Time-dependent Infection Rate Research Articles

Impact of time-dependent infection rate and self-isolation awareness on networked epidemic propagation

Impact of time-dependent infection rate and self-isolation awareness on networked epidemic propagation

Fractional SIZR model of Zombie infection

Abstract This research paper investigates the SIZR model related to Zombie infection outbreaks according to a time-dependent infection rate. The proposed model is extended to the fractional order using different fractional derivative operators. The solution of the proposed model by numerical schemes, is briefed. Graphical representations provide us with a better understanding of this mathematical model. Lastly, as observed in movies, we conclude that Zombie infections can generate the destruction and collapse of human development and it is of paramount importance to deal with Zombies as early as possible in order to avoid putting our civilization under threatening and destructive circumstances.

A Prospective Method for Generating COVID-19 Dynamics

Generating dynamic operators are constructed here from the cumulative case function to recover all state dynamics of a Susceptible–Exposed–Infectious–Recovered (SEIR) model for COVID-19 transmission. In this study, recorded and unrecorded EIRs and a time-dependent infection rate are taken into account to accommodate immeasurable control and intervention processes. Generating dynamic operators are built and implemented on the cumulative cases. All infection processes, which are hidden in this cumulative function, can be recovered entirely by implementing the generating operators. Direct implementation of the operators on the cumulative function gives all recorded state dynamics. Further, the unrecorded daily infection rate is estimated from the ratio between IFR and CFR. The remaining dynamics of unrecorded states are directly obtained from the generating operators. The simulations are conducted using infection data provided by Worldometers from ten selected countries. It is shown that the higher number of daily PCR tests contributed directly to reducing the effective reproduction ratio. The simulations of all state dynamics, infection rates, and effective reproduction ratios for several countries in the first and second waves of transmissions are presented. This method directly measures daily transmission indicators, which can be effectively used for the day-to-day control of the epidemic.

Physics‐informed machine learning for the COVID‐19 pandemic: Adherence to social distancing and short‐term predictions for eight countries

BackgroundThe analysis of COVID‐19 infection data through the eye of Physics‐inspired Artificial Intelligence leads to a clearer understanding of the infection dynamics and assists in predicting future evolution. The spreading of the pandemic during the first half of 2020 was curtailed to a larger or lesser extent through social distancing measures imposed by most countries. In the context of the standard Susceptible‐Infected‐Recovered (SIR) model, changes in social distancing enter through time‐dependent infection rates.MethodsIn this work we use machine learning and the infection dynamical equations of SIR to extract from the infection data the degree of social distancing and, through it, assess the effectiveness of the imposed measures.ResultsQuantitative machine learning analysis is applied to eight countries with infection data from the first viral wave. We find as two extremes Greece and USA where the measures were successful and unsuccessful, respectively, in limiting spreading. This physics‐based neural network approach is employed to the second wave of the infection, and by training the network with the new data, we extract the time‐dependent infection rate and make short‐term predictions with a week‐long or even longer horizon. This algorithmic approach is applied to all eight countries with good short‐term results. The data for Greece is analyzed in more detail from August to December 2020.ConclusionsThe model captures the essential spreading dynamics and gives useful projections for the spreading, both in the short‐term but also for a more intermediate horizon, based on specific social distancing measures that are extracted directly from the data.

Data-driven prediction of COVID-19 cases in Germany for decision making

BackgroundThe COVID-19 pandemic has led to a high interest in mathematical models describing and predicting the diverse aspects and implications of the virus outbreak. Model results represent an important part of the information base for the decision process on different administrative levels. The Robert-Koch-Institute (RKI) initiated a project whose main goal is to predict COVID-19-specific occupation of beds in intensive care units: Steuerungs-Prognose von Intensivmedizinischen COVID-19 Kapazitäten (SPoCK). The incidence of COVID-19 cases is a crucial predictor for this occupation.MethodsWe developed a model based on ordinary differential equations for the COVID-19 spread with a time-dependent infection rate described by a spline. Furthermore, the model explicitly accounts for weekday-specific reporting and adjusts for reporting delay. The model is calibrated in a purely data-driven manner by a maximum likelihood approach. Uncertainties are evaluated using the profile likelihood method. The uncertainty about the appropriate modeling assumptions can be accounted for by including and merging results of different modelling approaches. The analysis uses data from Germany describing the COVID-19 spread from early 2020 until March 31st, 2021.ResultsThe model is calibrated based on incident cases on a daily basis and provides daily predictions of incident COVID-19 cases for the upcoming three weeks including uncertainty estimates for Germany and its subregions. Derived quantities such as cumulative counts and 7-day incidences with corresponding uncertainties can be computed. The estimation of the time-dependent infection rate leads to an estimated reproduction factor that is oscillating around one. Data-driven estimation of the dark figure purely from incident cases is not feasible.ConclusionsWe successfully implemented a procedure to forecast near future COVID-19 incidences for diverse subregions in Germany which are made available to various decision makers via an interactive web application. Results of the incidence modeling are also used as a predictor for forecasting the need of intensive care units.

Second Quantization Approach to COVID-19 Epidemic

In this article, a stochastic SIR-type model for COVID-19 epidemic is built using the standard field theoretical language based on creation and annihilation operators. From the model, we derive the time evolution of the mean number of infectious (active cases) and deceased individuals. In order to capture the effects of lockdown and social distancing, we use a time-dependent infection rate. The results are in good agreement with the data for three different waves of epidemic activity in South Korea.

Estimating, monitoring, and forecasting COVID-19 epidemics: a spatiotemporal approach applied to NYC data

We propose a susceptible-exposed-infective-recovered-type (SEIR-type) meta-population model to simulate and monitor the (COVID-19) epidemic evolution. The basic model consists of seven categories, namely, susceptible (S), exposed (E), three infective classes, recovered (R), and deceased (D). We define these categories for n age and sex groups in m different spatial locations. Therefore, the resulting model contains all epidemiological classes for each age group, sex, and location. The mixing between them is accomplished by means of time-dependent infection rate matrices. The model is calibrated with the curve of daily new infections in New York City and its boroughs, including census data, and the proportions of infections, hospitalizations, and deaths for each age range. We finally obtain a model that matches the reported curves and predicts accurate infection information for different locations and age classes.

Simplified model on the timing of easing the lockdown

Lockdown procedures have been proven successful in mitigating the spread of the viruses in this COVID-19 pandemic, but they also have devastating impact on the economy. We use a modified Susceptible-Infectious-Recovered-Deceased model with time dependent infection rate to simulate how the infection is spread under lockdown. The economic cost due to the loss of workforce and incurred medical expenses is evaluated with a simple model. We find the best strategy, meaning the smallest economic cost for the entire course of the pandemic, is to keep the strict lockdown as long as possible.

Fitting an Epidemiological Model to Transmission Dynamics of COVID-19

A rapid increase in daily new cases was reported in the world from February 19 to April 3, 2020. In this study, a susceptible-infected-recovered-dead (SIRD) was developed to analyse the dynamics of the global spread of COVID-19 during the above-mentioned period of time. The values of the model parameters fitted the reported data were estimated by minimizing the sum of squared errors using the Levenberg-Marquardt optimization algorithm. A time-dependent infection rate was considered. The set of differential equations in the model was solved using the fourth order Runge-Kutta method. It was observed that a time-dependent parameter gives a better fit to a dynamic data. Based on the fitted model, the average value of basic reproduction number (\textit{R0}) for COVID-19 trasmission was estimated to be 2.8 which shows that the spread of COVID-19 disease in the world was growing exponentially. This may indicate that the control measures implemented worldwide could not decrease the COVID-19 transmission.

Analytical solution of the SIR-model for the temporal evolution of epidemics. Part A: time-independent reproduction factor

We revisit the susceptible-infectious-recovered/removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t = 0, where data is assumed to be available. We address the question of how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be relaxed in part B. Relaxing this assumption allows us to formulate initial conditions incompatible with the original SIR model.

Solvable delay model for epidemic spreading: the case of Covid-19 in Italy

We study a simple realistic model for describing the diffusion of an infectious disease on a population of individuals. The dynamics is governed by a single functional delay differential equation, which, in the case of a large population, can be solved exactly, even in the presence of a time-dependent infection rate. This delay model has a higher degree of accuracy than that of the so-called SIR model, commonly used in epidemiology, which, instead, is formulated in terms of ordinary differential equations. We apply this model to describe the outbreak of the new infectious disease, Covid-19, in Italy, taking into account the containment measures implemented by the government in order to mitigate the spreading of the virus and the social costs for the population.

Monitoring Italian COVID-19 spread by a forced SEIRD model.

Due to the recent evolution of the COVID-19 outbreak, the scientific community is making efforts to analyse models for understanding the present situation and for predicting future scenarios. In this paper, we propose a forced Susceptible-Exposed-Infected-Recovered-Dead (fSEIRD) differential model for the analysis and forecast of the COVID-19 spread in Italian regions, using the data from the Italian Protezione Civile (Italian Civil Protection Department) from 24/02/2020. In this study, we investigate an adaptation of fSEIRD by proposing two different piecewise time-dependent infection rate functions to fit the current epidemic data affected by progressive movement restriction policies put in place by the Italian government. The proposed models are flexible and can be quickly adapted to monitor various epidemic scenarios. Results on the regions of Lombardia and Emilia-Romagna confirm that the proposed models fit the data very accurately and make reliable predictions.

Studying the progress of COVID-19 outbreak in India using SIRD model.

We explore a standard epidemiological model, known as the SIRD model, to study the COVID-19 infection in India, and a few other countries around the world. We use (a) the stable cumulative infection of various countries and (b) the number of infection versus the tests carried out to evaluate the model. The time-dependent infection rate is set in the model to obtain the best fit with the available data. The model is simulated aiming to project the probable features of the infection in India, various Indian states, and other countries. India imposed an early lockdown to contain the infection that can be treated by its healthcare system. We find that with the current infection rate and containment measures, the total active infection in India would be maximum at the end of June or beginning of July 2020. With proper containment measures in the infected zones and social distancing, the infection is expected to fall considerably from August. If the containment measures are relaxed before the arrival of the peak infection, more people from the susceptible population will fall sick as the infection is expected to see a threefold rise at the peak. If the relaxation is given a month after the peak infection, a second peak with a moderate infection will follow. However, a gradual relaxation of the lockdown started well ahead of the peak infection, leads to a nearly twofold increase of the peak infection with no second peak. The model is further extended to incorporate the infection arising from the population showing no symptoms. The preliminary finding suggests that random testing needs to be carried out within the asymptomatic population to contain the spread of the disease. Our model provides a semi-quantitative overview of the progression of COVID-19 in India, with model projections reasonably replicating the current progress. The projection of the model is highly sensitive to the choice of the parameters and the available data.Electronic supplementary materialThe online version of this article (10.1007/s12648-020-01766-8) contains supplementary material, which is available to authorized users.

Parameter and State Estimation in a Cholera Model with Threshold Immunology: A Case Study of Senegal.

It is often impossible to measure all states affecting spread of a disease. In cholera, asymptomatic and cholera pathogen densities are not practically measurable despite playing a big role in its transmission. They are referred to as inaccessible states of the model and can only be manipulated using the measurable states of the given model. Our interest lies in estimating such states and the parameters catalyzing the spread. A mathematical model for cholera dynamics consisting of five compartments (susceptible, symptomatic, asymptomatic, recovered and bacteria population) with a minimum infection dose (MID) is considered. A method based on observer (from modern control theory) is proposed to estimate the state variables not accessible to measurement and the time-dependent parameters from real data of Senegal. We suppose that the total population of Senegal, the monthly reported cholera-induced deaths and the monthly recovered individuals are known inputs obtainable from real data, and the monthly new cholera cases the system output. An auxiliary system is used, an observer whose solutions converge exponentially to those of an original system and solely utilize known inputs and output of the model. Thus, the estimation of the unmeasured state variables like the pathogen density and the yearly asymptomatic population within the human community playing an important role in the spread of cholera is possible. We derive the expressions for time-dependent infection rate, induced cholera death rate and symptomatic recovery rate and their estimations done using real data. Numerical simulations are then performed for the validation of estimation results. The system together with the observer designed is detectable but is not observable. The observer delivered estimates reflect a close trend already ascertained by other researchers. They indicate the existence of bacteria and asymptomatic individuals in the population of Senegal at any single time for the duration of collection of the data. The ever existence of cholera pathogen explains the endemicity of Cholera in Senegal and other sub-Saharan-African countries owing to role played by the asymptomatic individual in the bacteria density. As such, the heath authorities in Senegal need to educate the general public on hygiene irrespective of observable symptoms to lower the possible number of new infections. We have analytically showed and numerically confirmed the exponential convergence to zero of the estimation errors resulting from the observer model hence the high quality of the estimates.

Parameters’ estimation, sensitivity analysis and model uncertainty for an influenza a mathematical model: case of Morocco

In this paper, we consider a SEIS model type compartmental model to explore the evolution of the Influenza epidemic in Morocco. To consider a more realistic model we consider the seasonality of parameters, by considering time-dependent infection rate, time-dependent recovery rate, and time-dependent intervention rate. The next-generation matrix method is used to compute the threshold of equilibria’s stability R0. Based on real data of Influenza epidemic A infections in Morocco from the month of December 2018 to the month of March 2019 published by the Influenza Laboratory Surveillance Information system, we estimate the model parameters. In order to identify the most influential parameter in the proposed model, we carry out the local sensitivity analysis. We calculate sensitivity indices based on the estimated parameters which identify the most influential parameters. We perform also the uncertainty analysis to determine the relationship between the different parameters of the model.

A model for porcine reproductive and respiratory syndrome with time-dependent infection rate: traveling wave solution

Porcine reproductive and respiratory syndrome (PRRS) gives rise to reproductive disorders in sows and problem with respiratory system in piglets and young pigs. This disease creates serious economic losses to major pork producing countries. The disease, which is characterized by high morbidity and significant mortality, combined with its potential for rapid spread, can devastate the pig industries of the affected countries. However, not much is known about the spatial transmission of PRRSV (porcine reproductive and respiratory syndrome virus) in growing pigs. In previous models, the infection rate has been assumed to be constant with time. Experimental studies on specific cases of this viral infection suggest that this assumption might not hold. A structured model for the spread of PRRSV has therefore been constructed, incorporating time and spatial dimensions as well as the decline of infection rate with time. Using the traveling wave coordinate and the modified extended hyperbolic tangent method, we derive analytical solutions to the model system. Stability and phase plane analyses are also carried out in order to gain insights into the spatial spread of PRRS as time progresses.

Non-periodic outbreaks of recurrent epidemics and its network modelling.

The study of recurrent epidemic outbreaks has been attracting great attention for decades, but its underlying mechanism is still under debate. Based on a large number of real data from different cities, we find that besides the seasonal periodic outbreaks of influenza, there are also non-periodic outbreaks, i.e. non-seasonal or non-annual behaviors. To understand how the non-periodicity shows up, we present a network model of SIRS epidemic with both time-dependent infection rate and a small possibility of persistent epidemic seeds, representing the influences from the larger annual variation of environment and the infection generated spontaneously in nature, respectively. Our numerical simulations reveal that the model can reproduce the non-periodic outbreaks of recurrent epidemics with the main features of real influenza data. Further, we find that the recurrent outbreaks of epidemic depend not only on the infection rate but also on the density of susceptible agents, indicating that they are both the necessary conditions for the recurrent epidemic patterns with non-periodicity. A theoretical analysis based on Markov dynamics is presented to explain the numerical results. This finding may be of significance to the control of recurrent epidemics.

Critical behavior of the SIS epidemic model with time-dependent infection rate

In this work we study a modified susceptible–infected–susceptible (SIS) model in which the infectionrate λ decays exponentially with the number of reinfectionsn, saturating aftern = l. We find a criticaldecaying rate ϵc(l) above which a finite fraction of the population becomes permanentlyinfected. From the mean-field solution and computer simulations onhypercubic lattices we find evidence that the upper critical dimension isd = 6, like in the SIR model, which can be mapped in ordinary percolation.

Stochastic dynamic model of SARS spreading.

Based upon the simulation of the stochastic process of infection, onset and spreading of each SARS patient, a system dynamic model of SRAS spreading is constructed. Data from Vietnam is taken as an example for Monte Carlo test. The preliminary results indicate that the time-dependent infection rate is the most inportant control factor for SARS spreading. The model can be applied to prediction of the course with fluctuations of the epidemics, if the previous history of the epidemics and the future infection rate under control measures are known.

A method for estimating the age-specific relative risk of HIV infection from AIDS incidence data

Using a multiplicative model, the age-specific relative risk of infection with the human immunodeficiency virus, HIV, and the time-dependent HIV infection rate can be estimated simultaneously from acquired immunodeficiency syndrome, AIDS, incidence data. Non-parametric estimates are arrived at by an EM-algorithm, with smoothing, applied to a back-projection model. The method gives estimates which lie on a smooth curve in accordance with strong prior expectations. The approach is illustrated with an application to Australian AIDS data. It is found that the HIV infection rate tends to be estimated more precisely when age is included as a covariate.