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.

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.

Analysis of Listeriosis Transmission Dynamics with Optimal Control

Listeriosis is an illness caused by the germ Listeria monocytogenes. Generally, humans are infected with listeriosis after eating contaminated food. Listeriosis mostly affects people with weakened immune systems, pregnant women and newborns. In this paper, a model describing the dynamics of Listeriosis is developed and analysed using ordinary differential equations. The model was analysed both quantitatively and qualitatively for its local and global stability, basic reproductive number and parameter contributions to the basic reproductive number to understand the impact of each parameter on the disease spread. The Listeriosis model has been extended to include time dependent control variables such as treatment of both humans and animals, vaccination and education of humans. Pontryagin’s Maximum Principle was introduced to obtain the best optimal control strategies required for curbing Listeriosis infections. Numerical simulation was performed and the results displayed graphically and discussed. Cost effectiveness analysis was conducted using the intervention averted ratio (IAR) concepts and it was revealed that the most effective intervention strategy is the treatment of infected humans and animals.

A Stochastic SVIR Model for Measles

In this article, we consider the construction of a SVIR (Susceptible, Vaccinated, Infected, Recovered) stochastic compartmental model of measles. We prove that the deterministic solution is asymptotically the average of the stochastic solution in the case of small population size. The choice of this model takes into account the random fluctuations inherent to the epidemiological characteristics of rural populations of Niger, notably a high prevalence of measles in children under 5, coupled with a very low immunization coverage.

Invasion reproductive numbers for periodic epidemic models

There are many cases within epidemiology where infections compete to persist within a population.In studying models for such cases,one of the goals is to determine which infections can invade a population and persist when other infections are already resident within the population.Invasion reproductive numbers(IRN),which are tied to the stability of boundary endemic equilibria,can address this question.By reinterpreting resident infections epidemiologically,this study extends methods for finding IRNs to periodic systems,and presents some examples which illustrate the often complex computations required.Results identify conditions under which a simple time-average can be used to derive IRNs,and apply the methods to examine how seasonal fluctuations in influenza incidence facilitate the year-round persistence of bacterial respiratory infections.

Simulation Model to the Zika Virus Considering Asymptomatic Population

A simulation model based on nonlinear ordinary differential equations to interpret the transmission dynamics of Zika Virus (ZIKV), is formulated and analyzed, integrating the asymptomatic human population and coupled to the Aedes aegypti dynamics, the epidemic threshold Basic Reproduction Number R0 is determined, as the spectral radius of Next-Generation Matrix and the system is simulated with MAPLE computing program taking the parameter values from literature.

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.

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.

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.

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.

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.

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.

Mathematical Modelling of the COVID-19 Epidemic in Northern Ireland in 2020

In this study, we investigate the dynamics of the COVID-19 epidemic in Northern Ireland from 1st March 2020 up to 25th December 2020, using several copies of a Susceptible-Exposed-Infectious-Recovered (SEIR) compartmental model, and compare it to a detailed publicly available dataset. We split the data into 10 time intervals and fit the models on the consecutive intervals to the cumulative number of confirmed positive cases on each interval. Using the fitted parameter estimates, we also provide estimates of the reproduction number. We also discuss the limitations and possible extensions of the employed model.

Approximations of Quasi-Stationary Distributions of the Stochastic <i>SVIR</i>Model for the Measles

In this paper, we analyze the quasi-stationary distribution of the stochastic SVIR (Susceptible, Vaccinated, Infected, Recovered) model for the measles. The quasi-stationary distributions, as discussed by Danoch and Seneta, have been used in biology to describe the steady state behaviour of population models which exhibit discernible stationarity before to become extinct. The stochastic SVIR model is a stochastic SIR (Susceptible, Infected, Recovered) model with vaccination and recruitment where the disease-free equilibrium is reached, regardless of the magnitude of the basic reproduction number. But the mean time until the absorption (the disease-free) can be very long. If we assume the effective reproduction number Rp < 1 or , the quasi-stationary distribution can be closely approximated by geometric distribution. β and δ stands respectively, for the disease transmission coefficient and the natural rate.

Fuzzy Global Stability Analysis of the Dynamics of Malaria with Fuzzy Transmission and Recovery Rates

In this paper, fuzzy techniques have been used to track the problem of malaria transmission dynamics. The fuzzy equilibrium of the proposed model was discussed for different amounts of parasites in the body. We proved that when the amounts of parasites are less than the minimum amounts required for disease transmission (), we reach the model disease-free equilibrium. Using Choquet integral, the fuzzy basic reproduction number through the expected value of fuzzy variable was introduced for the fuzzy Susceptible, Exposed, Infected, Recovered, susceptible-Susceptible, Exposed and Infected (SEIRS-SEI) malaria model. The fuzzy global stabilities were introduced and discussed. The disease-free equilibrium is globally asymptotically stable if or if the basic reproduction number is less than one (). When and , there exists a co-existing endemic equilibrium which is globally asymptotically stable in the interior of feasible set . Finally, the numerical simulation has been done for showing the effectiveness of our analytical results.