Next Generation Matrix Approach Research Articles

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.

Modelling the impact of human behavior using a two-layer Watts-Strogatz network for transmission and control of Mpox

PurposeThis study aims to evaluate the effectiveness of mitigation strategies and analyze the impact of human behavior on the transmission of Mpox. The results can provide guidance to public health authorities on comprehensive prevention and control for the new Mpox virus strain in the Democratic Republic of Congo as of December 2023.MethodsWe develop a two-layer Watts-Strogatz network model. The basic reproduction number is calculated using the next-generation matrix approach. Markov chain Monte Carlo (MCMC) optimization algorithm is used to fit Mpox cases in Canada into the network model. Numerical simulations are used to assess the impact of mitigation strategies and human behavior on the final epidemic size.ResultsOur results show that the contact transmission rate of low-risk groups and susceptible humans increases when the contact transmission rate of high-risk groups and susceptible humans is controlled as the Mpox epidemic spreads. The contact transmission rate of high-risk groups after May 18, 2022, is approximately 20% lower than that before May 18, 2022. Our findings indicate a positive correlation between the basic reproduction number and the level of heterogeneity in human contacts, with the basic reproduction number estimated at 2.3475 (95% CI: 0.0749–6.9084). Reducing the average number of sexual contacts to two per week effectively reduces the reproduction number to below one.ConclusionWe need to pay attention to the re-emergence of the epidemics caused by low-risk groups when an outbreak dominated by high-risk groups is under control. Numerical simulations show that reducing the average number of sexual contacts to two per week is effective in slowing down the rapid spread of the epidemic. Our findings offer guidance for the public health authorities of the Democratic Republic of Congo in developing effective mitigation strategies.

A fractal-fractional order Susceptible-Exposed-Infected-Recovered (SEIR) model with Caputo sense

This study explores the intricacies of the COVID-19 pandemic by employing a four-compartment model with a fractal-fractional derivative based on Caputo concept. The analysis hinges on Schauder fixed point theorem, used to qualitatively examine the solutions and ascertain their existence and uniqueness within the model. The fundamental reproduction number is determined through the next-generation matrix approach. This study delves into the stability of equilibrium points and conducts a sensitivity analysis of model parameters. The equilibrium without infections is locally and globally stable when the basic reproduction number is less than 1. Also, this equilibrium becomes unstable when the basic reproduction number exceeds 1. Applying Lyapunov principles and the Routh–Hurwitz criteria, it is established that the endemic equilibrium point is globally stable for the basic reproduction number values greater than 1. The proposed model incorporates Ulam-Hyers stability through nonlinear functional analysis. Lagrange interpolation method estimates solutions for the fractal-fractional order COVID-19 model. Numerical simulations are performed using MATLAB software to exemplify the model behavior in the context of the Italian case study. Furthermore, fractal-fractional calculus techniques hold significant promise for comprehending and predicting the pandemic’s global dynamics in other countries.

Analyzing dynamics and stability of single delay differential equations for the dengue epidemic model

This paper introduces a mathematical model that simulates the transmission of the dengue virus in a population over time. The model takes into account aspects such as delays in transmission, the impact of inhibitory effects, the loss of immunity, and the presence of partial immunity. The model has been verified to ensure the positivity and boundedness. The basic reproduction number R0 of the model is derived using the advanced next-generation matrix approach. An analysis is conducted on the stability criteria of the model, and equilibrium points are investigated. Under appropriate circumstances, it was shown that there is local stability in both the virus-free equilibrium and the endemic equilibrium points when there is a delay. Analyzing the global asymptotic stability of equilibrium points is done by using the appropriate Lyapunov function. In addition, the model exhibits a backward bifurcation, in which the virus-free equilibrium coexists with a stable endemic equilibrium. By using a sensitivity analysis technique, it has been shown that some factors have a substantial influence on the behavior of the model. The research adeptly elucidates the ramifications of its results by effortlessly validating theoretical concepts with numerical examples and simulations. Furthermore, our research revealed that augmenting the rate of inhibition on infected vectors and people leads to a reduction in the equilibrium point, suggesting the presence of an endemic state.

Analysis and comparative study of a deterministic mathematical model of SARS-COV-2 with fractal-fractional operators: a case study

In this paper, we investigate a fractal-fractional-order mathematical model with the influence of hospitalized patients and the impact of vaccination with fractal-fractional operators. The respective derivatives are considered in the Caputo, Caputo Fabrizio, and Atangana–Baleanu senses of fractional order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\alpha $$\\end{document} and fractal dimension τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ au $$\\end{document}. For the proposed problem, some results regarding basic reproduction number and stability are given. Using the next-generation matrix approach, we have investigated the global and local stability of several types of equilibrium points. We provide a detailed analysis of the existence and uniqueness of the solution. Moreover, we fit the model with the real data of Pakistan from June 01, 2020, till March 24, 2021. Then, we use the fractal-fractional derivative to find a numerical solution for the model. MATLAB software is used for numerical illustration. Graphical presentations corresponding to different parameteric values are given as well.

A mathematical model for varicella-zoster and HIV co-dynamic supported by numerical simulations.

The prevalence of the varicella-zoster virus (VZV) and its correlation underscore its impact on a significant segment of the population. Notably contagious, VZV serves as a risk factor for the manifestation of HIV/AIDS, with its reactivation often signaling the onset of immunodeficiency. Recognizing the concurrent existence of these two diseases, this study focuses on the co-infection dynamics through a deterministic mathematical model. The population is categorized into seven exclusive groups, considering the complexities arising from the interplay of HIV and Zoster. We establish the non-negativity and boundedness of solutions, examine equilibrium points, calculate basic reproduction numbers via the next-generation matrix approach, and analyze the existence and local stabilities of equilibriums using the Routh-Hurwitz stability criteria. The numerical simulations reveal that the model converges to an endemic equilibrium point when the reproduction number exceeds unity. The primary objectives of this study are to comprehensively understand the transmission dynamics of HIV and Zoster in a co-infected population and to provide valuable insights for developing effective intervention strategies. The findings emphasize the importance of addressing these co-infections to mitigate their impact on public health.

Optimal control and cost-effectiveness analysis for the human melioidosis model

In this work, we formulated and investigated an optimal control problem of the melioidosis epidemic to explain the effectiveness of time-dependent control functions in controlling the spread of the epidemic. The basic reproduction number (R0c) with control measures is obtained, using the next-generation matrix approach and the impact of the controls on R0c is illustrated numerically. The optimal control problem is analyzed using Pontryagin's maximum principle to derive the optimality system. The optimality system is simulated using the forward-backward sweep method based on the fourth-order Runge-Kutta method in the MATLAB program to illustrate the impact of all the possible combinations of the control interventions on the transmission dynamics of the disease. The numerical results indicate that among strategies considered, strategy C is shown to be the most effective in reducing the number of infectious classes compared to both strategy A and strategy B. Furthermore, we carried out a cost-effectiveness analysis to determine the most cost-effective strategy and the result indicated that the strategy B (treatment control strategy) should be recommended to mitigate the spread and impact of the disease regarding the costs of the strategies.

Mathematical simulations and sensitivity visualization of fractional order disease model describing human immunodeficiency

In this study, we explore a mathematical model of human immunodeficiency virus type 1 (HIV-1) infection in CD4+ T-cells, considering fractional-order dynamics in the Caputo sense. Fractional models are of great significance due to their capacity to predict disease outbreaks while incorporating memory and non-local characteristics. To establish the existence and uniqueness of solutions for the fractional model, a fixed-point theorem and an iterative technique are employed. Furthermore, it is confirmed that the solutions of the model are both positive and bounded. The basic reproduction number, denoted as R0, is calculated using the next-generation matrix approach, and the equilibrium points of the model, including disease-free and endemic equilibrium points, are presented. A sensitivity analysis of R0 is performed by varying different parameters. To analyze the local stability of equilibrium points, the Routh-Hurwitz technique is employed. The global stability of equilibrium points is demonstrated using the Lyapunov function and LaSalle's principle, as well as through UlamHyers (UH) stability and the generalized UH stability conditions. The proposed numerical scheme Adams-Bashforth predictor-corrector is validated by comparing it with the fourth-order Runge-Kutta (RK4) technique. We examine the influence of the fractional order (ξ) by conducting numerical simulations for different values of ξ using the Adams-Bashforth predictor-corrector approach. Furthermore, graphical presentations illustrate how different crucial model parameters impact disease dynamics. The results obtained from this study demonstrate that the adopted strategies significantly improve prediction accuracy.

SEIRS model for malaria transmission dynamics incorporating seasonality and awareness campaign

Malaria, a devastating disease caused by the Plasmodium parasite and transmitted through the bites of female Anopheles mosquitoes, remains a significant public health concern, claiming over 600,000 lives annually, predominantly among children. Novel tools, including the application of Wolbachia, are being developed to combat malaria-transmitting mosquitoes. This study presents a modified susceptible-exposed-infectious-recovered-susceptible (SEIRS) compartmental mathematical model to evaluate the impact of awareness-based control measures on malaria transmission dynamics, incorporating mosquito interactions and seasonality. Employing the next-generation matrix approach, we calculated a basic reproduction number (R0) of 2.4537, indicating that without robust control measures, the disease will persist in the human population. The model equations were solved numerically using fourth and fifth-order Runge-Kutta methods. The model was fitted to malaria incidence data from Kenya spanning 2000 to 2021 using least squares curve fitting. The fitting algorithm yielded a mean absolute error (MAE) of 2.6463 when comparing the actual data points to the simulated values of infectious human population (Ih). This finding indicates that the proposed mathematical model closely aligns with the recorded malaria incidence data. The optimal values of the model parameters were estimated from the fitting algorithm, and future malaria dynamics were projected for the next decade. The research findings suggest that social media-based awareness campaigns, coupled with specific optimization control measures and effective management methods, offer the most cost-effective approach to managing malaria.

Global Dynamics of a Within-Host Model for Usutu Virus

We propose a within-host mathematical model for the dynamics of Usutu virus infection, incorporating Crowley–Martin functional response. The basic reproduction number R0 is found by applying the next-generation matrix approach. Depending on this threshold, parameter, global asymptotic stability of one of the two possible equilibria is also established via constructing appropriate Lyapunov functions and using LaSalle’s invariance principle. We present numerical simulations to illustrate the results and a sensitivity analysis of R0 was also completed. Finally, we fit the model to actual data on Usutu virus titers. Our study provides new insights into the dynamics of Usutu virus infection.

Differential contagiousness of respiratory disease across the United States

The initial contagiousness of a communicable disease within a given population is quantified by the basic reproduction number, R0. This number depends on both pathogen and population properties. On the basis of compartmental models that reproduce Coronavirus Disease 2019 (COVID-19) surveillance data, we used Bayesian inference and the next-generation matrix approach to estimate region-specific R0 values for 280 of 384 metropolitan statistical areas (MSAs) in the United States (US), which account for 95% of the US population living in urban areas and 82% of the total population. We focused on MSA populations after finding that these populations were more uniformly impacted by COVID-19 than state populations. Our maximum a posteriori (MAP) estimates for R0 range from 1.9 to 7.7 and quantify the relative susceptibilities of regional populations to spread of respiratory diseases. One-Sentence SummaryInitial contagiousness of Coronavirus Disease 2019 varied over a 4-fold range across urban areas of the United States.

A next-generation matrix approach using Routh–Hurwitz criterion and quadratic Lyapunov function for modeling animal rabies with infective immigrants

Rabies remains a major public health concern in many regions of the world, particularly in poorer nations. This study proposes a mathematical model for the animal rabies transmission dynamics, considering infective immigrants and vaccination as potential control measures to address this issue. The effective reproduction number (Re) was calculated by using the next-generation matrix approach. By using Routh–Hurwitz Criterion, a disease-free equilibrium point (DFE) was obtained and proved to be locally asymptotically stable if Re<1 and unstable otherwise, and also, by using the quadratic Lyapunov function DFE was determined to be globally asymptotically stable. Moreover, sensitivity analysis of the model parameters on the (Re) was performed using the normalized forward sensitivity index method. For simulation analysis, the numerical analysis was performed with the help of the MATLAB software. The simulated data results revealed that the increase in vaccination rate and decreasing infective immigrants will reduce the spread of the decrease, which was depicted graphically.

Mathematical Model for Malaria Disease Transmission

Malaria is one of the fatal diseases caused by plasmodium parasites and transmitted to humans through biting of the female of {\it Anopheles} mosquitoes. We proposed a deterministic mathematical model for simulating Malaria disease transmission between humans and mosquitoes. The basic reproduction number $\mathcal{R}_{0}$ is determined by using the next-generation matrix approach. Stability conditions for the model equilibrium points with respect to $\mathcal{R}_{0}$ derived and we show that the forward bifurcation occurred. When $\mathcal{R}_{0} &lt;1$ or $\mathcal{R}_{0} &gt;1$ the Malaria disease dies out or spreads, respectively. The sensitivity analysis for the basic reproduction number $\mathcal{R}_{0}$ fulfilled locally and globally. The model simulation was found by using Runge--Kutta fourth order method in MATLAB. Furthermore, The effects of the important parameters were investigated, and the obtained results were presented in graphical forms. Also, we obtained that the simulation results agree with the stability analysis for $E_{def}$. We discussed the impacts of the Malaria disease control interventions on the important parameter for Malaria disease transmission. Recommendation for control and eradicating Malaria disease transmission provided.

A non-linear differential equation model of COVID-19 and seasonal influenza co-infection dynamics under vaccination strategy and immunity waning

This study presents a mathematical model of the transmission dynamics of COVID-19 and influenza co-infection. The potential impacts of the influenza vaccine only on the co-infection dynamics and the potential impacts of both vaccines on the co-infection dynamics are thoroughly studied. The basic reproduction number for the two diseases using the next-generation matrix approach and the stability of the sub-model is examined. The model assessed the scenario whereby both diseases’ waning immunity occurs concurrently to check the epidemic peaks. The numerical simulation results show that the diseases would continue to be endemic in the population if the immunity waning rates increase. The epidemic peak can be reduced by increasing vaccination and vaccine efficacy rates. The results show that the COVID-19 contact rate significantly increases the epidemic level more than the co-infection contact rate. A similar result was obtained when it was observed that the COVID-19 post-recovery waning rate has more significant effects on the epidemic peak than the co-infection post-recovery waning rate. A possible reason for this counter-intuitive occurrence is that two infections cannot have the same viral load nor the same within-host competitiveness. This means an infectious co-infected person will transmit the infection with the highest within-host competitiveness. Here, it is suspected that COVID-19 has a within-host competitive advantage over influenza in the co-dynamics.

Stability and optimal control analysis of a stage-structured SVIHRS model with isolation, infection based media awareness and media-induced vaccination

Sometimes vaccination is reported through media awareness and behavioral responses through different life stages on epidemic outbreaks. But how much awareness is required for effective implementation of vaccination is important from the public health standpoint. To validate this fact, we propose a deterministic SVIHRS stage-structured epidemic model with isolation, infection based media awareness and media-induced vaccination. In the absence of disease, no media campaign is required and hence the disease-free equilibrium exists under a sufficient condition, which is independent of the media awareness coefficient. To study the effect of awareness campaigns by media on the outbreak of an epidemic, a basic reproduction number RV using the next-generation matrix approach is determined, which shows that an endemic equilibrium point exists when RV>1. The local stability of both equilibrium points is analyzed by the corresponding characteristic equations and sufficient conditions are also determined for global stability using Lyapunov second method. A sensitivity analysis for the basic reproduction number is performed to inform target parameters to be controlled. Further, an optimal control problem is studied using Pontryagin’s minimum principle to identify an optimal implementation rate of awareness campaigns so that disease can be controlled with minimal possible expenditure on awareness campaigns. Finally, using MATLAB, some numerical simulations are carried out in support of the stability, sensitivity, and optimal control results.

Investigating the dynamics of a novel fractional-order monkeypox epidemic model with optimal control

In this paper, a novel fractional-order monkeypox epidemic model is introduced, wherefractional-order derivativesin the sense of Caputo are applied to achieve more realistic results for the proposednonlinear model. The newly developed model, which models the transmission and spread of monkeypox across the interacting populations of humans and rodents, is controlled by a 14-dimensional system of fractional-order differential equations. To comply with empirical and reported observations, the state variables of the proposed model are categorized into three main groups of state variables: the population who are at high risk of being infected, people with low infection probability, and finally, rodents who can carry and transmit the virus. The high-risk group represents individuals who might be more vulnerable to the virus due to their habits, workplace, or hygienic behaviors. The existence, uniqueness, non-negativity, and boundedness of the solution to the proposed model are proved. The next-generation matrix approach is used to determine the control monkeypox reproduction number, R0, and the equilibrium points for the proposed model are obtained. The effect of the main parameters in the model is thoroughly investigated to provide new insight into the new dynamics of the model. The region of stability of the disease-free points (DFE) is obtained in the space of parameters, and the effect of the parameters is examined. In addition, theoptimal control strategyis applied to the model to provide insight into some prevention control to stop the disease from spreading and to provide new control strategies during the monkeypox outbreak. Numerical simulations are performed to validate the theoretical results of the different optimal control strategies.

Analysis of Turberculosis-COVID-19 Coinfection Using Fractional Derivatives

Fractional-order derivative modeling continues to receive great interest among researchers across the globe. In this study, Tuberculosis-COVID-19 coinfection is studied using Atangana–Baleanu fractional-order derivatives defined in Caputo sense. We confirmed the existence and singularity of the solution and investigated the model’s equilibrium points. Additionally, we examined the model’s stability in terms of the Ulam–Hyers and generalized Ulam–Hyers stability criteria. The basic reproduction number R 0 was calculated using the next-generation matrix approach. We also looked into the model’s disease-free equilibrium point’s regional stability. Numerical scheme for simulating the fractional-order system with Mittag–Leffler Kernels are presented. Numerical simulations are given to validate the model. Results of the simulation showed a decline in the number of COVID-19 infections within the population when the fractional operator was reduced.

A mathematical model on the transmission dynamics of typhoid fever with treatment and booster vaccination

Typhoid fever is a potentially fatal illness that is caused by the bacteria Salmonella typhi. In this study, a deterministic mathematical model was formulated to look into transmission dynamics of typhoid fever with treatment and booster vaccination. The reproduction number R0 is calculated using the next-generation matrix approach. Then, a stability analysis on the equilibrium points was performed using Routh–Hurwitz criteria. It was revealed that the disease-free equilibrium point is locally asymptotically stable whenever R0 is less than 1 together with other conditions. We also showed that R0≤1 does not guarantee global stability of the typhoid-free equilibrium point and corroborated the result by showing the possible existence of backward bifurcation at R0=1. The model parameters in R0 were also subjected to sensitivity analysis, which revealed that the transmission rate, infection through an exposed person, and bacteria are the most influential parameters of the reproduction number R0. Numerical simulations were run to determine the impact of various parameters on the dynamics of typhoid.

Mathematical Model and Analysis on the Impacts of Vaccination and Treatment in the Control of the COVID-19 Pandemic with Optimal Control

In this study, a nonlinear deterministic mathematical model that evaluates two important therapeutic measures of the COVID-19 pandemic: vaccination of susceptible and treatment for infected people who are in quarantine, is formulated and rigorously analyzed. Some of the fundamental properties of the model system including existence and uniqueness, positivity, and invariant region of solutions are proved under a certain meaningful set. The model exhibits two equilibrium points: disease-free and endemic equilibrium points under certain conditions. The basic reproduction number, R 0 , is derived via the next-generation matrix approach, and the dynamical behavior of the model is explored in detail. The analytical analysis reveals that the disease-free equilibrium solution is locally as well as globally asymptotically stable when the associated basic reproduction number is less than unity which indicates that COVID-19 dies out in the population. Also, the endemic equilibrium point is globally asymptotically stable whenever the associated basic reproduction number exceeds a unity which implies that COVID-19 establishes itself in the population. The sensitivity analysis of the basic reproduction number is computed to identify the most dominant parameters for the spreading out as well as control of infection and should be targeted by intervention strategies. Furthermore, we extended the considered model to optimal control problem system by introducing two time-dependent variables that represent the educational campaign to susceptibles and continuous treatment for quarantined individuals. Finally, some numerical results are illustrated to supplement the analytical results of the model using MATLAB ode45.

Mathematical modeling of chickenpox in Phuket: Efficacy of precautionary measures and bifurcation analysis

In this paper, a mathematical model depicting the transmission dynamics of Chickenpox is developed by incorporating a new parameter denoting the rate of precautionary measures. The influence and the importance of following precautionary measures are showed by applying the real data collected at Phuket province, Thailand. The model analysis such as positivity and boundedness of the solutions are provided. The rate of precaution for the spread the of chickenpox was a factor that influenced the basic reproductive number, which was calculated using the next-generation matrix approach. The model’s equilibrium points are identified, and the condition for the disease-free equilibrium’s local and global asymptotic stability is established. The model also shows forward bifurcation. Numerical simulation is carried out to show the importance of considering the precautionary measures while controlling the disease spread and the influence of those introduced parameters are depicted graphically. Though our results, we concluded that the rate of precautionary measures plays an vital role at the same time it reduces the chance of getting infected by Chickenpox virus.