Next Generation Matrix Method Research Articles

Analysis of COVID-19’s Dynamic Behavior Using a Modified SIR Model Characterized by a Nonlinear Function

This study develops a modified SIR model (Susceptible–Infected–Recovered) to analyze the dynamics of the COVID-19 pandemic. In this model, infected individuals are categorized into the following two classes: Ia, representing asymptomatic individuals, and Is, representing symptomatic individuals. Moreover, accounting for the psychological impacts of COVID-19, the incidence function is nonlinear and expressed as Sg(Ia,Is)=βS(Ia+Is)1+α(Ia+Is). Additionally, the model is based on a symmetry hypothesis, according to which individuals within the same compartment share common characteristics, and an asymmetry hypothesis, which highlights the diversity of symptoms and the possibility that some individuals may remain asymptomatic after exposure. Subsequently, using the next-generation matrix method, we compute the threshold value (R0), which estimates contagiousness. We establish local stability through the Routh–Hurwitz criterion for both disease-free and endemic equilibria. Furthermore, we demonstrate global stability in these equilibria by employing the direct Lyapunov method and La-Salle’s invariance principle. The sensitivity index is calculated to assess the variation of R0 with respect to the key parameters of the model. Finally, numerical simulations are conducted to illustrate and validate the analytical findings.

Dynamics and Control of a Novel Discrete Internet Rumor Propagation Model in a Multilingual Environment

In the Internet age, the development of intelligent software has broken the limits of multilingual communication. Recognizing that the data collected on rumor propagation are inherently discrete, this study introduces a novel SIR discrete Internet rumor propagation model with the general nonlinear propagation function in a multilingual environment. Then, the propagation threshold R0 is obtained by the next-generation matrix method. Besides, the criteria determining the spread or demise of rumors are obtained by the stability theory of difference equations. Furthermore, combined with optimal control theory, prevention and refutation mechanisms are proposed to curb rumors. Finally, the validity and applicability of the model are demonstrated by numerical simulations and a real bilingual rumor case study.

A new approach to determine occupational accident dynamics by using ordinary differential equations based on SIR model

The motivation of this study is to develop and establish an occupational accident dynamical model (OA model) based on Susceptible-Infected-Recovered framework. In order to investigate the dynamics of the OA model, monthly occupational accident data from Turkey between 2013 and 2020 has been selected as dataset. The OA model is defined by a coupled first-order ordinary nonlinear differential equation with four variables. In addition, the relationships between these variables are described with ten parameters. The OA model’s characterization of the equilibrium points is analyzed by investigating the behaviors of these points according to the eigenvalues derived from the Jacobian matrix. Also, the stability of these points is obtained according to the eigenvalues. These results show the behavior of the system near equilibrium points. After that, the reproduction number is computed by using the next-generation matrix method. The calculated reproduction number for given parameters reveals that the OA model is unstable. The OA model is numerically solved in 96 steps, with a time interval of 1 month, using the ODE45 Matlab routine based on the explicit Runge–Kutta algorithm. In addition, a modified OA model is developed by adding the Occupational Health and Safety (OHS) re-training parameter to the OA model to observe a reducing effect on occupational accident numbers. The main results of this study provide a new approach about the future estimation of the number of occupational accidents. Furthermore, through the comparison of numerical results from both models, the study demonstrates that national safety policies, particularly those enhancing the efficacy of OHS training, can effectively mitigate accidents.

Dynamics and backward bifurcations of SEI tuberculosis models in homogeneous and heterogeneous populations

The main difference between tuberculosis (TB) and other infectious diseases is that the transmission of the bacterium should be considered not only as the development of a primary infection, but also as exogenous reinfection or endogenous reactivation. Moreover, individuals in the population may have heterogeneous contact rates, which can be described by complex networks. To this end, we present two differential equation-based TB models in homogeneous and heterogeneous populations. The first model assumes that the number of contacts per unit time is constant for the whole population, whereas the second model considers the heterogeneous number of contacts per unit time for each individual. We derive the basic reproduction number of each model using the next-generation matrix method, and analyze the dynamical properties of each model in detail. We find that the two models undergo backward bifurcations and have the same threshold condition for backward bifurcation. From this threshold condition, we see that the reduced rate of exogenous reinfection of individuals plays an important role in causing the backward bifurcation. Interestingly, the second model allows the threshold condition for backward bifurcation to be independent of network parameters. Thus, unlike other infectious disease models on complex networks, in controlling the spread of tuberculosis among populations with different numbers of contacts, we only need to focus on disease parameters during treatment. Finally, numerical simulations more intuitively demonstrate the impact of parameter changes on the prevalence of tuberculosis and reveal the model's richer and more interesting dynamical properties, such as bistability. Sensitivity analysis indicates that the basic reproduction number is highly correlated with both the relapse rate of latent individuals progressing to active infection and the probability of healthy individuals becoming infected after contact with the pathogen. Therefore, enhancing the detection and treatment of latent cases and reducing contact between infected and uninfected individuals are the most crucial public health interventions.

Minimal wave speed for a two-group epidemic model with nonlocal dispersal and spatial-temporal delay

In this paper, a two-group SIR reaction-diffusion epidemic model with nonlocal dispersal and spatial-temporal delay based on within-group and inter-group transmission mechanisms is investigated. The basic reproduction number R0 is calculated using the method of next-generation matrix. The critical wave speed cm* is determined by analyzing the distribution of roots of the characteristic equation. When R0>1 and wave speed c⩾cm*, the existence of traveling waves connecting disease-free and endemic steady states is obtained by constructing sub- and super-solutions and a Lyapunov functional, and applying Schauder’s fixed-point theorem and a limit argument. When R0>1 and 0<c<cm*, the nonexistence of traveling waves connecting disease-free and endemic steady states is proven by contradiction and two-sided Laplace transform. This indicates that the critical wave speed cm* is exactly the minimum wave speed. Numerical simulations are carried out to illustrate theoretical results. The dependence of the minimal speed cm* on time delay, diffusion rates and contact rates is discussed, showing that the longer the latent period and the lower the diffusion rates of infected individuals and the inter-group transmission rates between groups, the slower the spread of disease.

Complex dynamics of an SIHR epidemic model with variable hospitalization rate depending on unoccupied hospital beds

In this paper, we propose an susceptible-infectious-hospitalized-recovered (SIHR) epidemic model with a nonlinear hospitalization rate depending on the number of unoccupied hospital beds. Note that the number of all hospital beds is used as a measure of all available medical resources. The basic reproduction number is calculated using the next-generation matrix method. We analyze the existence of endemic equilibria and discuss the global stability of the disease-free equilibrium. Existence and stability of endemic equilibria indicate possible occurrences of bifurcations. We confirm the appearance of backward bifurcation, saddle–node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation using normal form theory and central manifold theory. Numerical simulations show that the dynamic behavior of the model undergoes a transition from forward bifurcation to backward bifurcation and saddle–node bifurcation when the number of total hospital beds is reduced. Our findings suggest that when the number of total hospital beds falls below a threshold, backward bifurcation will occur, meaning that the disease cannot be eliminated even if the basic reproduction number is below unity. Therefore, the number of hospital beds should be increased beyond the bed threshold during an outbreak of a disease, which has important implications for disease control.

Mathematical Modeling of the Dynamics of Psychoactive Drug Abuse with Intervention of Control Agencies

Drug abuse continues to pose a significant threat to both developed and developing countries. This paper examines the dynamics of psychoactive drug abuse with intervention of control agencies. The study established that the drug abuse model remains positively invariant over time, as indicated by the positivity of the solutions to the model equations. The study identified both drug abuse endemic and drug abuse free equilibrium states. Basic reproduction number was obtained using the method of next generation matrix. The local stability of the drug abuse free equilibrium state was analyzed using Jacobian method and the analysis showed that the drug abuse free equilibrium state is locally asymptotically stable since all the Eigen values are negative. Similarly, the global stability analysis of the drug abuse free equilibrium state confirmed that the system is globally asymptotically stable. The sensitivity index was computed. The results indicate that parameters with positive sensitivity indices increase the prevalence, while those with negative sensitivity indices decrease it. This suggests that these parameters have the ability to either increase or decrease the persistence of drug abuse in the population. The numerical simulation of the model was performed using MATLAB R2015a software and results show that enlightenment and law enforcement agencies play a vital role in reducing the spread of drug abuse. This research recommends intensification of early enlightenment to curtail the spread of drug abuse. Also making more policies, enactment and enforcement of more laws that will check the use of hard drugs

A fractional derivative model of the dynamic of dengue transmission based on seasonal factors in Thailand

Climate variability affects the changes in controlling diseases transferred by insects. An increase in the population, the growth of communities, and a lack of public health infrastructure bring about the return of diseases of which insects are carriers, one of the illness issues. Therefore, the disease control is significant to help reduce the burden on the government and strengthen the country's public health structure. This research proposes a novel approach to modeling dengue fever dynamics, we employ a fractional derivative model with the Atangana–Baleanu–Caputo derivative, which offers a more accurate representation of real-world disease dynamics compared to traditional integer-order models. Basic qualifications are proposed. Equilibrium points and basic reproduction numbers are analyzed. The next-generation matrix method is used to identify the transmission. Besides, parameter sensitivity analysis is performed to learn about factors affecting input parameter values' effects on the basic reproduction number. It was found that the most common parameter affecting the transmission was the biting rate of mosquitoes was 1. In addition, the existence and uniqueness of the solution are examined using the Banach fixed point theorem. The Toufik–Atangana method is used for the numerical examination of a fractional version of the proposed model. We compared different values of fractional-order α=0.965, 0.975, 0.985, 0.995 and 1 it was found that when the order of derivatives decreases, the transmission shall decrease accordingly. This research provides valuable insights for developing effective control strategies to reduce the burden of dengue fever and strengthen public health systems.

Modeling Rabies Dynamics: The Impact of Vaccination and Infectious Immigrants on Public Health

Public health is still seriously threatened by rabies in many parts of the world, particularly in poorer nations. In order to address this issue, this work suggests an equation describing the mechanisms of rabies transmission between animals, taking into account infectious immigrants and vaccination as possible preventive measures. The next-generation matrix (NGM) Method method was used to calculate the effective reproduction number (R0). Using the Routh-Hurwitz Criterion, a disease-free equilibrium point (DFE) was discovered and It was demonstrated to have local asymptotic stability if (R0 < 1), and unstable in every other case. Additionally, DFE was discovered to be quadratic Lyapunov stable and globally asymptotically stable. Additionally, the model parameters’ sensitivity analysis on the (R0) was carried out using the central manifold theory for the bifurcation analysis, and the analysis’s normalized forward sensitivity index method. MATLAB software was utilized to do numerical analysis for simulation analysis. The findings of the simulated data showed that a higher vaccination rate and fewer infectious immigrants would slow the development of the decline, as shown visually.

Dynamics of a susceptible-infected-recovered model on complex networks with interregional migration.

We present a susceptible-infected-recovered model based on a dynamic flow network that describes the epidemic process on complex metapopulation networks. This model views population regions as interconnected nodes and describes the evolution of each region using a system of differential equations. The next-generation matrix method is used to derive the global basic reproduction number for three cases: a general network with homogeneous infection rates in all regions, a fully connected network, and a star network with heterogeneous infection and recovery rates. For the homogeneous case, we show that this global basic reproduction number is independent of the migration rates between regions. However, the rate of convergence of each region to an equilibrium state exhibits a much larger variance in random (Erdős-Rényi) networks compared to small-scale (Barabási-Albert) networks. For the general heterogeneous case, we report interesting results, namely that the global basic reproduction number decays exponentially with respect to the smallest nonzero Laplacian eigenvalue (algebraic connectivity). Furthermore, we demonstrate both analytically and numerically that as the network's algebraic connectivity increases, either by increasing the average node degree of each region or the global migration rate, the global basic reproduction number decreases and converges to the ratio of the average local infection rate to the average local recovery rate, meaning that the lower bound of the global basic reproduction rate does not equal the mean of local basic reproduction rates.

A conformable mathematical model of Ebola Virus Disease and its stability analysis

Ebola Virus Disease (EVD) is a viral hemorrhagic fever that affects humans and other primates. It is characterized by rapid virus spread in a short period of time. The disease has the potential to spread to many different regions of the world. In this paper, we have developed a modified mathematical model of the Ebola virus, adding the quarantine population as a control strategy. The quarantine population F and parameters ρ3 represent the rate at which individuals enter the quarantine compartment, which is vital in controlling the virus spread within society. The conformable derivatives have been applied to the modified model to observe the behavior of individuals for fractional derivative values between 0.7 and 1. For a modified model, the threshold parameter (R0) has been determined using the Next-Generation Matrix (NGM) method. We have checked local and global stability at a disease-free equilibrium point using Routh-Herwitz (RH) criteria and Castillo-Chavez, respectively. Numerical results obtained through the Fourth-Order Runge Kutta Method (RK4) demonstrate, a decrease in the virus transmission rate after following the implementation of the quarantine strategy.

Mathematical modeling, sensitivity analysis, and optimal control of students awareness in mathematics education

This article presents a continuous-time mathematical model, EFSMK, integrating various educational categories, serving as a valuable tool for understanding students interactions with mathematics. The paper begins by exploring the necessary preliminaries to describe the model. Subsequently, we examine the model’s well-posedness, focusing on the positivity and boundedness of solutions. The basic reproduction number is then derived using the next-generation matrix method. The model exhibits two stable states: the mathematics-free equilibrium point and the learning equilibrium point for students who struggle with mathematics. The study shows that when the basic reproduction number is less than one, the mathematics-free equilibrium point is globally asymptotically stable. Conversely, when the basic reproduction number exceeds one, the learning equilibrium point for students who struggle with mathematics is globally asymptotically stable. Numerical simulations validate the theoretical findings. In the last part of the paper, we present a structured model involving two distinct strategies aimed at improving the mathematical skills of students who encounter difficulties in this subject. The first strategy consists of offering individualized or small-group tutoring sessions, focusing on challenging topics and using alternative teaching methods. The second strategy entails adjusting the overall program and pedagogical approach to better respond to diverse learning needs. This approach uses Pontryagin’s maximum principle and iterative analysis to determine optimal controls. The effectiveness of these strategies is evaluated using numerical simulations in various scenarios.

Mathematical insights into the [formula omitted] model with Allee and fear dynamics in the context of COVID-19

In this study, we introduce and investigate the SEIQRD COVID-19 model incorporating Allee and fear effects. Employing the next-generation matrix method, we derive the basic reproduction number, a crucial epidemiological metric. Our model exhibits two significant equilibrium points: one representing the disease-free state and the other representing the endemic state. Utilizing the Routh–Hurwitz criteria, we demonstrate that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number (R0) is less than 1, while the endemic equilibrium attains local stability when R0 exceeds 1. Furthermore, we establish global stability for the endemic equilibrium through the application of Poincare–Bendixson properties and Dulac’s criteria. To complement our theoretical results, we conduct comprehensive simulation studies, validating the practical implications of our findings.

Mathematical Model and Optimal Control Strategy for the Dynamics of Hepatitis B Virus Disease Incorporating Treatment Failure and Advanced Stage Compartments

In this paper, mathematical model for the dynamic of Hepatitis B virus infection was developed taking into cognizance the effect of rate of progression to the end stage (cirrhosis and HCC) by individuals who experience treatment failure in the treatment class and chronically infected class. In order to track the transmission of these diseases, the basic reproduction number which determines the rate of new secondary infection when an infected individual is introduced into a totally susceptible population was computed by next generation matrix method (NGM). The stability of the disease-free equilibrium point was also investigated using Routh-Hurwitz Criterion and the result proven to be Locally Asymptotically Stable (LAS) when the basic reproduction number less than unity. The global stability was also investigated by Castillo-Chavez method and the results show that the model is Globally Asymptotically (GAS) when reproduction number is less than unity. Sensitivity analysis was carried out on the parameters of the reproduction number, using forward normalized sensitivity index to ascertain those parameters of high impact on the reproduction number to aid the incorporation of the control variables appropriately. Numerical simulation was also carried using Forward-backward Runge-Kutta fourth order scheme to study the effect of these control strategies on the dynamics of the model and the results shows that the best strategy to implement to help eradicate HBV disease in the system is strategy E which is the combination of effective condom use, HBV vaccination, HBV Treatment and Behavioral change

Modelling the Time-varying Transmission of Wild Poliovirus in Pakistan

Aims/Objectives: The efforts to eradicate the wild poliovirus since 1988 have successfully reduced its global prevalence by 99%. However, as of 2024, Pakistan and Afghanistan remain the only two endemic countries facing the virus transmission. This study employs a transmission dynamic model to understand the persistence of wild poliovirus type 1 (WPV1) in Pakistan. Study Design: An ordinary differential equations-based deterministic type of mathematical model was developed. Place and Duration of Study: Department of Epidemiology and Public Health and Institute of Microbiology, University of Agriculture Faisalabad, March 2023 to August 2023. Methodology: We included the number of reported polio cases and the proportion of missed children by supplementary immunization activities (SIAs) across the country from 2017-2022. The model considered both human-human and environment-human virus transmission through sewage contamination and these are represented by time-dependent transmission rate parameters. The parameter estimation was done by model fitting to the reported data of WPV1 cases. The model simulations then predicted the future polio infections in Pakistan. Results: Our analysis identified that the asymptomatic infectious population missed by the SIAs is the major contributor to disease persistence in the country. Moreover, the environment can contribute to virus transmission in areas with poor WASH infrastructure and under-immunized populations. The estimated basic reproduction number through the next-generation matrix method was 1.61 while the estimated effective reproduction number was 0.12. The model showed a better predictive ability than constant transmission rate models. The numerical simulations considering the reduction in the virus-shedding rate by the asymptomatic population resulted in an 85% reduction in future polio cases in Pakistan. Conclusion: The results suggest that endemic virus transmission will continue subject to the current higher vaccination coverage across the country. The model can be further utilized to guide eradication efforts for targeted allocation of preemptive measures.

Impact of Yoga Awareness on Transmission Dynamics of Communicable Diseases: SIR Model Analysis

Communicable diseases are major health problems that affect the whole economy of the na-tion. So it becomes the prime agenda of developed and developing countries to educate peo-ple about disease dynamics and control strategies. Due to changes in people’s living styles, disease treatment modality has been changed. So, we have introduced Yoga awareness as control strategy which includes Ahar, Vihar, Achar and Vichar. They are means for achiev-ing the physical, mental, social and spiritual well-being. Our main aim is to model the role of Yoga awareness in controlling the disease dynamics. It maintains the physical fitness of practitioners and improve the whole metabolism system of the human body. In this work, the previous SIR model is modified by incorporating a new awareness transmission rate β 1 to the existing model. Real life situation data of Yoga aware and aware infected individuals were col-lected from different Yoga centers of Sudurpashchim province, Nepal and analyzed by using mathematical techniques. Stability analysis of governing by ordinary differential equations showed that model is stable. Yoga awareness reproduction number Ra was calculated by next-generation matrix method. Sensitivity analysis of Ra and concerning parameters indicate that Ra decreases with an increase in Yoga awareness coverage level. Also, the recovery rate has the opposite relation with Ra which indicates that the recovery period also decreases with an increase in awareness coverage. Local and global stability analysis showed that disease-free equilibrium exists when Ra < 1 and endemic equilibrium exists when Ra > 1. Numerical simulations also support the analytical results and suggest that Yoga awareness has a positive influence on controlling disease d ynamics. The increase in the coverage of awareness leads to reduced susceptibility and infectivity. So propagation of disease can be controlled by Yoga awareness.

Study on SEAI Model of COVID-19 Based on Asymptomatic Infection

In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R0 and calculate the equilibrium point. Secondly, when R0<1, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When R0>1, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of R0 is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease.

Analyzing Global Stability of M-Pox Disease Dynamics: Mathematical Insights into Detection and Treatment Strategies

In this research, a mathematical model is constructed to scrutinize the transmission patterns of monkeypox (mpox), with a specific emphasis on integrating the early detection of infected undetected individuals to curb its transmission. This research takes into account a range of factors influencing the propagation of monkeypox, encompassing population demographics, contact dynamics, and the efficacy of early detection of unidentified infected individuals. Employing the next-generation matrix method, the basic reproduction number (R0) is computed, revealing that the disease-free equilibrium state is locally asymptotically stable when (R0<1). This suggests that containment of monkeypox is achievable within a human populace where (R0) remains below one (1), yet it transitions to an endemic state when (R0) exceeds this critical value one (1). Furthermore, a sensitivity analysis is conducted to evaluate the robustness of our findings to variations in model parameters. Utilizing numerical simulations conducted via MAPLE 18, we demonstrate the significance of prompt identification and immediate intervention for infected individuals who may otherwise go undetected, in effectively diminishing the dynamic propagation of monkeypox. The results underscore the pivotal role of early detection in mitigating monkeypox outbreaks and curtailing transmission rates.

Mathematical Modelling of COVID-19 with Vaccination, Quarantine and Non-Pharmaceutical Interventions

This study formulated and analyzed a deterministic mathematical model of ten compartments for the transmission dynamics of COVID-19 infection using a system of non-linear ordinary differential equations. The system has disease-free and endemic equilibrium points. The basic reproduction number was obtained using the next-generation matrix method and the stability of the equilibrium points were analyzed. From the qualitative analysis, the disease-free equilibrium point is both locally and globally asymptotically stable. Finally, numerical simulations of the model were carried out using MATLAB R2021a. Based on the result obtained, it was concluded that the implementation of vaccination, non-pharmaceutical intervention as well as contact tracing and quarantine can lead to effective control or elimination of the COVID-19 pandemic. And hence, we recommend that the responsible agencies should create a way of enlightening the public on the need of being vaccinated for COVID-19 so as to prevent themselves from getting infected; also, adequate and effective implementation of vaccination and non-pharmaceutical interventions is required in order to prevent or curtail the COVID-19 infection completely. COVID-19, Non-pharmaceutical intervention, Quarantine, Stability analysis, Vaccination

A Mathematical Model for Transmission of Taeniasis and Neurocysticercosis

In this study, we present a mathematical model for the codynamics of taeniasis and neurocysticercosis and rigorously analyze it. To understand the underlying dynamics of the proposed model, basic system properties such as the positivity and boundedness of solutions are investigated through the completing differential process. The basic reproduction number was calculated using the next-generation matrix method, and the analysis showed that when R0<1, the disease in the community eventually dies out, and when R0>1, the diseases persist. Local stability of the equilibria was analyzed using the Jacobian matrix, and Lyapunov function techniques were used to determine the global analysis, which showed that the endemic equilibrium point was globally stable when R0>1. On the other hand, the disease-free equilibrium was determined to be globally stable when R0<1. To identify the most influential parameters of the proposed model, partial correlation coefficient techniques were used. The numerical results depict that the model aligns well with the transmission dynamics, which goes through two populations: humans and pigs, whereby the model system stabilizes after some time, showing the validity of the proposed model. Furthermore, the simulations of the proposed model revealed that the shedding habit of infected humans with taeniasis and the bad cooking habit or eating of raw or undercooked pork products have a higher impact on the spread of neurocysticercosis and taeniasis in the community. Hence, this study proposes that in order to control taeniasis and neurocysticercosis, effective disease control measures should primarily prioritize hygienic behaviour and proper cooking of pork meat to the required temperature.