ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES

Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.

SEIHCRD Model for COVID-19 Spread Scenarios,Disease Predictions and Estimates the Basic Reproduction Number,Case Fatality Rate,Hospital,and ICU Beds Requirement

We have proposed a new mathematical method,the SEIHCRD model,which has an excellent potential to predict the incidence of COVID-19 diseases.Our proposed SEIHCRD model is an extension of the SEIR model.Three-compartments have added death,hospitalized,and critical,which improves the basic understanding of disease spread and results.We have studiedCOVID-19 cases of six countries,where the impact of this disease in the highest are Brazil,India,Italy,Spain,the United Kingdom,and the United States.After estimating model parameters based on available clinical data,the modelwill propagate and forecast dynamic evolution.Themodel calculates the Basic reproduction number over time using logistic regression and the Case fatality rate based on the selected countries’age-category scenario.Themodel calculates two types of Case fatality rate one is CFR daily,and the other is total CFR.The proposed model estimates the approximate time when the disease is at its peak and the approximate time when death cases rarely occur and calculate how much hospital beds and ICU beds will be needed in the peak days of infection.The SEIHCRD model outperforms the classic ARXmodel and the ARIMA model.RMSE,MAPE,andRsquaredmatrices are used to evaluate results and are graphically represented using Taylor and Target diagrams.The result shows RMSE has improved by 56%–74%,and MAPE has a 53%–89%improvement in prediction accuracy.

Preliminary prediction of the control reproduction number of COVID-19 in Shaanxi Province,China

Objectives Firstly,according to the characteristics of COVID-19 epidemic and the control measures of the government of Shaanxi Province,a general population epidemic model is es-tablished.Then,the control reproduction number of general population epidemic model is obtained.Based on the epidemic model of general population,the epidemic model of general population and college population is further established,and the control reproduction number is also obtained.Methods For the established epidemic model,firstly,the expression of the control reproduc-tion number is obtained by using the next generation matrix.Secondly,the real-time reported data of COVID-19 in Shaanxi Province is used to fit the epidemic model,and the parameters in the model are estimated by least square method and MCMC.Thirdly,the Latin hypercube sampling method and partial rank correlation coefficient(PRCC)are adopted to analyze the sensitivity of the model.Conclusions The control reproduction number remained at 3 from January 23 to January 31,then gradually decreased from 3 to slightly greater than 0.2 by using the real-time reports on the number of COVID-19 infected cases from Health Committee of Shaanxi Province in China.In order to further control the spread of the epidemic,the following measures can be taken:(i)reducing infection by wearing masks,paying attention to personal hygiene and limiting travel;(i)improving isolation of suspected patients and treatment of symptomatic individuals.In particular,the epidemic model of the collge population and the general population is estab-lished,and the control reproduction number is given,which will provide theoretical basis for the prevention and control of the epidemic in the colleges.

On the Reproduction Number and a Presentation of Results for Infectious Diseases Models

The classical Kermack-McKendrick homogeneous SIR （susceptible, infected and removed） model is well known, Its general solution is a function of the unique parameter （the reproduction number） that is equal to a mean number of secondary cases produced by a typical infected individual in a completely susceptible population. If the reproduction number is more than one （the threshold value） its value describes an epidemic scope： larger values correspond to more severe epidemics. In the more complex compartment SIR models the population is divided into several non-overlapping groups. It allows us to partly remove assumptions of the classical model. It is well known that for this kind of models, just as for the classical model there is the threshold parameter R0. Usually it is called by the same name--the reproduction number--though the physical meaning of this parameter has changed. The main purpose of the paper is to show that this new parameter is a not unique measure of an epidemic severity for any compartment SIR model. In particular it means that for such models comparison of the severity of two epidemics by simple comparing values of their reproduction numbers is incorrect. For compartment models these statements were proved with the help of the corresponding ODEs analysis. Very popular now individual-based models （IBMs） are more complex in comparison with the compartment ones since they use overlapping groups （school children are members of families also, for example）. In such a case Diekmann＇s calculation method for the reproduction number used in many papers is inapplicable as well as a presentation the simulation results obtained as functions of this parameter.

Spatial analysis of Dengue through the reproduction numbers relating to socioeconomic features:Case studies on two Brazilian urban centers

The study of the propagation of infectious diseases in urban centers finds a close connection with their population's social characteristics and behavior.This work performs a spatial analysis of dengue cases in urban centers based on the basic reproduction numbers,R0,and incidence by planning areas(PAs),as well as their correlations with the Human Development Index(HDI)and the number of trips.We analyzed dengue epidemics in 2002 at two Brazilian urban centers,Belo Horizonte(BH)and Rio de Janeiro(RJ),using PAs as spatial units.Our results reveal heterogeneous spatial scenarios for both cities,with very weak correlations between R0 and both the number of trips and the HDI;in BH,the values of R0 show a less spatial heterogeneous pattern than in RJ.For BH,there are moderate correlations between incidence and both the number of trips and the HDI;meanwhile,they weakly correlate for RJ.Finally,the absence of strong correlations between the considered measures indicates that the transmission process should be treated considering the city as a whole.

EpiMix:A novel method to estimate effective reproduction number

Transmission potential of a pathogen,often quantified by the time-varying reproduction number R t,provides the current pace of infection for a disease and indicates whether an emerging epidemic is under control.In this study,we proposed a novel method,EpiMix,for R t estimation,wherein we incorporated the impacts of exogenous factors and random effects under a Bayesian regression framework.Using Integrated Nested Laplace Approx-imation,EpiMix is able to efficiently generate reliable,deterministic R t estimates.In the simulations and case studies performed,we further demonstrated the method's robust-ness in low-incidence scenarios,together with other merits,including its flexibility in selecting variables and tolerance of varying reporting rates.All these make EpiMix a potentially useful tool for real-time R t estimation provided that the serial interval distri-bution,time series of case counts and external influencing factors are available.

Estimating the instantaneous reproduction number(R_(t))by using particle filter

Background:Monitoring the transmission of coronavirus disease 2019(COVID-19)requires accurate estimation of the effective reproduction number(Rt).However,existing methods for calculating Rt may yield biased estimates if important real-world factors,such as delays in confirmation,pre-symptomatic transmissions,or imperfect data observation,are not considered.Method:To include real-world factors,we expanded the susceptible-exposed-infectiousrecovered(SEIR)model by incorporating pre-symptomatic(P)and asymptomatic(A)states,creating the SEPIAR model.By utilizing both stochastic and deterministic versions of the model,and incorporating predetermined time series of Rt,we generated simulated datasets that simulate real-world challenges in estimating Rt.We then compared the performance of our proposed particle filtering method for estimating Rt with the existing EpiEstim approach based on renewal equations.Results:The particle filtering method accurately estimated Rt even in the presence of data with delays,pre-symptomatic transmission,and imperfect observation.When evaluating via the root mean square error(RMSE)metric,the performance of the particle filtering method was better in general and was comparable to the EpiEstim approach if perfectly deconvolved infection time series were provided,and substantially better when Rt exhibited short-term fluctuations and the data was right truncated.Conclusions:The SEPIAR model,in conjunction with the particle filtering method,offers a reliable tool for predicting the transmission trend of COVID-19 and assessing the impact of intervention strategies.This approach enables enhanced monitoring of COVID-19 transmission and can inform public health policies aimed at controlling the spread of the disease.

Estimating effective reproduction number revisited

Accurately estimating the effective reproduction number is crucial for characterizing the transmissibility of infectious diseases to optimize interventions and responses during epidemic outbreaks.In this study,we improve the estimation of the effective reproduction number through two main approaches.First,we derive a discrete model to represent a time series of case counts and propose an estimation method based on this framework.We also conduct numerical experiments to demonstrate the effectiveness of the proposed discretization scheme.By doing so,we enhance the accuracy of approximating the underlying epidemic process compared to previous methods,even when the counting period is similar to the mean generation time of an infectious disease.Second,we employ a negative binomial distribution to model the variability of count data to accommodate overdispersion.Specifically,given that observed incidence counts follow a negative binomial distribution,the posterior distribution of secondary infections is obtained as a Dirichlet multinomial distribution.With this formulation,we establish posterior uncertainty bounds for the effective reproduction number.Finally,we demonstrate the effectiveness of the proposed method using incidence data from the COVID-19 pandemic.

Reproduction numbers of infectious disease models

This primer article focuses on the basic reproduction number,ℛ0,for infectious diseases,and other reproduction numbers related toℛ0 that are useful in guiding control strategies.Beginning with a simple population model,the concept is developed for a threshold value ofℛ0 determining whether or not the disease dies out.The next generation matrix method of calculatingℛ0 in a compartmental model is described and illustrated.To address control strategies,type and target reproduction numbers are defined,as well as sensitivity and elasticity indices.These theoretical ideas are then applied to models that are formulated for West Nile virus in birds(a vector-borne disease),cholera in humans(a disease with two transmission pathways),anthrax in animals(a disease that can be spread by dead carcasses and spores),and Zika in humans(spread by mosquitoes and sexual contacts).Some parameter values from literature data are used to illustrate the results.Finally,references for other ways to calculateℛ0 are given.These are useful for more complicated models that,for example,take account of variations in environmental fluctuation or stochasticity.

Estimation of reproduction numbers of COVID-19 in typical countries and epidemic trends under different prevention and control scenarios

The coronavirus disease 2019(COVID-19)has become a life-threatening pandemic.The epidemic trends in different countries vary considerably due to different policy-making and resources mobilization.We calculated basic reproduction number(R0)and the time-varying estimate of the effective reproductive number(Rt)of COVID-19 by using the maximum likelihood method and the sequential Bayesian method,respectively.European and North American countries possessed higher (R0)and unsteady Rt fluctuations,whereas some heavily affected Asian countries showed relatively low (R0)and declining Rt now.The numbers of patients in Africa and Latin America are still low,but the potential risk of huge outbreaks cannot be ignored.Three scenarios were then simulated,generating distinct outcomes by using SEIR(susceptible,exposed,infectious,and removed)model.First,evidence-based prompt responses yield lower transmission rate followed by decreasing Rt.Second,implementation of effective control policies at a relatively late stage,in spite of huge casualties at early phase,can still achieve containment and mitigation.Third,wisely taking advantage of the time-window for developing countries in Africa and Latin America to adopt adequate measures can save more people’s life.Our mathematical modeling provides evidence for international communities to develop sound design of containment and mitigation policies for COVID-19.

Basic reproduction number and predicted trends of coronavirus disease 2019 epidemic in the mainland of China

Background Coronavirus disease 2019(COVID-19)has caused a serious epidemic around the world,but it has been effectively controlled in the mainland of China.The Chinese government limited the migration of people almost from all walks of life.Medical workers have rushed into Hubei province to fight against the epidemic.Any activity that can increase infection is prohibited.The aim of this study was to confirm that timely lockdown,large-scale case-screening and other control measures proposed by the Chinese government were effective to contain the spread of the virus in the mainland of China.Methods Based on disease transmission-related parameters,this study was designed to predict the trend of COVID-19 epidemic in the mainland of China and provide theoretical basis for current prevention and control.An SEIQR epidemiological model incorporating asymptomatic transmission,short term immunity and imperfect isolation was constructed to evaluate the transmission dynamics of COVID-19 inside and outside of Hubei province.With COVID-19 cases confirmed by the National Health Commission(NHC),the optimal parameters of the model were set by calculating the minimum Chi-square value.Results Before the migration to and from Wuhan was cut off,the basic reproduction number in China was 5.6015.From 23 January to 26 January 2020,the basic reproduction number in China was 6.6037.From 27 January to 11 February 2020,the basic reproduction number outside Hubei province dropped below 1,but that in Hubei province remained 3.7732.Because of stricter controlling measures,especially after the initiation of the large-scale case-screening,the epidemic rampancy in Hubei has also been contained.The average basic reproduction number in Hubei province was 3.4094 as of 25 February 2020.We estimated the cumulative number of confirmed cases nationwide was 82186,and 69230 in Hubei province on 9 April 2020.Conclusions The lockdown of Hubei province significantly reduced the basic reproduction number.The large-scale case-screening also showed the effectiveness in the epidemic control.This study provided experiences that could be replicated in other countries suffering from the epidemic.Although the epidemic is subsiding in China,the controlling efforts should not be terminated before May.

Assessment of the SARS-CoV-2 basic reproduction number,R0,based on the early phase of COVID-19 outbreak in Italy

As of March 12th Italy has the largest number of SARS-CoV-2 cases in Europe as well as outside China.The infections,first limited in Northern Italy,have eventually spread to all other regions.When controlling an emerging outbreak of an infectious disease it is essential to know the key epidemiological parameters,such as the basic reproduction number R0,i.e.the average number of secondary infections caused by one infected individual during his/her entire infectious period at the start of an outbreak.Previous work has been limited to the assessment of R0 analyzing data from the Wuhan region or China's Mainland.In the present study the R0 value for SARS-CoV-2 was assessed analyzing data derived from the early phase of the outbreak in Italy.In particular,the spread of SARS-CoV-2 was analyzed in 9 cities(those with the largest number of infections)fitting the well-established SIR-model to available data in the interval between February 25–March 12,2020.The findings of this study suggest that R0 values associated with the Italian outbreak may range from 2.43 to 3.10,confirming previous evidence in the literature reporting similar R0 values for SARS-CoV-2.

Estimating effective reproduction number using generation time versus serial interval,with application to COVID-19 in the Greater Toronto Area,Canada

BACKGROUND.The effective reproduction number Re(t)is a critical measure of epidemic potential.Re(t)can be calculated in near real time using an incidence time series and the generation time distribution:the time between infection events in an infector-infectee pair.In calculating Re(t),the generation time distribution is often approximated by the serial interval distribution:the time between symptom onset in an infector-infectee pair.However,while generation time must be positive by definition,serial interval can be negative if transmission can occur before symptoms,such as in COVID-19,rendering such an approximation improper in some contexts.METHODS.We developed a method to infer the generation time distribution from parametric definitions of the serial interval and incubation period distributions.We then compared estimates of Re(t)for COVID-19 in the Greater Toronto Area of Canada using:negative-permitting versus non-negative serial interval distributions,versus the inferred generation time distribution.RESULTS.We estimated the generation time of COVID-19 to be Gamma-distributed with mean 3.99 and standard deviation 2.96 days.Relative to the generation time distribution,non-negative serial interval distribution caused overestimation of Re(t)due to larger mean,while negative-permitting serial interval distribution caused underestimation of Re(t)due to larger variance.IMPLICATIONS.Approximation of the generation time distribution of COVID-19 with non-negative or negative-permitting serial interval distributions when calculating Re(t)may result in over or underestimation of transmission potential,respectively.

A comparative analysis of three different methods for the estimation of the basic reproduction number of dengue

The basic reproduction number,R0,is defined as the expected number of secondary cases of a disease produced by a single infection in a completely susceptible population,and can be estimated in several ways.For example,from the stability analysis of a compartmental model;through the use of the matrix of next generation,or from the final size of an epidemic,etc.In this paper we applied the method for estimating R0 of dengue fever from the initial growth phase of an outbreak,without assuming exponential growth of cases,a common assumption in many studies.We used three different methods of calculating R0 to compare the techniques'details and to evaluate how these techniques estimate the value of R0 of dengue using data from the city of Ribeir^ao Preto(SE of Brazil)in two outbreaks.The results of the three methods are numerically different but,when we compare them using a system of differential equations developed for modeling only the first generation time,we can observe that the methods differ little in the initial growth phase.We conclude that the methods predict that dengue will spread in the city studied and the analysis of the data shows that the estimated values of R0 have an equal pattern overtime.

Replicating and projecting the path of COVID-19 with a model-implied reproduction number

We demonstrate a methodology for replicating and projecting the path of COVID-19 using a simple epidemiology model.We fit the model to daily data on the number of infected cases in China,Italy,the United States,and Brazil.These four countries can be viewed as representing different stages,from later to earlier,of a COVID-19 epidemic cycle.We solve for a model-implied effective reproduction number R t each day so that the model closely replicates the daily number of currently infected cases in each country.For out-of-sample projections,we fit a behavioral function to the in-sample data that allows for the endogenous response of R t to movements in the lagged number of infected cases.We show that declines in measures of population mobility tend to precede declines in the model-implied reproduction numbers for each country.This pattern suggests that mandatory and voluntary stay-at-home behavior and social distancing during the early stages of the epidemic worked to reduce the effective reproduction number and mitigate the spread of COVID-19.

Disease momentum: Estimating the reproduction number in the presence of superspreading

A primary quantity of interest in the study of infectious diseases is the average number of new infections that an infected person produces.This so-called reproduction number has significant implications for the disease progression.There has been increasing literature suggesting that superspreading,the significant variability in number of new infections caused by individuals,plays an important role in the spread of SARS-CoV-2.In this paper,we consider the effect that such superspreading has on the estimation of the reproduction number and subsequent estimates of future cases.Accordingly,we employ a simple extension to models currently used in the literature to estimate the reproduction number and present a case-study of the progression of COVID-19 in Austria.Our models demonstrate that the estimation uncertainty of the reproduction number increases with superspreading and that this improves the performance of prediction intervals.Of independent interest is the derivation of a transparent formula that connects the extent of superspreading to the width of credible intervals for the reproduction number.This serves as a valuable heuristic for understanding the uncertainty surrounding diseases with superspreading.

A Study on the Transmission Dynamics of the Omicron Variant of COVID-19 Using Nonlinear Mathematical Models

This research examines the transmission dynamics of the Omicron variant of COVID-19 using SEIQIcRVW and SQIRV models,considering the delay in converting susceptible individuals into infected ones.The significant delays eventually resulted in the pandemic’s containment.To ensure the safety of the host population,this concept integrates quarantine and the COVID-19 vaccine.We investigate the stability of the proposed models.The fundamental reproduction number influences stability conditions.According to our findings,asymptomatic cases considerably impact the prevalence of Omicron infection in the community.The real data of the Omicron variant from Chennai,Tamil Nadu,India,is used to validate the outputs.

Modelling Foot and Mouth Disease in the Context of Active Immigrants

This study employs mathematical modeling to analyze the impact of active immigrants on Foot and Mouth Disease (FMD) transmission dynamics. We calculate the reproduction number (R0) using the next-generation matrix approach. Applying the Routh-Hurwitz Criterion, we establish that the Disease-Free Equilibrium (DFE) point achieves local asymptotic stability when R0 α1 and α2) are closely associated with reduced susceptibility in animal populations, underscoring the link between immigrants and susceptibility. Furthermore, our findings emphasize the interplay of disease introduction with population response and adaptation, particularly involving incoming infectious immigrants. Swift interventions are vital due to the limited potential for disease establishment and rapid susceptibility decline. This study offers crucial insights into the complexities of FMD transmission with active immigrants, informing effective disease management strategies.

Analysis of radiation diffusion of COVID-19 driven by social attributes

This paper first estimated the infectious capacity of COVID-19 based on the time series evolution data of confirmed cases in multiple countries. Then, a method to infer the cross-regional spread speed of COVID-19 was introduced in this paper, which took the gross domestic product(GDP) of each region as one of the factors that affect the spread speed of COVID-19 and studied the relationship between the GDP and the infection density of each region(China's Mainland, the United States, and EU countries). In addition, the geographic distance between regions was also considered in this method and the effect of geographic distance on the spread speed of COVID-19 was studied. Studies have shown that the probability of mutual infection of these two regions decreases with increasing geographic distance. Therefore, this paper proposed an epidemic disease spread index based on GDP and geographic distance to quantify the spread speed of COVID-19 in a region. The analysis results showed a strong correlation between the epidemic disease spread index in a region and the number of confirmed cases. This finding provides reasonable suggestions for the control of epidemics. Strengthening the control measures in regions with higher epidemic disease spread index can effectively control the spread of epidemics.

Mathematical Modeling of the Co-Infection Dynamics of HIV and Tuberculosis Incorporating Inconsistency in HIV Treatment

A non-linear HIV-TB co-infection has been formulated and analyzed. The positivity and invariant region has been established. The disease free equilibrium and its stability has been determined. The local stability was determined and found to be stable under given conditions. The basic reproduction number was obtained and according to findings, co-infection diminishes when this number is less than unity, and persists when the number is greater than unity. The global stability of the endemic equilibrium was calculated. The impact of HIV on TB was established as well as the impact of TB on HIV. Numerical solution was also done and the findings indicate that when the rate of HIV treatment increases the latent TB increases while the co-infected population decreases. When the rate of HIV treatment decreases the latent TB population decreases and the co-infected population increases. Encouraging communities to prioritize the consistent treatment of HIV infected individuals must be emphasized in order to reduce the scourge of HIV-TB co-infection.