While there are many models of epidemic evolution perhaps the basis for such models finds itself in the lumped behavior expressed through the so-called SIR model (Susceptible, Infectious, Recovered) from which spring many related models. This paper discusses multiple analytic solutions to that equation including those that are available in closed analytic form and those for which at least one final integral has to be done numerically, so-called quasi-analytic solutions. The solutions are intrinsically time-dependent of course. The hope is that such an investigation will lead to a better understanding of when and how models can be of use in studying the dynamical evolution of diseases including, perhaps, the great influenza pandemic of 1918 together with later pandemics and epidemics not excluding the Covid-19 pandemic of the present day.

The original purpose of this article was to modify the original SIR equations to allow for a direct source of infection (without which the original equations would have no solutions unless one starts with an already infected population) and also to see to what extent one could obtain multiple outbreaks of an infectious disease. In the course of developing the basic ideas several other factors arose to take prominent roles.Perhaps one of the more salient factors is the point that choosing an arbitrary time to change conditions from say a lock-down for the population to a less stringent social behavior, such as allowing partial or complete opening of businesses and schools, etc. should be based on knowledge of the disease and its evolution. Such decisions are usually made by politicians who have less than full information concerning the consequences of their actions. Several examples are given to illustrate these points.

In this paper, Lie symmetry and Painlevé Techniques are applied to the SIRD (Susceptible, Infected, Recovered and Dead) model. A demonstration of the integrability of the model is provided to present an exact solution. The study revealed that nonlinear system passes Painlevé test and does not possesses complex chaotic behaviour. However, the system fails to pass the Painlevé test while constraints reach values equivalent to the corresponding complex chaotic behaviour. The two-dimensional Lie symmetry algebra and the commutator table of the infinitesimal generators are obtained. Lie symmetry analysis serves to linearize the nonlinear system and find the corresponding invariant solution.

A major challenge for the dissemination, replication, and reuse of epidemiological forecasting studies during COVID-19 pandemics is the lack of clear guidelines and platforms to exchange models in a Findable, Accessible, Interoperable, and Reusable (FAIR) manner, facilitating reproducibility of research outcomes. During the beginning of pandemics, models were developed in diverse tools that were not interoperable, opaque without traceability and semantics, and scattered across various platforms - making them hard to locate, infer and reuse. In this work, we demonstrate that implementing the standards developed by the systems biology community to encode and share COVID-19 epidemiological models can serve as a roadmap to implement models as a tool in medical informatics, in general. As a proof-of-concept, we encoded and shared 24 epidemiological models using the standard format for model exchange in systems biology, annotated them with cross-references to data resources, packed up all associated files in COMBINE archives for easy sharing, and finally, disseminated the models through BioModels repository to significantly enhance their reproducibility and repurposing potential. We recommend the use of systems biology standards to encode and share models of epidemic and pandemic forecasts to improve their findability, accessibility, interoperability, reusability, and reproducibility.

The crux of the paper is to present a detailed analysis of COVID-19 data which is available on global basis. This analysis is performed using some specific package of R software. It provides various insights from the data and help to understand the current status of this pandemic in India so that effective measures can be formulated by policymakers. These insights include global summary of this disease, growth rate of this pandemic and performance of SIR model for the given global data. The analysis has been presented in different tables and graphs to understand the outputs of the problem in a more detailed point of view.

This article describes a simple Susceptible Infected Recovered (SIR) model fitting with COVID-19 data for the month of march 2020 in New York (NY) state. The model is a classical SIR, but is non-autonomous; the rate of susceptible people becoming infected is adjusted over time in order to fit the available data. The death rate is also secondarily adjusted. Our fitting is made under the assumption that due to limiting number of tests, a large part of the infected population has not been tested positive. In the last part, we extend the model to take into account the daily fluxes between New Jersey (NJ) and NY states and fit the data for both states. Our simple model fits the available data, and illustrates typical dynamics of the disease: exponential increase, apex and decrease. The model highlights a decrease in the transmission rate over the period which gives a quantitative illustration about how lockdown policies reduce the spread of the pandemic. The coupled model with NY and NJ states shows a wave in NJ following the NY wave, illustrating the mechanism of spread from one attractive hot spot to its neighbor.

In this work, we consider two SIR patterns with demography: the classical pattern and a modified pattern with a linear transmission coefficient of the infection. By reformulating of each first order differential systems as a system with two second-order differential equations, we investigate the nonlinear dynamics of the system from the Jacobi stability point of view by using the KCC geometric theory. We will study the intrinsic geometric properties of the systems by determining the geometric associated objects: the zero-connection curvature tensor, the nonlinear connection, the Berwald connection, and the five KCC invariants: the first invariant - the external force εi, the second invariant - the deviation curvature tensor Pji, the third invariant - the torsion tensor Pjki, the fourth invariant - the Riemann-Christoffel curvature tensor Pjkli, and the fifth invariant - the Douglas tensor Djkli. In order to obtain necessary and sufficient conditions for the Jacobi stability near the equilibrium points, the deviation curvature tensor will be determined at each equilibrium points. Furthermore, we will compare the Jacobi stability with the classical linear stability, inclusive by diagrams related to the values of parameters of the system.

The dramatic outbreak of the coronavirus disease 2019 (COVID-19) pandemics and its ongoing progression boosted the scientific community's interest in epidemic modelling and forecasting. The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the community to derive explicit solution. The present work demonstrates that this is a non-trivial task. Notably, it is proven that the explicit solution of the model requires the introduction of a new transcendental special function, related to the Wright's Omega function. The present manuscript reports new analytical results and numerical routines suitable for parametric estimation of the SIR model. The manuscript introduces iterative algorithms approximating the incidence variable, which allows for estimation of the model parameters from the numbers of observed cases. The numerical approach is exemplified with data from the European Centre for Disease Prevention and Control (ECDC) for several European countries in the period Jan 2020 -- Jun 2020.

Human civilizations are under enormous threats due to the outbreak of novel coronavirus (COVID-19) originated from Wuhan, China. The asymptomatic carriers are the potential spreads of this novel virus. Since, guaranteed antiviral treatments have not been available in the market so far, it is really challenging to fight against this contagious disease. To save the living mankind, it is urgent to know more about how the virus transmits itself from one to another quite rapidly and how we can predict future infections. Scientists and Researchers are working hard in investigating to understand its high infection rate and transmission process. One possible way to know is to use our existing COVID-19 infection data and prepare a useful model to predict the future trend. Mathematical modelling is very useful to understand the basic principle of COVID-19 transmission and provide necessary guidelines for future prediction. Here, we have reviewed 9 distinct commonly used models based on Mathematical implementations for COVID-19 transmission and dig into the deep head to head comparison of each model. Finally, we have discussed interesting key behaviour of each model, relevant upcoming important issues, challenges and future directions.

Background: Flattening the curve refers to community isolation measures that keep the daily number of disease cases at a manageable level for medical providers. Efforts to completely contain the COVID – 19 have failed. As there is currently no vaccine or specific medication to treat COVID-19, the only way to flatten the curve is through collective action of mitigation efforts. Objectives: The study was carried out with aim to study the effect of mitigation efforts in reducing the overall COVID -19 attack rate and flattening the curve of COVID-19 in Egypt. Methods: Secondary data in the situation reports of WHO, Worldometer and Egyptian MoHP Report about the COVID-19 epidemic in Egypt were analyzed till June 27, 2020. Survival analysis was carried out to determine the cumulative proportional survival of COVID-19. The Susceptible Infected Recovered (SIR) model was used to study the epidemic dynamics of COVID-19 pandemic. Results: In Egypt in the fourth week after exposure β was 0.17 and γ = 0.07. The fraction of infectious individuals will grow exponentially as the exponential growth rate was 0.10, with an epidemic time of 7 days. R in 22 ^nd June was 0.80 (less than one person infected) where β=0.21 and γ =0.26. There was reduction in the actual overall attack rate at 22^nd June of 96 % of its value without intervention with a cumulative proportional survival of 0.89. There was flattening of the curve. Conclusion and Recommendation: Mitigation efforts of Egypt had succeeded in flattening the curve of COVID-19 Pandemic. Even though, strengthening all the mitigation efforts to keep the exposure probability, as well as the transmission rate as low as possible is mandatory for containment of the epidemic.

The COVID-19 pandemic caused by the SARS-CoV-2 virus has inflicted significant mortality and morbidity worldwide. Continuous virus mutations have led to the emergence of new variants. The Omicron BA.1 sub-lineage prevailed as the dominant variant globally at the beginning of 2022 but was subsequently replaced by BA.2 in numerous countries. Wastewater-based epidemiology (WBE) offers an efficient tool for capturing viral shedding from infected individuals, enabling early detection of potential pandemic outbreaks without relying solely on community cooperation and clinical testing resources. This study integrated RT-qPCR assays for detecting general SARS-CoV-2 and its variants levels in wastewater into a modified triple susceptible-infected-recovered-susceptible (SIRS) model. The emergence of the Omicron-BA.1 variant was observed, replacing the presence of its predecessor, the Delta variant. Comparative analysis between the wastewater data and the modified SIRS model effectively described the BA.1 and subsequent BA.2 waves, with the decline of the Delta variant aligning with its diminished presence below the detection threshold in wastewater. This study demonstrates the potential of WBE as a valuable tool for future pandemics. Furthermore, by analyzing the sensitivity of different variants to model parameters, we are able to deduce real-life values of cross-variant immunity probabilities, emphasizing the asymmetry in their strength.

We review and assess the classic SIR and SEIR epidemiological models regarding possible applications to the COVID-19 pandemic. In spite of numerous more complicated models, we show how the qualitative features of the solution to the SIR and SEIR models continue to provide valuable public health insights in some scenarios. Using estimated COVID-19 data as of this date, the SEIR model shows that if it were possible to reduce R₀ from 2.5 to 1.25 through social distancing and other measures, the maximum fraction of the population that would become infected at any particular time would drop from 17% to 4%, provided that all of the model assumptions are satisfied. Finally, we compare the classic SIR model with a recent stochastic model with favorable results. Since this comparison underscores the importance of underlying connectivity assumptions, we conclude with Monte-Carlo simulations with specific connectivity that reproduce the classical SIR model with standard incidence.

The goal of this paper is to contribute the perspective of a systems engineer to the effort to fight pandemics. The availability of low latency case data and effectiveness of social distancing suggest there is sufficient control for successful smoothing and targeting almost any desired level of low or high cases and immunity. This control proceeds from spontaneous public reaction to caseloads and news as well as government mediated recommendations and orders. We simulate multi-step and intermittent-with-feedback partial unlock of social distancing for rapidly-spreading moderate-mortality epidemics and pandemics similar to COVID-19. Optimized scenarios reduce total cases and therefore deaths typically 8% and up to 30% by controlling overshoot as groups cross the herd immunity threshold, or lower thresholds to manage medical resources and provide economic relief. We analyze overshoot and provide guidance on how to damp it. However, we find overshoot damping, whether from expert planning or natural public self-isolation, increases the likelihood of transition to an endemic disease. An SIR model is used to evaluate scenarios that are intended to function over a wide variety of parameters. The end result is not a case trajectory prediction, but a prediction of which strategies produce near-optimal results over a wide range of epidemiological and social parameters. Overshoot damping perversely increases the chance a pathogen will transition to an endemic disease, so we briefly describe the undershoot conditions that promote transition to endemic status.

On February 24, 2022 Russia started the full-scale invasion of Ukraine, which created an unprecedented number of refugees. To estimate the influence of this humanitarian disaster on the COVID-19 pandemic dynamics, the averaged daily numbers of cases and reproduction numbers in Ukraine, the UK, Poland, Germany, the Republic of Moldova, and in the whole world were calculated for the period February-April 2022. The registered numbers of cases were compared with ones calculated with the use of the generalized SIR-model and corresponding parameter identification procedure for the previous epidemic waves in Ukraine, Poland, Germany, and the world. Since before February 24, 2022 the estimation of the number of infectious persons per capita in Ukraine 3.6 times exceeded the global figure, the increase of the number the new cases and the pandemic duration was expected. In March 2022 the increase of the averaged number of new cases in the UK, Germany, and worldwide was visible. A simple formula to estimate the effective reproduction number based on the smoothed accumulated numbers of cases is proposed. The results of calculations agree with the figures presented by John Hopkins University and demonstrate a short-term growth of the reproduction number in the UK, Poland, Germany, Moldova, and worldwide in March 2022.