Value Of Basic Reproduction Number Research Articles

Stability and Bifurcation Analysis of Measles Model

A six compartmental deterministic mathematical model, governed by a system of ordinary differential equations for measles was formulated in other to study and analyze the transmission dynamics of measles in human population. The model was shown to be mathematically and epidemiologically meaningful. The basic reproduction number of the model was obtained and global stability of the disease-free equilibrium and endemic equilibrium were obtained and shown to be asymptotically stable, whenever the basic reproduction and unstable if otherwise. More so, if then the endemic equilibrium of the model equation is globally asymptotically stable The effect of some parameters of the model relatives to the basic reproduction number was calculated using the normalized forward sensitivity indices, and it was shown that increase in the parameters with negative indices will reduce the value of the basic Reproduction number, while increase in those with positive indices will increase the value of basic reproduction number. The bifurcation analysis was also carried out and the model was shown to exhibit backward bifurcation which indicates that is no longer sufficient for effective disease control. The numerical result shows that isolation of infective plays a major role in reducing the transmission of the disease in the population.

Global dynamics of a multiscale model for hepatitis C virus infection

Hepatitis C virus (HCV) can establish infection via two distinct modes: virus-to-cell infection and cell-to-cell transmission. These infections prompt the activation of two types of adaptive immune responses: the cytotoxic T lymphocyte response and the antibody response. In this paper, we study HCV dynamics by developing a multiscale model that incorporates both modes of infection as well as the two types of immune responses. We derive both the basic reproduction number of virus and four immune reproduction numbers for the model. We identify five equilibria, the existence of which depends on the values of basic reproduction number of virus and the immune reproduction numbers. We also establish the global asymptotic stability of the equilibria by employing Lyapunov functions, which further underscores the profound impact of the aforementioned reproduction numbers on the model’s overall stability.

Long-time dynamics and semi-wave of a delayed nonlocal epidemic model with free boundaries

This paper is concerned with a nonlocal reaction–diffusion system with double free boundaries and two time delays. The free boundary problem describes the evolution of faecally–orally transmitted diseases. We first show the well-posedness of global solution, and then establish the monotonicity and asymptotic property of basic reproduction number for the epidemic model without delays, which is defined by spectral radius of the next infection operator. By introducing the generalized principal eigenvalue defined in general domain, we obtain an upper bound of the limit value of basic reproduction number. We discuss the spreading and vanishing phenomena in terms of the basic production number. By employing the perturbed approximation method and monotone iteration method, we establish the existence, uniqueness and monotonicity of solution to semi-wave problem. When spreading occurs, we determine the asymptotic spreading speeds of free boundaries by constructing suitable upper and lower solutions from the semi-wave solutions. Moreover, spreading speeds for partially degenerate diffusion case are provided in a similar way.

Within-host models of dengue virus transmission with immune response

AbstractDengue fever is an infectious viral fever. The complex behavior of the virus within the body can be explained through mathematical models to understand the virus’s dynamics. We propose two different with-in host models of dengue virus transmission with humoral immune response. The proposed models differ from one another because one of the models assumes that newly formed viruses infect healthy cells again. To understand the dynamics of the proposed models, we perform a comparative study of stability analysis, numerical simulation, and sensitivity analysis. The basic reproduction number (BRN) of the two models is computed using next-generation matrix method. The local stability (l.s) analysis is discussed using the linearization method. The Lyapunov’s direct method is used to check the global stability (g.s) of the models. It has been found that both the equilibrium states for both the models, namely, virus-free equilibrium state and endemic equilibrium state, are globally stable, based on the value of BRN. Results show the influence of immune response on the cell dynamics and virus particles. The virus neutralization rate by antibodies and rate that affects the antibody growth are highly sensitive for the two models. Optimal control is applied to explore the possible control strategies to prevent virus spread in the host system. It is evident from the results that the strategy to administrate antibiotic drugs and home remedies slow down the virus spread in the host.

Analysis Of Local Stability Of The Model On COVID-19 Spread In DKI Jakarta Province

The province of DKI Jakarta in Indonesia has an advanced amount of COVID-19 incidents. Hence its dispersion must be restrained. The SEAIQHRD (Susceptible, Exposed, Asymptomatic, Infected, Quarantined, Hospitalized, Recovery, Deceased) model for the dispersion of COVID-19 was evolved in this article. Next, using NGM method to compute basic reproduction number and employing Routh-Hurwitz criterion method to analyze its local stability. Further, two equilibrium points, namely: endemic and disease-free equilibrium points, are obtained. The value of basic reproduction number is used to determine stability analysis. If basic reproduction number less than one, then the endemic equilibrium point is considered asymptotically stable. Based on the sensitivity analysis, the recovery rate of those who are symptomatic subpopulations can help stop the propagation of COVID-19 illness. This article employs data from the DKI Jakarta Province in numerical simulations to depict the dynamics of the COVID-19 dispersion model. According to the analysis's findings, the COVID-19 dispersion model is asymptotically stable at the endemic equilibrium point with ℜ0=2,1966. This indicate that the average of each infected individual can infect two susceptible persons so that the number of infected person will increase over time and cause an outbreak, which means that COVID-19 will remain in the community.

Global stability analysis of viral infection model with logistic growth rate, general incidence function and cellular immunity

It is well known that the mathematical biology and dynamical systems give very important information for the study and research of viral infection models such as HIV, HBV, HCV, Ebola and Influenza. This paper deals with the global dynamics of generalized virus model with logistic growth rate for target cells, general incidence rate and cellular immunity. The results will be obtained by using Lyapunov’s second method and LaSalle’s invariance principle. We prove the global stability of the rest points of the system by the value of basic reproduction number (R0) and the immune response reproduction number (RCTL). We have found that if R0<1, then the infection-free equilibrium is globally asymptotically stable. For R0>1 and RCTL<1, under certain conditions on incidence rate function, immune-free equilibrium is globally asymptotically stable. Finally, we prove that if R0>1 and RCTL>1, then under certain conditions on incidence rate function the endemic equilibrium is globally asymptotically stable. Since the logistic growth rate for target cells and general incidence rate have been included in this manuscript, our obtained results are the generalization of those in the previous literatures. Moreover, the results have been obtained with weaker assumptions in comparison with the previous ones. Numerical simulations are presented to support and illustrate our analytical results.

Analysis of COVID-19 using a modified SLIR model with nonlinear incidence

Infectious diseases kill millions of people each year, and they are the major public health problem in the world. This paper presents a modified Susceptible-Latent-Infected-Removed (SLIR) compartmental model of disease transmission with nonlinear incidence. We have obtained a threshold value of basic reproduction number (R0) and shown that only a disease-free equilibrium exists when R0<1 and endemic equilibrium when R0>1. With the help of the Lyapunov-LaSalle Invariance Principle, we have shown that disease-free equilibrium and endemic equilibrium are both globally asymptotically stable. The study has also provided the model calibration to estimate parameters with month wise coronavirus (COVID-19) data, i.e. reported cases by worldometer from March 2020 to May 2021 and provides prediction until December 2021 in China. The Partial Rank Correlation Coefficient (PRCC) method was used to investigate how the model parameters’ variation impact the model outcomes. We observed that the most important parameter is transmission rate which had the most significant impact on COVID-19 cases. We also discuss the epidemiology of COVID-19 cases and several control policies and make recommendations for controlling this disease in China.

Global Stability Analysis of Dengue Transmission Model with Awareness, Vector Control and Time Delays

In this paper, a mathematical model for a single-strain dengue virus transmission, incorporating vector control, disease awareness of susceptible humans, and both the latent delays for human and mosquitoes, is proposed and studied. The global stability properties of disease-free equilibrium and endemic equilibrium are completely established through Lyapunov functionals and LaSalle’s invariance principle. The global dynamics of the equilibrium points are characterized by the value of basic reproductive number R 0. If R 0 < 1, then the disease-free equilibrium is globally asymptotically stable. If R 0 > 1, then the disease-free equilibrium is unstable, and the endemic equilibrium exists which is globally asymptotically stable. Lastly, this paper presents numerical simulations and possible recommendations for future works.

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.

Estimation of the reproduction number and identification of periodicity for HFMD infections in northwest China

Repeated outbreaks of Hand, foot and mouth disease (HFMD) infections have been observed in recent decades and dominated by various enteroviral serotypes. In particular, enterovirus 71 (EV-A71), coxsackievirus A16 (CV-A16) and coxsackievirus A6 (CV-A6) dominated the prevalence of HFMD infections alternatively in recent years with various outbreak sizes in Baoji, a city of Shaanxi Province in Northwest China. Estimating the reproduction number for various enteroviruses serotypes in northwest China (north temperate zone) and identification of cyclicity of HFMD infections are therefore an issue of great importance for future epidemics prediction and control. The basic/effective reproduction numbers for EV-A71, CV-A16 and CV-A6 were estimated based on daily new cases in 2010, 2011 and 2018, respectively, in which the corresponding pathogen dominated the epidemic. Two different methods based on serial interval were adopted and the basic reproduction number were estimated to be in the range of (1.33, 1.46) for CV-A16, (1.20, 1.29) for EV-A71, and (1.38, 1.59) for CV-A6, respectively. The estimated daily effective reproduction numbers significantly fluctuated before June or after July but varied mildly in (0.5,2) in around June to July for three serotypes. The weekly effective reproduction number for HFMD was estimated based on weekly new cases from year 2010 to 2018, and in most years it peaked in the range of (1.6,2.0) in February to March as well as in the range of (1.0,1.2) in September to October. The wavelet analysis based on the time series of HFMD cases from 2008 to 2018 showed obvious annual and semi-annual cyclicity, while the inter-annual cycles are infeasible. In this study we found that CV-A6 shows the greatest transmission ability among these three pathogens while EV-A71 exhibits the weakest ability of transmission, and moreover, the estimated values of basic reproduction number in northwest China are lower than those in Singapore, Hongkong and Guangdong, which may be due to different climatic circumstances.

Analyze The Effects of Quarantine And Vaccination on Malware Propagation in Wireless Sensor Network

Malware (worm, virus, malicious signals, etc.) propagation in Wireless Sensor Network (WSN) is one of the important concern. The WSN becomes unstable due to presence of malicious signals. Vulnerability of WSN is very high because of the structural constraint of sensor nodes. The attackers target a sensor node of WSN for malware attack. A single infected node starts to spread the malware in the entire network through neighbouring nodes. Therefore, for controlling of malware propagation in WSN a mathematical model is developed. The developed model is based on epidemic theory. The developed model consist of five states such as Susceptible-Infectious-Quarantine-Vaccination-Dead (SIQVD). The quarantine is a method through which to cease the infection spread in WSN. And through vaccination eliminate the malware from the network. The combination of quarantine and vaccination technique improves the network stability. This technique prevents malware propagation in WSN. The basic reproduction number ( ) of the model is deduced. The stability of the network depends on the value of basic reproduction number. It is found that if the value of is less than one the network system exist in malware-fee state, otherwise in endemic state. The equilibrium points of the system is obtained. The effects of quarantine and vaccination has been analyzed on system performance. The theoretical findings are verified by simulation results. Attack Epidemic model Equilibrium point Malware propagation Security Wireless Sensor Network

Education campaign and family understanding affect stability and qualitative behavior of an online game addiction model for children and youth in Thailand

Online game addiction has become a large problem worldwide, and it could give negative impacts on children in many ways such as physical health, learning, emotion, and behavior. Online game addiction is mostly found in children and youth at the age of 15 to 24 years. Therefore, in this paper, we have developed a mathematical model of online game addiction to explore the effects of education campaign and family understanding on online game addiction in Thailand. Analysis of this model reveals two main equilibria, addiction‐free (AFE) and addiction‐present (APE) ones. Results show that the AFE is locally asymptotically stable when the value of basic reproduction number (R0) is less than the unity; otherwise, there is a unique endemic equilibrium point, and it is locally stable when satisfies the Routh‐Hurwitz criterion. Further, the conditions of AFE to be globally stable are demonstrated. Finally, the results of sensitivity analysis and numerical simulation show that the effectiveness of both education campaign and family understanding is an important factor in reducing the value of R0 and holds great promise for lowering the number of children and youth who addict to online game in Thailand.

Role of Optimal Screening and Treatment on Infectious Diseases Dynamics in Presence of Self-protection of Susceptible

In this study, a nonlinear SEIR model has been proposed and analysed that accounts for the screening of exposed population, limited treatment of infective and self-protection in susceptible population. We observe that for the basic reproduction number ( $$\mathcal {R}_0$$ ) below one, a backward bifurcation exists due to saturation in treatment. Also, $$\mathcal {R}_0$$ depends on self-protection parameter and hence self-protection can help in reducing it. Further to understand the impact of screening and treatment on disease dynamics, we extend this model to an optimal control problem. Using Pontryagin’s Maximum Principle, optimal control paths are obtained analytically corresponding to the proposed cost functional. Our numerical experimentations suggest that between only screening and only treatment control policies, screening is more effective and economically viable than the only treatment policy. The combined usage of both screening and treatment is found highly effective and least expensive. Hence, we conclude that implementation of screening along with treatment not only reduces the infection level but also minimizes the associated economic burden. Also, when optimal controls are applied, the cumulative count of infective reduces with the increase in self-protection, therefore, self-protection helps in reducing the disease burden. The epidemic peak as well as cost is also shown to be reduced when self-protection is high for various values of basic reproduction number.

Parameter estimation of modeling schistosomiasis transmission for four provinces in China.

According to monitoring data of Anhui, Jiangxi, Hubei, Jiangsu Provinces, in this paper the transmission of schistosomiasis is studied based on Barbour's mathematical model. The values of the basic reproduction number and key parameters are obtained with two methods. The first method is to calculate directly by using parameter values in references. The second one is to estimate parameter values by the methods of noise measurement and data smoothing and then to obtain new values of basic reproduction number. Comparing these two methods, we found the second method is good to fit the data. This parameter values can be used as reference values in other provinces. Finally, some numerical simulations are carried out to discuss the development trend of prevalence of humans and snails in each province. It is found that schistosomiasis in four provinces is expected to be eliminated with the improvement or maintenance of the standards of prevention and control. Furthermore, the time needed is different in the different four provinces.

A Comparative Review of Prevention of Rabies Incursion between Japan and Other Rabies-Free Countries or Regions.

Although rabies still kills many people, the global eradication of human rabies is considered to be feasible. Progress towards eradication may differ among regions with differing socio-economic statuses; therefore, states that successfully eradicate this disease must be vigilant for rabies re-emergence. Here, we discuss challenges that remain concerning current rabies prevention measures and risk assessment results concerning possible rabies introduction and spread in rabies-free Japan. We summarize the preventative measures undertaken by representative rabies-free countries and regions. Our risk assessment results show that the risk of rabies reintroduction under current circumstances is very low, and that subsequent spread of the disease would be minimal because of quite low value of basic reproduction number. Similar assessments conducted in other rabies-free areas also showed limited risks of introduction. The majority of rabies-free countries maintain their rabies-free status through strict import quarantine of carnivorous animals, efficient surveillance of animal rabies including wildlife, quick emergency responses, and raising public awareness of the disease. To maintain the current rabies-free status in Japan, we strongly recommend maintaining the current quarantine system and reinforcing stakeholder compliance for those involved in international movement of dogs. Moreover, sustainable surveillance systems targeting wildlife are indispensable.

Modeling and Analysis of an Imprecise Epidemic System with Optimal Treatment and Vaccination Control

In this paper, we propose and analyze a Susceptible-Vaccinated-Exposed-Infected-Recovered (SVEIR) type infectious disease model with imprecise parameters. Introducing the interval numbers in functional form, the SVEIR model is proposed and formulated. The existence of possible equilibrium points with their feasibility criteria and an explicit value of basic reproduction number is obtained. The asymptotic stability of the system at different equilibrium points are also discussed. Next by considering treatment and vaccination as two control parameters, an optimal control problem is formulated and solved. Finally, some computer simulation works are given in support of our analytical results.

Estimation of epidemiological parameters for historicalm ship outbreaks of influenza

Periodic Influenza epidemics are a cause of concern world-wide due to heavy burden of disease consequently leading to economic distress mortality. In modern era, rapid international travel between populations makes the impact of air-borne diseases like Influenza more dramatic, as observed in the last pandemic due to Swine origin Influenza A/H1N1 2009. Though transmission of Influenza in humans has been studied in various settings, studies on ship outbreaks are sparse. The current research aims to analyze the historical Influenza outbreaks on sailing ships. Study revealed the pattern of transmission in isolated population and estimates the epidemiological parameters viz. basic reproduction number (R0), epidemic growth rate (r), transmission rate (β). High clustering lead to intense transmission with 0 high value of basic reproductive number, R0. Also, the study brought to light the limitations of analyzing 0 historical data.

The Preventive Control of a Dengue Disease Using Pontryagin Minimum Principal

Behaviour analysis for host-vector model without control of dengue disease is based on the value of basic reproduction number obtained using next generation matrices. Furthermore, the model is further developed involving a preventive control to minimize the contact between host and vector. The purpose is to obtain an optimal preventive strategy with minimal cost. The Pontryagin Minimum Principal is used to find the optimal control analytically. The derived optimality model is then solved numerically to investigate control effort to reduce infected class.

Dynamical Behavior of a Malaria Model with Discrete Delay and Optimal Insecticide Control

In this paper we have proposed and analyzed a simple three-dimensional mathematical model related to malaria disease. We consider three state variables associated with susceptible human population, infected human population and infected mosquitoes, respectively. A discrete delay parameter has been incorporated to take account of the time of incubation period with infected mosquitoes. We consider the effect of insecticide control, which is applied to the mosquitoes. Basic reproduction number is figured out for the proposed model and it is shown that when this threshold is less than unity then the system moves to the disease-free state whereas for higher values other than unity, the system would tend to an endemic state. On the other hand if we consider the system with delay, then there may exist some cases where the endemic equilibrium would be unstable although the numerical value of basic reproduction number may be greater than one. We formulate and solve the optimal control problem by considering insecticide as the control variable. Optimal control problem assures to obtain better result than the noncontrol situation. Numerical illustrations are provided in support of the theoretical results.

Partial Immunity and Vaccination for Influenza

In this article, we estimated the basic reproductive numbers by mathematical modeling and computer simulation using the hospitalization data of influenza type A (H3N2) from the United States as provided by the Centers for Disease Control and Prevention (CDC) from the 2001 to 2006 influenza seasons, respectively. The mean value of basic reproductive number from the 2001-2002 to 2005-2006 influenza seasons is 1.2440, with a 95% confidence interval of 1.1170-1.3710. Our model predicts that the proportion of vaccination of susceptible is 20% and 60 million doses of vaccines should be prepared for each influenza season in the United States. The chi-square test of goodness of fit indicates that our model fits the data reasonably well.