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.

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.

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.

Asymptotic behavior of the principal eigenvalue and the basic reproduction ratio for periodic patch models

This paper is devoted to the study of the asymptotic behavior of the principal eigenvalue and the basic reproduction ratio associated with periodic population models in a patchy environment for small and large dispersal rates.We first deal with the eigenspace corresponding to the zero eigenvalue of the connectivity matrix.Then we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem as the dispersal rate goes to zero and infinity,respectively.We further establish the asymptotic behavior of the basic reproduction ratio in the case of small and large dispersal rates.Finally,we apply these results to a periodic Ross-Macdonald patch model.

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.

Taenia solium taeniasis and cysticercosis:extinction or outbreak

Taenia solium taeniasis and cysticercosis are neglected zoonotic diseases that affect human health and economies of developing countries.In this work,we formulate and analyze deterministic and continuous time Markov chain(CTMC)stochastic models to determine parameters that drive Taenia solium taeniasis and cysticercosis and the likelihood of their extinction.The basic reproduction number R0 is computed by the next generation matrix approach,sensitivity index of each parameter in R0 is derived by the normalized forward sensitivity index and the likelihood of diseases’extinction is computed by the multitype branching process.The analysis shows that humans with Taenia solium taeniasis,infectious pork and Taenia solium eggs in the environment play an important role in the transmission of Taenia solium taeniasis and cysticercosis,and the model exhibits forward bifurcation at R0=1.This implies that R0<1 is a sufficient condition to eliminate Taenia solium taeniasis and cysticercosis.For CTMC model,analysis shows that the probability of Taenia solium taeniasis and cysticercosis extinction is high if the diseases emerge from humans with Taenia solium cysticercosis and there is an outbreak if the diseases emerge from either humans with Taenia solium taeniasis or infectious pork or Taenia solium eggs in the environment.To control Taenia solium taeniasis and cysticercosis,the intervention strategies should focus on improving hygiene and sanitation for reducing shedding rate of Taenia solium eggs in the environment,inspection of pork for reducing the rate of acquiring Taenia solium taeniasis and spraying of insecticides for killing Taenia solium eggs in the environment.

Optimal Treatment Strategy for Infectious Diseases with Two Treatment Stages

In this paper, a disease transmission model with two treatment stages is proposed and analyzed. The results indicate that the basic reproduction number is a critical threshold for the prevalence of the disease. If the basic reproduction number is less than one, the disease free equilibrium is globally asymptotically stable. Otherwise, the endemic equilibrium is globally asymptotically stable. Therefore, besides the basic reproduction number, a new marker for characterizing the seriousness of the disease, named as dynamical final infective size, is proposed, which differs from traditional final size because the proposed model includes the natural birth and death. Finally, optimization strategies for limited medical resources are obtained from the perspectives of basic reproduction number and dynamical final infective size, and the real-world disease management scenarios are given based on these finding.

Dynamic analysis of a sexually transmitted disease model on complex networks

In this paper,a sexually transmitted disease model is proposed on complex networks,where contacts between humans are treated as a scale-free social network.There are three groups in our model,which are dangerous male,non-dangerous male,and female.By mathematical analysis,we obtain the basic reproduction number for the existence of endemic equilibrium and study the effects of various immunization schemes about different groups.Furthermore,numerical simulations are undertaken to verify more conclusions.

Stability Analysis of Information Spreading on SNS Based on Refined SEIR Model

Investigating the stability of information spreading over SNS helps to understand the principles inherent in the spreading behavior.This paper explores the mechanisms of information spreading including stifling mechanism,latent mechanism and forgetting mechanism,establishes a refined SEIR model,and builds the corresponding mean-field equations.The methods of the differential dynamics and the next generation matrix are used to calculate the equilibriums and the basic reproductive number,and the asymptotical stability of the network equilibriums are proved theoretically.Simulation experiments are carried out to analyze the effect of the spreading mechanisms on the information spreading process and the results support our conclusions.

DYNAMICS OF A NONLOCAL DISPERSAL FOOT-AND-MOUTH DISEASE MODEL IN A SPATIALLY HETEROGENEOUS ENVIRONMENT

Foot-and-mouth disease is one of the major contagious zoonotic diseases in the world.It is caused by various species of the genus Aphthovirus of the family Picornavirus,and it always brings a large number of infections and heavy financial losses.The disease has become a major public health concern.In this paper,we propose a nonlocal foot-and-mouth disease model in a spatially heterogeneous environment,which couples virus-to-animals and animals-to-animals transmission pathways,and investigate the dynamics of the disperal.The basic reproduction number R_(0)is defined as the spectral radius of the next generation operator R(x)by a renewal equation.The relationship between R_(0)and a principal eigenvalue of an operator L_(0)is built.Moreover,the proposed system exhibits threshold dynamics in terms of R_(0),in the sense that R_(0)determines whether or not foot-and-mouth disease invades the hosts.Through numerical simulations,we have found that increasing animals'movements is an effective control measure for preventing prevalence of the disease.

Mathematical Study of Dengue Disease Transmission in Multi-Patch Environment

Dengue disease is the most common vector borne infectious disease transmitted to humans by infected adult female Aedes mosquitoes. Over the past several years the disease has been increasing remarkably and it has become a major public health concern. Dengue viruses have increased their geographic range into new human population due to travel of humans from one place to the other. In the present paper, we have proposed a multi patch SIR-SI model to study the host-vector dynamics of dengue disease in different patches including the travel of human population among the patches. We have considered different disease prevalences in different patches and different travel rates of humans. The dimensionless number, basic reproduction number R0 which shows that the disease dies out if R0 < 1 and the disease takes hold if R0 ≥ 1, is calculated. Local and global stability of the disease free equilibrium are analyzed. Simulations are observed considering the two patches only. The results show that controlling the travel of infectious hosts from high disease dominant patch to low disease dominant patch can help in controlling the disease in low disease dominant patch while high disease dominant becomes even more disease dominant. The understanding of the effect of travel of humans on the spatial spread of the disease among the patches can be helpful in improving disease control and prevention measures. In the present study, a patch may represent a city, a village or some biological habitat.

Sensitivity Analysis of COVID-19 in Mediterranean Island

The aim of this study is to examine the progress of the worldwide pandemic Covid-19.As authors,we have decided to analyze the situation of COVID-19 onMediterranean islandwith accurate data.For this purpose,amathematical model is constructed and proposed by dividing the whole population into sensible and suitable compartments.The study captures the dates February 01 till May 15,2021.For the control of the spread of disease,vaccination and infection rates are compared and calculated.During calculations and comparison,MatLab software is used.All of the data that are used are taken from the Ministry of Health.The effect of parameters is examined with sensitivity analysis.Furthermore,with this analysis,values of parameters are obtained.Afterwards,by using the constructed model,the effect of vaccine on infected individuals is analyzed separately.As a result,it is concluded that the studied part of the island is late for the control of the disease via vaccine.This can be explained by two main reasons;vaccinating the people that are not inmobilitymost of the time(aged people and people with chronic diseases)and getting the vaccine late.Hence,the results showed that this rate and distribution of vaccines would not be enough to control the pandemic on the island.

Awareness as the Most Effective Measure to Mitigate the Spread of COVID-19 in Nigeria

A mathematical model consisting of a system of four nonlinear ordinary differential equations is constructed.Our aim is to study the dynamics of the spread of COVID-19 in Nigeria and to show the effectiveness of awareness and the need for relevant authorities to engage themselves more in enlightening people on the significance of the available control measures in mitigating the spread of the disease.Two equilibrium solutions;Disease free equilibrium and Endemic equilibrium solutions were calculated and their global stability analysis was carried out.Basic reproduction ratio(R0)was also obtained,in this research𝑅𝑅0=3.0784.Data obtained for Nigeria is used to conduct numerical simulations in order to support the analytic result and to show the significance of awareness in controlling the disease spread.From the simulation result,it was shown that to mitigate the spread of COVID-19 in Nigeria there is need for serious awareness programs to enlighten people on the available control measures;social distancing,self-isolation,use of personal protective equipment(such as face mask,hand globes,overall gown,etc.),regular hand washing using soap or sanitizer,avoiding having contact with person showing the symptoms and reporting any suspected case.

A Binomial Model Approach: Comparing the R0 Values of SARS-CoV-2 rRT-PCR Data from Laboratories across Northern Cyprus

Northern Cyprus has implemented relatively strict measures in the battle against the outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The measures were introduced at the beginning of the COVID-19pandemic, in order to prevent the spread of the disease. One of these measures was the use of two separate realtime reverse transcription polymerase chain reaction (rRT-PCR) tests for SARS-CoV-2 referred to as the doublescreening procedure, which was adopted following the re-opening of the sea, air and land borders for passengersafter the first lockdown. The rRT-PCR double screening procedure involved reporting a negative rRT-PCR testwhich was carried in 72 to 120 h before departure whilst presenting no known symptoms of the COVID-19and performing a second rRT-PCR test at the point of arrival. This study compares the results of SARS-CoV-2rRT-PCR tests performed on incoming flight passengers from the 1st July to 9th of September 2020 to NorthernCyprus. The rRT-PCR test results collected by the Near East University (NEU) DESAM COVID-19 laboratory werecompared with the rRT-PCR test results collected by the Ministry of Health and/or private COVID-19 laboratoriesin Northern Cyprus. This comparative study was conducted using binomial distribution. In addition, by applyingthe Susceptible-Exposed-Infected-Removed (SEIR) model to Northern Cyprus, overall basic reproduction number(R0) value of the COVID-19 was analysed for the same time period to act as a threshold for this comparison.In both the statistical and SEIR mathematical model, R0 was calculated. It was assumed that, the more similarthe R0 results of NEU DESAM COVID-19 laboratory and other laboratories were with the overall R0 value ofNorthern Cyprus, the more reliable the results would be. We calculated that the median R0 values of the NEUDESAM COVID-19 laboratory and other laboratories performing the SARS-CoV-2 rRT-PCR on air passengersduring the studied period to be 0.96 and 1.29, respectively, compared to Northern Cyprus median R0 value whichwas 0.99. The rRT-PCR screening results from the NEU DESAM COVID-19 laboratory were closely aligned withthe screening results of Northern Cyprus whereas the screening results reported by other laboratories were not in afit with the regional pattern. This study also aimed to point out the importance of the rRT-PCR screening procedure since asymptomatic positive SARS-CoV-2 cases entery to Northern Cyprus was inhibited and this prevented thespread of the disease within the population.

Direct immune-SCIR public-opinion propagation model based on real-time online users

Current public-opinion propagation research usually focused on closed network topologies without considering the fluctuation of the number of network users or the impact of social factors on propagation. Thus, it remains difficult to accurately describe the public-opinion propagation rules of social networks. In order to study the rules of public opinion spread on dynamic social networks, by analyzing the activity of social-network users and the regulatory role of relevant departments in the spread of public opinion, concepts of additional user and offline rates are introduced, and the direct immune-susceptible, contacted, infected, and refractory (DI-SCIR) public-opinion propagation model based on real-time online users is established. The interventional force of relevant departments, credibility of real information, and time of intervention are considered, and a public-opinion propagation control strategy based on direct immunity is proposed. The equilibrium point and the basic reproduction number of the model are theoretically analyzed to obtain boundary conditions for public-opinion propagation. Simulation results show that the new model can accurately reflect the propagation rules of public opinion. When the basic reproduction number is less than 1, public opinion will eventually disappear in the network. Social factors can significantly influence the time and scope of public opinion spread on social networks. By controlling social factors, relevant departments can analyze the rules of public opinion spread on social networks to suppress the propagate of negative public opinion and provide a powerful tool to ensure security and stability of society.

Global Asymptotical Stability of an SEIR Model with Random Perturbation

Stochasticity is introduced into a susceptible-exposed but not infectious-infectious-removed （SEIR） model describing epidemics＇ transmission, via the technique of parameter perturbation which is standard in stochastic population modeling. The existence and uniqueness of the model have been proved in this paper. And E detailed analysis on global asymptotic stability is also carried out.

Stability Analysis of SIQS Epidemic Model with Saturated Incidence Rate

A SIQS epidemic model with saturated incidence rate is studied. Two equilibrium points exist for the system, disease-free and endemic equilibrium. The stability of the disease-free equilibrium and endemic equilibrium exists when the basic reproduction number R0, is less or greater than unity respectively. The global stability of the disease-free and endemic equilibrium is proved using Lyapunov functions and Poincare-Bendixson theorem plus Dulac’s criterion respectively.

Analysis the Dynamics of SIHR Model: Covid-19 Case in Djibouti

The Covid-19 epidemic is an emerging infectious disease of the viral zoonosis type caused by the coronavirus strain SARS-CoV-2, it is classified as a human-to-human communicable disease and is currently a pandemic worldwide. In this paper, we propose conceptual mathematical models of the epidemic dynamics of four compartments. We have collected data from the Djibouti health ministry. We study the positivity, boundedness, existence and uniqueness of the weak solution. Next, we define the Basic reproduction number by the method of the DFE and EEP. Then, we study the local and global stability and the bifurcation analysis of equilibrium to examine its epidemiological relevance. Finally, we analyze the fit of the data in comparison with the result of our mathematical results, to validate the model and estimate the important model parameters and prediction about the disease. We consider the real cases of Djibouti from 15th March to 15th May 2021.