptible, addicted and recovered populations. While existing research has focused on drug abuse models in general and opioid models with one prescription compartment, no previous work has been done comparing the roles that the most commonly prescribed opioids have had on the crisis. By combining data from the Substance Abuse and Mental Health Services Administration (which tracked the proportion of people who used or misused one of the four individual opioids) with data from the Centers of Disease Control and Prevention (which counted the total number of prescriptions), we estimate prescription rates and probabilities of addiction for the four most commonly prescribed opioids. Additionally, we perform a sensitivity analysis and reallocate prescriptions to determine which opioid has the largest impact on the epidemic. Our results indicate that oxycodone prescriptions are both the most likely to lead to addiction and have the largest impact on the size of the epidemic, while hydrocodone prescriptions had the smallest impact.

Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives

The first reported case of coronavirus disease (COVID-19) in Brazil was confirmed on 25 February 2020 and then the number of symptomatic cases produced day by day. In this manuscript, we studied the epidemic peaks of the novel coronavirus (COVID-19) in Brazil by the successful application of Predictor–Corrector (P-C) scheme. For the proposed model of COVID-19, the numerical solutions are performed by a model framework of the recent generalized Caputo type non-classical derivative. Existence of unique solution of the given non-linear problem is presented in terms of theorems. A new analysis of epidemic peaks in Brazil with the help of parameter values cited from a real data is effectuated. Graphical simulations show the obtained results to classify the importance of the classes of projected model. We observed that the proposed fractional technique is smoothly work in the coding and very easy to implement for the model of non-linear equations. By this study we tried to exemplify the roll of newly proposed fractional derivatives in mathematical epidemiology. The main purpose of this paper is to predict the epidemic peak of COVID-19 in Brazil at different transmission rates. We have also attempted to give the stability analysis of the proposed numerical technique by the reminder of some important lemmas. At last we concluded that when the infection rate increases then the nature of the diseases changes by becoming more deathly to the population.

Parameter Estimation of Compartmental Epidemiological Model Using Harmony Search Algorithm and Its Variants

Epidemiological models play a vital role in understanding the spread and severity of a pandemic of infectious disease, such as the COVID-19 global pandemic. The mathematical modeling of infectious diseases in the form of compartmental models are often employed in studying the probable outbreak growth. Such models heavily rely on a good estimation of the epidemiological parameters for simulating the outbreak trajectory. In this paper, the parameter estimation is formulated as an optimization problem and a metaheuristic algorithm is applied, namely Harmony Search (HS), in order to obtain the optimized epidemiological parameters. The application of HS in epidemiological modeling is demonstrated by implementing ten variants of HS algorithm on five COVID-19 data sets that were calibrated with the prototypical Susceptible-Infectious-Removed (SIR) compartmental model. Computational experiments indicated the ability of HS to be successfully applied to epidemiological modeling and as an efficacious estimator for the model parameters. In essence, HS is proposed as a potential alternative estimation tool for parameters of interest in compartmental epidemiological models.

Using a COVID-19 Model in Various Classroom Settings to Assess Effects of Interventions

ABSTRACT The COVID-19 pandemic has made mathematical epidemiology a topic of critical importance, providing mathematics educators with an unparalleled opportunity. This opportunity is accompanied by a challenge: how do mathematics educators, some of whom have little personal experience with mathematical modeling, teach mathematical epidemiology to their students in courses ranging from precalculus to differential equations, and do so in a way that builds understanding of epidemic disease dynamics as well as mathematical methods? We address this issue with a collection of materials that allow students to conduct virtual experiments with a COVID-19 model to assess the effects of public health policies and community behavior. The materials are designed to require only a bare minimum of coding by students so as to focus students' efforts on interpretation of results.

Are college campuses superspreaders? A data-driven modeling study

The COVID-19 pandemic continues to present enormous challenges for colleges and universities and strategies for save reopening remain a topic of ongoing debate. Many institutions that reopened cautiously in the fall experienced a massive wave of infections and colleges were soon declared as the new hotspots of the pandemic. However, the precise effects of college outbreaks on their immediate neighborhood remain largely unknown. Here we show that the first two weeks of instruction present a high-risk period for campus outbreaks and that these outbreaks tend to spread into the neighboring communities. By integrating a classical mathematical epidemiology model and Bayesian learning, we learned the dynamic reproduction number for 30 colleges from their daily case reports. Of these 30 institutions, 14 displayed a spike of infections within the first two weeks of class, with peak seven-day incidences well above 1,000 per 100,000, an order of magnitude larger than the nation-wide peaks of 70 and 150 during the first and second waves of the pandemic. While most colleges were able to rapidly reduce the number of new infections, many failed to control the spread of the virus beyond their own campus: Within only two weeks, 17 campus outbreaks translated directly into peaks of infection within their home counties. These findings suggests that college campuses are at risk to develop an extreme incidence of COVID-19 and become superspreaders for neighboring communities. We anticipate that tight test-trace-quarantine strategies, flexible transition to online instruction, and–most importantly–compliance with local regulations will be critical to ensure a safe campus reopening after the winter break.

Impact analysis of COVID-19 on the imminent tightness of hospital beds using the mathematical model of infectious disease

Impact analysis of COVID-19 on the imminent tightness of hospital beds using the mathematical model of infectious disease

Mathematical Models for Infectious Diseases and COVID-19

Mathematical Models for Infectious Diseases and COVID-19

Dynamical Behaviors of Nonlinear Coronavirus (COVID 9) Model with Numerical Studies

The development of mathematical modeling of infectious diseases is a key research area in various fields including ecology and epidemiology. One aim of these models is to understand the dynamics of behavior in infectious diseases. For the new strain of coronavirus (COVID-19), there is no vaccine to protect people and to prevent its spread so far. Instead, control strategies associated with health care, such as social distancing, quarantine, travel restrictions, can be adopted to control the pandemic of COVID-19. This article sheds light on the dynamical behaviors of nonlinear COVID-19 models based on two methods: the homotopy perturbation method (HPM) and the modified reduced differential transform method (MRDTM). We invoke a novel signal flow graph that is used to describe the COVID-19 model. Through our mathematical studies, it is revealed that social distancing between potentially infected individuals who are carrying the virus and healthy individuals can decrease or interrupt the spread of the virus. The numerical simulation results are in reasonable agreement with the study predictions. The free equilibrium and stability point for the COVID-19 model are investigated. Also, the existence of a uniformly stable solution is proved.

Solutions of transmission dynamics model for HBV using HPM with residual errors

The transmission dynamics mathematical model of infectious disease is a significant disease controlling technique, which is being used on the occurrence of hepatitis B to value the varying immunization strategies. In this paper, we analyze the transmission dynamics models through mathematical model using Homotopy perturbation method (HPM) which defines how to control the impact of Hepatitis B virus (HBV). To get the solution for nonlinear ordinary differential equations, Homotopy Perturbation Method (HPM) has been used. We have discussed the numerical simulations up to six order approximation and residual error analysis with the help of Matlab software in this paper.

Homotopy Perturbation Method for Mathematical Modelling of Dengue Fever

Dengue fever is a viral mosquito-transmitted infection that has become a major international infection in recent years. The leading cause of disease and death in tropical and sub-tropical regions is a public health concern. Models from mathematical epidemiology, such as the classical SIR-model and its variants, are used to characterize the spread of Dengue in a given population. The mathematical modelling of Dengue Fever is formulated into a first-order nonlinear differential equation. Homotopy Perturbation Method approaches the analytical solution of the model (HPM), and also simulation results are identified. Finally, the analytical solutions, simulation results are compared, and satisfactory agreement is noted.