Reproduction Number Research Articles

Enhancing infectious diseases early warning: A deep learning approach for influenza surveillance in China

ObjectiveThis study aimed to develop a universally applicable, feedback-informed Self-Excitation Attention Residual Network (SEAR) model. This model dynamically adapts to evolving disease trends and surveillance system changes, accommodating various scenarios. Thereby enhancing the effectiveness of early warning systems. MethodsSurveillance data on influenza-like illness (ILI) was collected from various regions including Northern China, Southern China, Beijing, and Yunnan. The reproduction number (Rt) was estimated to determine the threshold for issuing warnings. The Self-Excitation Attention Residual Network (SEAR) was devised employing deep learning algorithms and was trained, validated, and tested. The SEAR model's efficacy was assessed based on five metrics: accuracy rate, recall rate, F1 score, confusion matrix, and the receiver operating characteristic curve. ResultsWith an advance warning set at three days, the SEAR model outperformed five primary models − logistic regression, support vector machine, random forest, Extreme Gradient Boosting, and Long Short-Term Memory model − in all five evaluation metrics. Notably, the model's warning performance declined with an increase in the early warning value and the number of warning days, albeit maintaining a ROC value over 0.7 in all scenarios. ConclusionThe SEAR model demonstrated robust early warning performance for influenza in diverse Chinese regions with high accuracy and specificity. This novel model, augmenting traditional systems, supports widespread application for respiratory disease outbreak monitoring. Future evaluations could incorporate alternative indicators, with the model continuously updating through data feedback, thus enhancing its universal applicability. Ongoing optimization, using iterative feedback and expert judgment, heralds a transformative approach to surveillance-based early warning strategies.

Modeling the dynamics of the Hepatitis B virus via a variable-order discrete system

We investigate the dynamics of the hepatitis B virus by integrating variable-order calculus and discrete analysis. Specifically, we utilize the Caputo variable-order difference operator in this study. To establish the existence and uniqueness results of the model, we employ a fixed-point technique. Furthermore, we prove that the model exhibits bounded and positive solutions. Additionally, we explore the local stability of the proposed model by determining the basic reproduction number. Finally, we present several numerical simulations to illustrate the richness of our results.

Finite Difference Method for Infection Model of HPV with Cervical Cancer under Caputo Operator

In this paper, a fractional model in the Caputo sense is used to characterize the dynamics of HPV with cervical cancer. Generalized mean value theorem has been used to examine whether the infection model has a unique positive solution. The model has two equilibrium points: the disease-free point and the endemic point. The examination of the system’s local and global stability is provided in terms of the basic reproductive number Rp°. The global stability analysis has been carried out using an appropriate Lyapunov function and the LaSalle invariant principle. The results demonstrate that in the infection model, if Rp°<1, then the solution converges to the disease-free equilibrium, which is both locally and globally asymptotically stable. Whilst Rp°>1, the endemic equilibrium is considered to exist. Simulations are implemented via a finite difference method with Grünwald-Letnikov discretization approach for Caputo derivative operator to define how changes in parameters impact the dynamic behavior of the system using Matlab.

Asymptotic profiles of a nonlocal dispersal SIS epidemic model with saturated incidence

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

Neural network approach for cholera dynamics: Integrating deterministic and stochastic insights

In this article, a comprehensive qualitative analysis framework is presented that integrates stochastic and deterministic models to explore the transmission dynamics of Cholera disease. The significance of the reproduction number in understanding the spread of the disease and in formulating effective control strategies is emphasized. The sensitivity analysis of model parameters is conducted to identify those that critically influence transmission. By converting the deterministic model into a stochastic form, Brownian motion and stochastic parameters are incorporated to capture the inherent randomness in disease dynamics. This approach is further enriched by the application of neural networks (NN) to simulate and validate the complex interactions involved in Cholera transmission. The analysis extends to the global positive solution, stochastic reproduction number, and virus-free stochastic dynamics, providing an improved understanding of the disease's behavior. The critical conditions necessary for the extinction and long-term elimination of the virus are also addressed, highlighting the importance of controlling its spread. The stationary distribution of the disease is examined through the lens of a regular Markov process, enhancing the understanding of the stochastic model's dynamics. The results verified by neural networks, compared with those obtained from conventional numerical methods, underscore the accuracy and versatility of neural networks in epidemiological research, offering new perspectives and methodologies for tackling this global health challenge.

Dynamics and Control Strategies for SLBRS Model of Computer Viruses Based on Complex Networks

Dynamics and Control Strategies for SLBRS Model of Computer Viruses Based on Complex Networks

A simulation-based approach for estimating the time-dependent reproduction number from temporally aggregated disease incidence time series data

Tracking pathogen transmissibility during infectious disease outbreaks is essential for assessing the effectiveness of public health measures and planning future control strategies. A key measure of transmissibility is the time-dependent reproduction number, which has been estimated in real-time during outbreaks of a range of pathogens from disease incidence time series data. While commonly used approaches for estimating the time-dependent reproduction number can be reliable when disease incidence is recorded frequently, such incidence data are often aggregated temporally (for example, numbers of cases may be reported weekly rather than daily). As we show, commonly used methods for estimating transmissibility can be unreliable when the timescale of transmission is shorter than the timescale of data recording. To address this, here we develop a simulation-based approach involving Approximate Bayesian Computation for estimating the time-dependent reproduction number from temporally aggregated disease incidence time series data. We first use a simulated dataset representative of a situation in which daily disease incidence data are unavailable and only weekly summary values are reported, demonstrating that our method provides accurate estimates of the time-dependent reproduction number under such circumstances. We then apply our method to two outbreak datasets consisting of weekly influenza case numbers in 2019–20 and 2022–23 in Wales (in the United Kingdom). Our simple-to-use approach will allow accurate estimates of time-dependent reproduction numbers to be obtained from temporally aggregated data during future infectious disease outbreaks.

Kalman filter based on a fractional discrete-time stochastic augmented CoVid-19 model

In this paper, we study the dynamics of the CoVid-19 outbreak in Semarang, Indonesia, using a fractional CoVid-19 model. We first determine the effects of the isolation rate ∊ and infection rate β on the reproduction number R0 and infected number V. We find that R0 is directly proportional to β and inversely proportional to ∊. For V, the effect of physical distancing is not as significant as changing ∊. As ∊ increases, V decreases, the number of susceptible individuals increases, the number of quarantined individuals decreases sharply, and the number of recovered individuals decreases. Moreover, the effect of vaccination is also considered. The combination of physical distancing, isolation, and vaccination has a significant impact on reducing the number of infected individuals. Analysis of dynamical systems allows us to understand the characteristics of our model, such as its boundedness and non-negativity, the existence of equilibrium points, the existence and uniqueness of solutions, and the local and global stability. To validate our fractional CoVid-19 model, we introduce the fractional extended Kalman filter (FracEKF) as a prediction method and compare the results against reported CoVid-19 data. FracEKF is a modified version of the basic extended Kalman filter with a time-fractional memory effect. The prediction results illustrate the accuracy of this model in terms of the root mean square error (RMSE), normalized root mean square error (NRMSE), and mean absolute percentage error (MAPE) for each fractional-order. Varying ∊ reproduces the trends observed in the reported data for the number of infected individuals, i.e., when ∊ increases, the infected number decreases. Moreover, a higher fractional-order results in higher model accuracy. Furthermore, higher values of the process noise Qf give smaller errors, whereas higher values of the observation noise Rf produce higher errors. Qf and the fractional-order α are inversely proportional to RMSE,NRMSE, and MAPE, whereas Rf is directly proportional to RMSE,NRMSE, and MAPE.

Dynamical behavior of a hepatitis B epidemic model and its NSFD scheme

AbstractHepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number $$\mathcal {R}_0$$ R 0 , stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when $$\mathcal {R}_0=1$$ R 0 = 1 . It is concluded that the endemic equilibrium is globally asymptotically stable when $$\mathcal {R}_0>1$$ R 0 > 1 , by utilizing Dulac’s criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained.

Wiener and Lévy processes to prevent disease outbreaks: Predictable vs stochastic analysis

The study considers stochastic behavior along with modeling, mathematical analysis, theory development, and numerical simulation of the COVID-19 virus. To evaluate current trends and make future projections regarding the basic reproduction number and infection case, we have taken the modified five-compartment SEIRD mathematical model. Since the basic reproduction number R0 is not sufficient to predict the outbreak, we applied the Itô stochastic differential equations (SDEs) along with the Weiner process and Lévy jump to investigate the nature of the disease outbreak. Since COVID-19 is an infectious disease caused by the severe acute respiratory syndrome coronavirus 2 “(SARS-CoV-2)”, that has common symptoms including fever, cough, and difficulty breathing, we have illustrated the boundedness and positivity of solutions and investigated the stability of endemic and disease-free equilibrium points. The findings show that the dynamics of disease epidemics are influenced by contact patterns. For all the models, the threshold quantity, R0 is calculated which is the key reason to prove the global and local stability analysis of disease-free equilibrium and endemic equilibrium points. Using the least squares method for data fitting, we have performed a case study on COVID-19 data in Italy for this article. A sensitivity analysis is part of our investigation to find important factors. This paper investigates a stochastic epidemic model (SDE) using Lévy jumps and Weiner processes. Local and global stability in a stochastic setting are examined in the analysis. To view the analytical study, a multitude of numerical results are achieved. The study is important to comprehend the stochastic behavior of the virus in the contagious model in order to prevent similar disease outbreaks in the future.

Linking Spontaneous Behavioral Changes to Disease Transmission Dynamics: Behavior Change Includes Periodic Oscillation.

Behavior change significantly influences the transmission of diseases during outbreaks. To incorporate spontaneous preventive measures, we propose a model that integrates behavior change with disease transmission. The model represents behavior change through an imitation process, wherein players exclusively adopt the behavior associated with higher payoff. We find that relying solely on spontaneous behavior change is insufficient for eradicating the disease. The dynamics of behavior change are contingent on the basic reproduction number corresponding to the scenario where all players adopt non-pharmaceutical interventions (NPIs). When , partial adherence to NPIs remains consistently feasible. We can ensure that the disease stays at a low level or maintains minor fluctuations around a lower value by increasing sensitivity to perceived infection. In cases where oscillations occur, a further reduction in the maximum prevalence of infection over a cycle can be achieved by increasing the rate of behavior change. When , almost all players consistently adopt NPIs if they are highly sensitive to perceived infection. Further consideration of saturated recovery leads to saddle-node homoclinic and Bogdanov-Takens bifurcations, emphasizing the adverse impact of limited medical resources on controlling the scale of infection. Finally, we parameterize our model with COVID-19 data and Tokyo subway ridership, enabling us to illustrate the disease spread co-evolving with behavior change dynamics. We further demonstrate that an increase in sensitivity to perceived infection can accelerate the peak time and reduce the peak size of infection prevalence in the initial wave.

Corruption dynamics: a mathematical model and analysis

This study proposes and analyzes a deterministic mathematical model to describe the dynamics of corruption transmission. We began by proving that the solution to the model is bounded and positive. The next-generation matrix approach is used to compute the basic reproduction number (R0) in relation to corruption-free equilibrium. The Jacobian and Lyapunov functions are used to show that corruption-free equilibrium is asymptotically stable in both locally and globally when R0<1, and otherwise, an endemic corruption equilibrium develops. Furthermore, the sensitivity of the model's parameters was investigated. The findings demonstrate that religious precepts govern public education. The two sectors most susceptible to corruption control are education and corrections. The study recommends investing more in the provision of public education to citizens by creating awareness among all and including it in the education curriculum and religious leaders to teach their followers seriously about the impact of corruption as well as the use of jail as punishment. The numerical simulation results agreed with the analytical results.

Mathematical Modeling of the Transmission Dynamics of Gumboro Disease

Gumboro disease is a viral poultry disease that causes immune suppression on the infected birds leading to poor production, mortality, and exposure to secondary infections, hence a major threat in the poultry industry worldwide. A mathematical model of the transmission dynamics of Gumboro disease is developed in this paper having four compartments of chicken population and one compartment of Gumboro pathogen population. The basic reproduction number Rog is derived, and the dynamical behaviors of both the disease-free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary differential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever Rog<1 and an asymptotic stable EE whenever Rog>1. The numerical simulation to verify the theoretical results was carried out using MATLAB ode45 solver, and the results were found to be consistent with the theoretical findings.

Flip bifurcation analysis and mathematical modeling of cholera disease by taking control measures

To study the dynamical system, it is necessary to formulate the mathematical model to understand the dynamics of various diseases which are spread in the world wide. The objective of the research study is to assess the early diagnosis and treatment of cholera virus by implementing remedial methods with and without the use of drugs. A mathematical model is built with the hypothesis of strengthening the immune system, and a ABC operator is employed to turn the model into a fractional-order model. A newly developed system SEIBR, which is examined both qualitatively and quantitatively to determine its stable position as well as the verification of flip bifurcation has been made for developed system. The local stability of this model has been explored concerning limited observations, a fundamental aspect of epidemic models. We have derived the reproductive number using next generation method, denoted as “R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{0}$$\\end{document}”, to analyze its impact rate across various sub-compartments, which serves as a critical determinant of its community-wide transmission rate. The sensitivity analysis has been verified according to its each parameters to identify that how much rate of change of parameters are sensitive. Atangana–Toufik scheme is employed to find the solution for the developed system using different fractional values which is advanced tool for reliable bounded solution. Also the error analysis has been made for developed scheme. Simulations have been made to see the real behavior and effects of cholera disease with early detection and treatment by implementing remedial methods without the use of drugs in the community. Also identify the real situation the spread of cholera disease after implementing remedial methods with and without the use of drugs. Such type of investigation will be useful to investigate the spread of virus as well as helpful in developing control strategies from our justified outcomes.

A model for the control of transmission dynamics of human monkeypox disease in Sub-Saharan Africa

The Human Monkeypox virus has received significant research interest in recent times. While few researchers have studied the effects of vaccination on human-to-animal or animal-to-human Monkeypox transmission, others just studied the effects of treatment on human Monkeypox. In this article, we made the proposition of a deterministic vaccine model that deals with the dynamics of the effects of vaccination and treatment on human Monkeypox in sub-Saharan Africa. We investigated the effects of vaccination on the various epidemiological classes qualitatively. The findings from the analysis of the model are that the model possesses two equilibria, locally asymptotically stable disease-free equilibrium (DFE) when an epidemiological threshold - the effective reproductive number is less than unity, and locally asymptotically stable endemic equilibrium when the number is greater than unity. We then corroborated the theoretical findings with numerical simulations, which reveal that when the rate of vaccination is increased resulting in many newly born persons in the populace being vaccinated, the prevalence of the deadly scourge is significantly reduced, while newly born individuals that miss vaccination experience a drastic torment of the deadly disease that often occasion death. Further revelation from the simulations is that when greater efforts

Modelling the co-infection of malaria and zika virus disease

In this work, a new model for the co-infection of malaria and zika virus disease incorporating vaccination, treatment and vector control using sterile-insect technology (SIT) is formulated. The importance of this study is to highlight the possibility of the co-infection of humans with malaria and zika virus disease in any environment where both diseases co-circulate. Also, to suggest a new and comprehensive method for controlling the individual diseases and their co-infection. Through stability analysis, we showed that the disease-free equilibrium, (DFE) point of the co-infection model is locally asymptotically stable when the basic reproduction numbers, Rmz is less than one, and unstable otherwise. But, the DFE failed to be globally stable when Rmz < 1 which is an indication of existence of backward bifurcation in the model. This shows that bringing down the reproduction number, Rmz to less than one is not enough to eradicate the co-infection of the two diseases. Furthermore, it is shown that the two diseases have positive impact on the spread of each other, which could be attributed to misdiagnoses of one disease as the other. We also showed that effective treatment of infectious humans, increasing the rate of vaccination and employing sterile-insect technique to control the vectors significantly helped to control the individual diseases as well as the co-infection. From the results obtained in the study, it can be concluded that effective control of malaria and zika virus disease requires measures that will control their spread in both human and mosquito populations.

Analysis of a diffusive two-strain malaria model with the carrying capacity of the environment for mosquitoes

We propose a malaria model involving the sensitive and resistant strains, which is described by reaction-diffusion equations. The model reflects the scenario that the vector and host populations disperse with distinct diffusion rates, susceptible individuals or vectors cannot be infected by both strains simultaneously, and the vector population satisfies the logistic growth. Our main purpose is to get a threshold type result on the model, especially the interaction effect of the two strains in the presence of spatial structure. To solve this issue, the basic reproduction number (BRN) R0i and invasion reproduction number (IRN) Rˆ0i of each strain (i = 1 and 2 are for the sensitive and resistant strains, respectively) are defined. Furthermore, we investigate the influence of the diffusion rates of populations and vectors on BRNs and IRNs.

Bayesian estimation of the time-varying reproduction number for pulmonary tuberculosis in Iran: A registry-based study from 2018 to 2022 using new smear-positive cases

IntroductionTuberculosis (TB) is one of the most prevalent infectious diseases in the world, causing major public health problems in developing countries. The rate of TB incidence in Iran was estimated to be 13 per 100,000 in 2021. This study aimed to estimate the reproduction number and serial interval for pulmonary tuberculosis in Iran. Material and methodsThe present national historical cohort study was conducted from March 2018 to March 2022 based on data from the National Tuberculosis and Leprosy Registration Center of Iran's Ministry of Health and Medical Education (MOHME). The study included 30,762 tuberculosis cases and 16,165 new smear-positive pulmonary tuberculosis patients in Iran. We estimated the reproduction number of pulmonary tuberculosis in a Bayesian framework, which can incorporate uncertainty in estimating it. Statistical analyses were accomplished in R software. ResultsThe mean age at diagnosis of patients was 52.3 ± 21.2 years, and most patients were in the 35–63 age group (37.1%). Among the data, 9121 (56.4%) cases were males, and 7044 (43.6%) were females. Among patients, 7459 (46.1%) had a delayed diagnosis between 1 and 3 months. Additionally, 3039 (18.8%) cases were non-Iranians, and 2978 (98%) were Afghans. The time-varying reproduction number for pulmonary tuberculosis disease was calculated at an average of 1.06 ± 0.05 (95% Crl 0.96–1.15). ConclusionsIn this study, the incidence and the time-varying reproduction number of pulmonary tuberculosis showed the same pattern. The mean of the time-varying reproduction number indicated that each infected person is causing at least one new infection over time, and the chain of transmission is not being disrupted.

Investigating the Dynamics of Bayoud Disease in Date Palm Trees and Optimal Control Analysis

The fungus Fusarium oxysporum (f.sp. albedinis) causes Bayoud disease. It is one of the epiphytotic diseases that affects a wide range of palm species and has no known cure at present. However, preventive measures can be taken to reduce the effects of the disease. Bayoud disease has caused enormous economic losses due to decreased crop yield and quality. Therefore, it is essential to develop a mathematical model for the dynamics of the disease to propose some affordable methods for disease management. In this study, we propose a novel mathematical model that describes the transmission dynamics of the disease in date palm trees. The model incorporates various factors such as the contact rate of the fungi with date palm trees, the utilization of fungicides, and the introduction of a quarantine compartment to prevent disease dissemination. We first prove a few key properties of the proposed model to ensure that the model is well-posed and suitable for numerical investigations. We establish that the model has a unique positive solution that is bounded and stable over time. We use sensitivity analysis to identify the parameters that have the greatest effect on the reproduction number R0 and illustrate this effect graphically. We then formulate an optimal control problem to identify the most suitable and cost-effective disease control approaches. As a first approach, we solely focus on the application of fungicide to susceptible trees and determine the best spray rates for a greater decrease in exposed and infected trees. Secondly, we emphasize quarantining exposed and infected trees at optimal quarantine rates. Finally, we explore the combined effect of fungicide spraying and isolating infected trees on disease control. The findings of the last approach turn out to be the most rewarding and cost-effective for minimizing infections in date palm trees.

Social contacts in Switzerland during the COVID-19 pandemic: Insights from the CoMix study

To mitigate the spread of SARS-CoV-2, the Swiss government enacted restrictions on social contacts from 2020 to 2022. In addition, individuals changed their social contact behavior to limit the risk of COVID-19. In this study, we aimed to investigate the changes in social contact patterns of the Swiss population. As part of the CoMix study, we conducted a survey consisting of 24 survey waves from January 2021 to May 2022. We collected data on social contacts and constructed contact matrices for the age groups 0–4, 5–14, 15–29, 30–64, and 65 years and older. We estimated the change in contact numbers during the COVID-19 pandemic to a synthetic pre-pandemic contact matrix. We also investigated the association of the largest eigenvalue of the social contact and transmission matrices with the stringency of pandemic measures, the effective reproduction number (Re), and vaccination uptake. During the pandemic period, 7084 responders reported an average number of 4.5 contacts (95% confidence interval, CI: 4.5–4.6) per day overall, which varied by age and survey wave. Children aged 5–14 years had the highest number of contacts with 8.5 (95% CI: 8.1–8.9) contacts on average per day and participants that were 65 years and older reported the fewest (3.4, 95% CI: 3.2–3.5) per day. Compared with the pre-pandemic baseline, we found that the 15–29 and 30–64 year olds had the largest reduction in contacts. We did not find statistically significant associations between the largest eigenvalue of the social contact and transmission matrices and the stringency of measures, Re, or vaccination uptake. The number of social contacts in Switzerland fell during the COVID-19 pandemic and remained below pre-pandemic levels after contact restrictions were lifted. The collected social contact data will be critical in informing modeling studies on the transmission of respiratory infections in Switzerland and to guide pandemic preparedness efforts.