THE FIRST EPIDEMIC MODEL: A HISTORICAL NOTE ON P.D. EN'KO

Predictive algorithm for the regional spread of coronavirus infection across the Russian Federation

BACKGROUND

Outbreaks of infectious diseases are a complex phenomenon with many interacting factors. Regional health authorities need prognostic modeling of the epidemic process.

METHODS

For these purposes, various mathematical algorithms can be used, which are a useful tool for studying the infections spread dynamics. Epidemiological models act as evaluation and prognosis models. The authors outlined the experience of developing a short-term predictive algorithm for the spread of the COVID-19 in the region of the Russian Federation based on the SIR model: Susceptible (vulnerable), Infected (infected), Recovered (recovered). The article describes in detail the methodology of a short-term predictive algorithm, including an assessment of the possibility of building a predictive model and the mathematical aspects of creating such forecast algorithms.

RESULTS

Findings show that the predicted results (the mean square of the relative error of the number of infected and those who had recovered) were in agreement with the real-life situation: σ(I) = 0.0129 and σ(R) = 0.0058, respectively.

CONCLUSIONS

The present study shows that despite a large number of sophisticated modifications, each of which finds its scope, it is advisable to use a simple SIR model to quickly predict the spread of coronavirus infection. Its lower accuracy is fully compensated by the adaptive calibration of parameters based on monitoring the current situation with updating indicators in real-time.

The transmission mechanism theory of disease dynamics: Its aims, assumptions and limitations

Most of the progress in the development of single scale mathematical and computational models for the study of infectious disease dynamics which now span over a century is build on a body of knowledge that has been developed to address particular single scale descriptions of infectious disease dynamics based on understanding disease transmission process. Although this single scale understanding of infectious disease dynamics is now founded on a body of knowledge with a long history, dating back to over a century now, that knowledge has not yet been formalized into a scientific theory. In this article, we formalize this accumulated body of knowledge into a scientific theory called the transmission mechanism theory of disease dynamics which states that at every scale of organization of an infectious disease system, disease dynamics is determined by transmission as the main dynamic disease process. Therefore, the transmission mechanism theory of disease dynamics can be seen as formalizing knowledge that has been inherent in the study of infectious disease dynamics using single scale mathematical and computational models for over a century now. The objective of this article is to summarize this existing knowledge about single scale modelling of infectious dynamics by means of a scientific theory called the transmission mechanism theory of disease dynamics and highlight its aims, assumptions and limitations.

Mathematical modeling in perspective of vector-borne viral infections: a review

Background

Viral diseases are highly widespread infections caused by viruses. These viruses are passing from one human to other humans through a certain medium. The medium might be mosquito, animal, reservoir and food, etc. Here, the population of both human and mosquito vectors are important.

Main body of the abstract

The main objectives are here to introduce the historical perspective of mathematical modeling, enable the mathematical modeler to understand the basic mathematical theory behind this and present a systematic review on mathematical modeling for four vector-borne viral diseases using the deterministic approach. Furthermore, we also introduced other mathematical techniques to deal with vector-borne diseases. Mathematical models could help forecast the infectious population of humans and vectors during the outbreak.

Short conclusion

This study will be helpful for mathematical modelers in vector-borne diseases and ready-made material in the review for future advancement in the subject. This study will not only benefit vector-borne conditions but will enable ideas for other illnesses.

Evaluation of COVID-19 pandemic spreading using computational analysis on nonlinear SITR model

The main purpose of present paper is to investigate the nonlinear model of COVID-19 (novel coronavirus) computationally. The SITR model is designed according to four classifications of Susceptible (S), Infectious (I), Treatment (T) and Recovered (R). Two convenient and effective numerical techniques namely the Adams-Bashforth Method (ABM) and Milne-Simpson Method (MSM) are employed to analyze the epidemic model. The influences of the contact rate parameter (β), recovery parameter (μ) and death parameter (α) on the variables including S, I and R are studied comprehensively. The obtained findings indicate that by increasing the contact rate parameter the infectious and recovered categories enhance but the susceptible mechanism decreases.

The Hybrid Incidence Susceptible-Transmissible-Removed Model for Pandemics : Scaling Time to Predict an Epidemic's Population Density Dependent Temporal Propagation

The susceptible-transmissible-removed (STR) model is a deterministic compartment model, based on the susceptible-infected-removed (SIR) prototype. The STR replaces 2 SIR assumptions. SIR assumes that the emigration rate (due to death or recovery) is directly proportional to the infected compartment’s size. The STR replaces this assumption with the biologically appropriate assumption that the emigration rate is the same as the immigration rate one infected period ago. This results in a unique delay differential equation epidemic model with the delay equal to the infected period. Hamer’s mass action law for epidemiology is modified to resemble its chemistry precursor—the law of mass action. Constructing the model for an isolated population that exists on a surface bounded by the extent of the population’s movements permits compartment density to replace compartment size. The STR reduces to a SIR model in a timescale that negates the delay—the transmissible timescale. This establishes that the SIR model applies to an isolated population in the disease’s transmissible timescale. Cyclical social interactions will define a rhythmic timescale. It is demonstrated that the geometric mean maps transmissible timescale properties to their rhythmic timescale equivalents. This mapping defines the hybrid incidence (HI). The model validation demonstrates that the HI-STR can be constructed directly from the disease’s transmission dynamics. The basic reproduction number (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}}_0$$\end{document}R0) is an epidemic impact property. The HI-STR model predicts that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}}_0 \propto \root \mathfrak{B} \of {\rho_n}$$\end{document}R0∝ρnB where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_n$$\end{document}ρn is the population density, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{B}}$$\end{document}B is the ratio of time increments in the transmissible- and rhythmic timescales. The model is validated by experimentally verifying the relationship. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}}_0$$\end{document}R0’s dependence on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_n$$\end{document}ρn is demonstrated for droplet-spread SARS in Asian cities, aerosol-spread measles in Europe and non-airborne Ebola in Africa.

Swarming morlet wavelet neural network procedures for the mathematical robot system

The task of this work is to present the solutions of the mathematical robot system (MRS) to examine the positive coronavirus cases through the artificial intelligence (AI) based Morlet wavelet neural network (MWNN). The MRS is divided into two classes, infected I ( θ ) and Robots R ( θ ) . The design of the fitness function is presented by using the differential MRS and then optimized by the hybrid of the global swarming computational particle swarm optimization (PSO) and local active set procedure (ASP). For the exactness of the AI based MWNN-PSOIPS, the comparison of the results is presented by using the proposed and reference solutions. The reliability of the MWNN-PSOASP is authenticated by extending the data into 20 trials to check the performance of the scheme by using the statistical operators with 10 hidden numbers of neurons to solve the MRS.

Numerical treatment on the new fractional-order SIDARTHE COVID-19 pandemic differential model via neural networks

In this study, modeling the COVID-19 pandemic via a novel fractional-order SIDARTHE (FO-SIDARTHE) differential system is presented. The purpose of this research seemed to be to show the consequences and relevance of the fractional-order (FO) COVID-19 SIDARTHE differential system, as well as FO required conditions underlying four control measures, called SI,  SD,  SA, and SR. The FO-SIDARTHE system incorporates eight phases of infection: susceptible (S), infected (I), diagnosed (D), ailing (A), recognized (R), threatening (T), healed (H), and extinct (E). Our objective of all these investigations is to use fractional derivatives to increase the accuracy of the SIDARTHE system. A FO-SIDARTHE system has yet to be disclosed, nor has it yet been treated using the strength of stochastic solvers. Stochastic solvers based on the Levenberg-Marquardt backpropagation methodology (L-MB) and neural networks (NNs), specifically L-MBNNs, are being used to analyze a FO-SIDARTHE problem. Three cases having varied values under the same fractional order are being presented to resolve the FO-SIDARTHE system. The statistics employed to provide numerical solutions toward the FO-SIDARTHE system are classified as obeys: 72% toward training, 18% in testing, and 10% for authorization. To establish the accuracy of such L-MBNNs utilizing Adams-Bashforth-Moulton, the numerical findings were compared with the reference solutions.

A box, a trough and marbles: How the Reed-Frost epidemic theory shaped epidemiological reasoning in the 20th century

The article takes the renewed popularity and interest in epidemiological modelling for Covid-19 as a point of departure to ask how modelling has historically shaped epidemiological reasoning. The focus lies on a particular model, developed in the late 1920s through a collaboration of the former field-epidemiologists and medical officer, Wade Hampton Frost, and the biostatistician and population ecologist Lowell Reed. Other than former approaches to epidemic theory in mathematical formula, the Reed-Frost epidemic theory was materialised in a simple mechanical analogue: a box with coloured marbles and a wooden trough. The article reconstructs how the introduction of this mechanical model has reshaped epidemiological reasoning by shifting the field from purely descriptive to analytical practices. It was not incidental that the history of this model coincided with the foundation of epidemiology as an academic discipline, as it valorised and institutionalised new theoretical contributions to the field. Through its versatility, the model shifted the field's focus from mono-causal explanations informed by bacteriology, eugenics or sanitary perspectives towards the systematic consideration of epidemics as a set of interdependent and dynamic variables.

A unifying nonlinear probabilistic epidemic model in space and time

Covid-19 epidemic dramatically relaunched the importance of mathematical modelling in supporting governments decisions to slow down the disease propagation. On the other hand, it remains a challenging task for mathematical modelling. The interplay between different models could be a key element in the modelling strategies. Here we propose a continuous space-time non-linear probabilistic model from which we can derive many of the existing models both deterministic and stochastic as for example SI, SIR, SIR stochastic, continuous-time stochastic models, discrete stochastic models, Fisher-Kolmogorov model. A partial analogy with the statistical interpretation of quantum mechanics provides an interpretation of the model. Epidemic forecasting is one of its possible applications; in principle, the model can be used in order to locate those regions of space where the infection probability is going to increase. The connection between non-linear probabilistic and non-linear deterministic models is analyzed. In particular, it is shown that the Fisher-Kolmogorov equation is connected to linear probabilistic models. On the other hand, a generalized version of the Fisher-Kolmogorov equation is derived from the non-linear probabilistic model and is shown to be characterized by a non-homogeneous time-dependent diffusion coefficient (anomalous diffusion) which encodes information about the non-linearity of the probabilistic model.

Computational Intelligent Paradigms to Solve the Nonlinear SIR System for Spreading Infection and Treatment Using Levenberg–Marquardt Backpropagation

The current study aims to design an integrated numerical computing-based scheme by applying the Levenberg–Marquardt backpropagation (LMB) neural network to solve the nonlinear susceptible (S), infected (I) and recovered (R) (SIR) system of differential equations, representing the spreading of infection along with its treatment. The solutions of both the categories of spreading infection and its treatment are presented by taking six different cases of SIR models using the designed LMB neural network. A reference dataset of the designed LMB neural network is established with the Adam numerical scheme for each case of the spreading infection and its treatment. The approximate outcomes of the SIR system based on the spreading infection and its treatment are presented in the training, authentication and testing procedures to adapt the neural network by reducing the mean square error (MSE) function using the LMB. Studies based on the proportional performance and inquiries based on correlation, error histograms, regression and MSE results establish the efficiency, correctness and effectiveness of the proposed LMB neural network scheme.

Estimating and interpreting secondary attack risk: Binomial considered biased

The household secondary attack risk (SAR), often called the secondary attack rate or secondary infection risk, is the probability of infectious contact from an infectious household member A to a given household member B, where we define infectious contact to be a contact sufficient to infect B if he or she is susceptible. Estimation of the SAR is an important part of understanding and controlling the transmission of infectious diseases. In practice, it is most often estimated using binomial models such as logistic regression, which implicitly attribute all secondary infections in a household to the primary case. In the simplest case, the number of secondary infections in a household with m susceptibles and a single primary case is modeled as a binomial(m, p) random variable where p is the SAR. Although it has long been understood that transmission within households is not binomial, it is thought that multiple generations of transmission can be neglected safely when p is small. We use probability generating functions and simulations to show that this is a mistake. The proportion of susceptible household members infected can be substantially larger than the SAR even when p is small. As a result, binomial estimates of the SAR are biased upward and their confidence intervals have poor coverage probabilities even if adjusted for clustering. Accurate point and interval estimates of the SAR can be obtained using longitudinal chain binomial models or pairwise survival analysis, which account for multiple generations of transmission within households, the ongoing risk of infection from outside the household, and incomplete follow-up. We illustrate the practical implications of these results in an analysis of household surveillance data collected by the Los Angeles County Department of Public Health during the 2009 influenza A (H1N1) pandemic.

The household secondary attack risk (SAR), often called the secondary attack rate or secondary infection risk, is the probability of infectious contact from an infectious household member A to a given household member B, where we define infectious contact to be a contact sufficient to infect B if he or she is susceptible. The most common statistical models used to estimate the SAR are binomial models such as logistic regression, which implicitly assume that all secondary infections in a household are infected by the primary case. Here, we use analytical calculations and simulations to show that estimation of the SAR must account for multiple generations of transmission within households. As an example, we show that binomial models and statistical models that account for multiple generations of within-household transmission reach different conclusions about the household SAR for 2009 influenza A (H1N1) in Los Angeles County, with the latter models fitting the data better. In an epidemic, accurate estimation of the SAR allows rigorous evaluation of the effectiveness of public health interventions such as social distancing, prophylaxis or treatment, and vaccination.

Transmission dynamics and control methodology of COVID-19: A modeling study

The coronavirus disease 2019 (COVID-19) has grown up to be a pandemic within a short span of time. To investigate transmission dynamics and then determine control methodology, we took epidemic in Wuhan as a study case. Unfortunately, to our best knowledge, the existing models are based on the common assumption that the total population follows a homogeneous spatial distribution, which is not the case for the prevalence occurred both in the community and in hospital due to the difference in the contact rate. To solve this problem, we propose a novel epidemic model called SEIR-HC, which is a model with two different social circles (i.e., individuals in hospital and community). Using the model alongside the exclusive optimization algorithm, the spread process of COVID-19 epidemic in Wuhan city is reproduced and then the propagation characteristics and unknown data are estimated. The basic reproduction number of COVID-19 is estimated to be 7.9, which is far higher than that of the severe acute respiratory syndrome (SARS). Furthermore, the control measures implemented in Wuhan are assessed and the control methodology of COVID-19 is discussed to provide guidance for limiting the epidemic spread.

Modelling Excess Mortality in Covid-19-Like Epidemics

We develop an agent-based model to assess the cumulative number of deaths during hypothetical Covid-19-like epidemics for various non-pharmaceutical intervention strategies. The model simulates three interrelated stochastic processes: epidemic spreading, availability of respiratory ventilators and changes in death statistics. We consider local and non-local modes of disease transmission. The first simulates transmission through social contacts in the vicinity of the place of residence while the second through social contacts in public places: schools, hospitals, airports, etc., where many people meet, who live in remote geographic locations. Epidemic spreading is modelled as a discrete-time stochastic process on random geometric networks. We use the Monte-Carlo method in the simulations. The following assumptions are made. The basic reproduction number is R0=2.5 and the infectious period lasts approximately ten days. Infections lead to severe acute respiratory syndrome in about one percent of cases, which are likely to lead to respiratory default and death, unless the patient receives an appropriate medical treatment. The healthcare system capacity is simulated by the availability of respiratory ventilators or intensive care beds. Some parameters of the model, like mortality rates or the number of respiratory ventilators per 100,000 inhabitants, are chosen to simulate the real values for the USA and Poland. In the simulations we compare 'do-nothing' strategy with mitigation strategies based on social distancing and reducing social mixing. We study epidemics in the pre-vacine era, where immunity is obtained only by infection. The model applies only to epidemics for which reinfections are rare and can be neglected. The results of the simulations show that strategies that slow the development of an epidemic too much in the early stages do not significantly reduce the overall number of deaths in the long term, but increase the duration of the epidemic. In particular, a hybrid strategy where lockdown is held for some time and is then completely released, is inefficient.

Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic

Nowadays, COVID-19 has put a significant responsibility on all of us around the world from its detection to its remediation. The globe suffer from lockdown due to COVID-19 pandemic. The researchers are doing their best to discover the nature of this pandemic and try to produce the possible plans to control it. One of the most effective method to understand and control the evolution of this pandemic is to model it via an efficient mathematical model. In this paper, we propose to model COVID-19 pandemic by fractional order SIDARTHE model which did not appear in the literature before. The existence of a stable solution of the fractional order COVID-19 SIDARTHE model is proved and the fractional order necessary conditions of four proposed control strategies are produced. The sensitivity of the fractional order COVID-19 SIDARTHE model to the fractional order and the infection rate parameters are displayed. All studies are numerically simulated using MATLAB software via fractional order differential equation solver.

Estimating Parameters in Mathematical Model for Societal Booms through Bayesian Inference Approach

In this study, based on our previous study in which the proposed model is derived based on the SIR model and E. M. Rogers’s Diffusion of Innovation Theory, including the aspects of contact and time delay, we examined the mathematical properties, especially the stability of the equilibrium for our proposed mathematical model. By means of the results of the stability in this study, we also used actual data representing transient and resurgent booms, and conducted parameter estimation for our proposed model using Bayesian inference. In addition, we conducted a model fitting to five actual data. By this study, we reconfirmed that we can express the resurgences or minute oscillations of actual data by means of our proposed model.

SBDiEM: A new mathematical model of infectious disease dynamics

A worldwide multi-scale interplay among a plethora of factors, ranging from micro-pathogens and individual or population interactions to macro-scale environmental, socio-economic and demographic conditions, entails the development of highly sophisticated mathematical models for robust representation of the contagious disease dynamics that would lead to the improvement of current outbreak control strategies and vaccination and prevention policies. Due to the complexity of the underlying interactions, both deterministic and stochastic epidemiological models are built upon incomplete information regarding the infectious network. Hence, rigorous mathematical epidemiology models can be utilized to combat epidemic outbreaks. We introduce a new spatiotemporal approach (SBDiEM) for modeling, forecasting and nowcasting infectious dynamics, particularly in light of recent efforts to establish a global surveillance network for combating pandemics with the use of artificial intelligence. This model can be adjusted to describe past outbreaks as well as COVID-19. Our novel methodology may have important implications for national health systems, international stakeholders and policy makers.

Stochastic and spatio-temporal analysis of the Middle East Respiratory Syndrome outbreak in South Korea, 2015

South Korea was free of the Middle East Respiratory Syndrome (MERS) until 2015. The MERS outbreak in South Korea during 2015 was the largest outbreak of the Coronavirus outside the Middle East. The major characteristic of this outbreak is inter- or intra-hospital transmission. This recent MERS outbreak in South Korea is examined and assessed in this paper. The main objectives of the study is to characterize the pattern of the MERS outbreak in South Korea based on a basic reproductive ratio, the probability of ultimate extinction of the disease, and the spatio-temporal proximity of occurrence between patients. The survival function method and stochastic branching process model are adapted to calculate the basic reproductive ratio and the probability of ultimate extinction of the disease. We further investigate the occurrence pattern of the outbreak using a spatio-temporal autocorrelation function.

Deep learning for supervised classification of spatial epidemics

In an emerging epidemic, public health officials must move quickly to contain the spread. Information obtained from statistical disease transmission models often informs the development of containment strategies. Inference procedures such as Bayesian Markov chain Monte Carlo allow researchers to estimate parameters of such models, but are computationally expensive. In this work, we explore supervised statistical and machine learning methods for fast inference via supervised classification, with a focus on deep learning. We apply our methods to simulated epidemics through two populations of swine farms in Iowa, and find that the random forest performs well on the denser population, but is outperformed by a deep learning model on the sparser population.

Mathematical epidemiology: Past, present, and future

We give a brief outline of some of the important aspects of the development of mathematical epidemiology.

Mathematical modeling of infectious disease dynamics

Over the last years, an intensive worldwide effort is speeding up the developments in the establishment of a global surveillance network for combating pandemics of emergent and re-emergent infectious diseases. Scientists from different fields extending from medicine and molecular biology to computer science and applied mathematics have teamed up for rapid assessment of potentially urgent situations. Toward this aim mathematical modeling plays an important role in efforts that focus on predicting, assessing, and controlling potential outbreaks. To better understand and model the contagious dynamics the impact of numerous variables ranging from the micro host–pathogen level to host-to-host interactions, as well as prevailing ecological, social, economic, and demographic factors across the globe have to be analyzed and thoroughly studied. Here, we present and discuss the main approaches that are used for the surveillance and modeling of infectious disease dynamics. We present the basic concepts underpinning their implementation and practice and for each category we give an annotated list of representative works.

Stability analysis and optimal vaccination of an SIR epidemic model

Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.

Compartmental Models in Epidemiology

We describe and analyze compartmental models for disease transmission. We begin with models for epidemics, showing how to calculate the basic reproduction number and the final size of the epidemic. We also study models with multiple compartments, including treatment or isolation of infectives. We then consider models including births and deaths in which there may be an endemic equilibrium and study the asymptotic stability of equilibria. We conclude by studying age of infection models which give a unifying framework for more complicated compartmental models.

A generalized chain binomial model with application to HIV infection

The original Reed-Frost formulation of the chain binomial model is mathematically equivalent to a stochastic model allowing a Poisson number of effective contacts in a time interval. Their formulation cannot accommodate survey data that necessarily correspond to more complex distributions of partners or contacts, or to large populations where complete random mixing is unlikely. This paper generalizes the Reed-Frost model to accommodate these situations in both the one- and two-population settings. The extension to multiple populations is also outlined. Using the model to predict HIV incidence in San Francisco's homosexual population, we show that the total number of contacts over all partners is more important than the distribution of contacts among partners in determining the number of infected.