Basic Reproduction Number R0 Research Articles

Titik Equilibrium Bebas dari Penyebaran Hoax Pemilu Lewat Media Sosial Tiktok

Election issues are sensitive information among the society because they relate to people's survival in terms of work, economy and education. Election hoaxes where information whose truth cannot be ascertained becomes a weapon used by certain parties to bring down political opponents. Social media such as TikTok is a media that is often used to spread hoaxes because TikTok social media users come from various ages and there is no distance or time between users. In this paper, a mathematical model is presented to see how the equilibrium point is to be free from hoax based on the basic reproduction number R0. The spread of hoaxes on social media occurs if R0 > 1 and does not occur if R0 < 1. The model approach used is the SEIR mathematical model because it is seen that there are people who receive hoax videos (exposed) and then spread them without finding out the truth (infected), and users with this condition it needs to be recovered. The method used in this research is a literature review. The results obtained show that the mathematical model presented shows that the basic reproduction number R0 is zero (R0 < 1). This means that there will be no spread of election hoaxes

Modeling Ebola Dynamics with a Φ-Piecewise Hybrid Fractional Derivative Approach

Ebola virus disease (EVD) is a severe and often fatal illness posing significant public health challenges. This study investigates EVD transmission dynamics using a novel fractional mathematical model with five distinct compartments: individuals with low susceptibility (S1), individuals with high susceptibility (S2), infected individuals (I), exposed individuals (E), and recovered individuals (R). To capture the complex dynamics of EVD, we employ a Φ-piecewise hybrid fractional derivative approach. We investigate the crossover effect and its impact on disease dynamics by dividing the study interval into two subintervals and utilize the Φ-Caputo derivative in the first interval and the Φ-ABC derivative in the second interval. The study determines the basic reproduction number R0, analyzes the stability of the disease-free equilibrium and investigates the sensitivity of the parameters to understand how variations affect the system’s behavior and outcomes. Numerical simulations support the model and demonstrate consistent results with the theoretical analysis, highlighting the importance of fractional calculus in modeling infectious diseases. This research provides valuable information for developing effective control strategies to combat EVD.

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.

Dynamics of an age-structured SIS epidemic model with local dispersal and general incidence functions

In this paper, an age-structured SIS epidemic model with local dispersal and general incidence functions is formulated. We first analytically derive the well-posedness of solutions, the basic reproduction number R0 and the existence of steady states. Then, we show that the disease-free steady state is global asymptotically stable if R0≤1 and disease in the model is uniformly persistent if R0>1. Moreover, we show global asymptotic stability of the endemic steady state under some cases. Finally, infection age, media coverage and spatial diffusion on the epidemic of infectious diseases are demonstrated by numerical simulations.

Dynamics of a nonlinear infection viral propagation model with one fixed boundary and one free boundary

In this paper we study a nonlinear infection viral propagation model with diffusion, in which, the left boundary is fixed and with homogeneous Dirichlet boundary conditions, while the right boundary is free. We find that the habitat always expands to the half line [0,∞), and that the virus and infected cells always die out when the Basic Reproduction NumberR0≤1, while the virus and infected cells have persistence properties when R0>1. To obtain the persistence properties of virus and infected cells when R0>1, the most work of this paper focuses on the existence and uniqueness of bounded positive equilibrium solutions for subsystems and the existence of positive equilibrium solutions for the entire system.

Research on safety strategies for nucleic acid testing in sudden epidemic conditions based on a SEIARD dynamic model

Infectious diseases have caused enormous disasters in human society, and asymptomatic carriers are an important challenge in our epidemic prevention and control process. Nucleic acid testing has played an important role in rapid testing for asymptomatic individuals. How to carry out nucleic acid testing in a scientific manner is a practical problem encountered in normal production and life. Based on the real COVID-19 epidemic data from Shanghai, we established a susceptible-exposed-infected-asymptomatic-recovered-death (SEIARD) dynamic model. The least squares method was used to fit the data and estimate the unknown parameters β and E(0) in the model, and MATLAB software was employed to simulate the development of the epidemic. The data fitting results indicated that the SEIARD model can better describe the early development patterns of the epidemic (R2 = 0.98; MAPE = 2.54%). We calculated the basic reproduction number of the Shanghai epidemic as R0 = 2.86. As the frequency of nucleic acid testing increased, the basic reproduction number R0 continued to decrease. When there is one latent carrier and one asymptomatic carrier in the nucleic acid testing team, the number of queues is directly proportional to the number of infected individuals, the nucleic acid testing team increases by 50 people, and the number of new asymptomatic cases increases by approximately 4 people. If both susceptible individuals (S) and asymptomatic patients (A) are not wearing masks, the infection rate reaches approximately 7%; after wearing masks, the final infection rate is less than 1% at 1.5 m between two people. The queue spacing is inversely proportional to the number of infected individuals. With a distance of d = 1 m, a nucleic acid testing team of 100 people added 8% of the infected individuals; when d = 1.5 m, fewer than 2% of the newly infected individuals. The results confirmed that controlling the queue size for nucleic acid testing, strictly wearing masks, and maintaining a queue spacing of more than 1.5 m are safe and effective nucleic acid testing strategies. Our findings are also applicable to the prevention of other newly emerging infectious diseases.

Dynamic analysis and optimal control of a mosquito-borne infectious disease model under the influence of biodiversity dilution effect

In this paper, biodiversity and a saturation treatment rate are introduced into a type of mosquito-borne infectious disease model with an incubation time delay. Through the dynamical analysis of the model, conditions for the existence and stability of disease-free and endemic equilibria are determined. The basic reproduction number of the model is calculated, and the condition for the existence of backward bifurcation is outlined. The study finds that under the dual influence of biodiversity and saturation treatment, the threshold characteristic of the basic reproduction number becomes invalidated. When 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} is less than 1, the model may exhibit four equilibrium states, with both the disease-free equilibrium and the endemic equilibrium being locally stable. In this scenario, whether the virus will become extinct depends on the initial conditions. The study also finds that when the basic reproduction number 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} is greater than 1, the stability of the model is influenced by the time delay, with Hopf bifurcation occurring at a specific time delay. In addition, another novel contribution of this paper is the formulation of an optimal control problem that takes into account the minimization of damage caused by humans to biodiversity. Based on the Pontryagin’s maximum principle, the specific characteristics of the optimal control measures are given, and the optimal strategy is derived by comparing five groups of control strategies. The optimal control results highlight the synergistic effect of multiple control measures with biodiversity. Under optimal control, a significant complementary effect between medical inputs and the dilution effect of biodiversity is evident. The findings imply that maintaining high biodiversity levels can decrease the demand for medical resources in mosquito-borne disease control efforts.

A cholera model coupling within-host and between-host with age-dependent and asymptomatic infections

In this paper, a coupling cholera epidemic model is proposed, in which the pathogens could both spread within-host and between-host, meanwhile, there exist age-dependent infection between the asymptomatic and symptomatic infected people. For the fast-time subsystem, the infection-free and endemic equilibria are both globally asymptotically stable. For the slow-time subsystem, the basic reproduction number R0 is determined, by which we observe that the disease-free equilibrium is globally asymptotically stable (the absence of pathogens in environment) if R0<1, while the endemic equilibrium is globally asymptotically stable (the presence of pathogens in environment) when R0>1. The theoretical results are illustrated by numerical simulations, by which we find that age-dependent and asymptomatic infections may further promote cholera spread. Besides, the linking of pathogen transmission at the individual level with infection at the population level could result in forward or backward bifurcation.

A spatio-temporal infection epidemic model with fractional order, general incidence, and vaccination analysis

The paper investigates a spatio-temporal SEIR epidemic model on a global level with fractional order. It employs four partial differential equations incorporating fractional derivatives and account for diffusion to characterize infection dynamics. By applying fixed-point theorem results, the paper establishes the existence, uniqueness, and boundedness of the solution. Equilibrium points are determined based on vaccination values and the basic reproduction number R0. Global stability of each equilibrium is confirmed using the Lyapunov direct method. Through simulations with a predictor–corrector algorithm, key insights into epidemiological dynamics are provided, elucidating the impact of vaccination on reducing disease transmission and altering R0. Trajectories of various compartments closely align with theoretical equilibrium points, affirming the model’s predictive precision. Furthermore, simulations indicate the potential for attaining disease-free equilibria with heightened vaccination rates, underscoring the pivotal role of vaccination strategies in epidemic control and disease eradication. It was shown that the fractional derivative order has no effect on the equilibrium stability but rather only on the convergence speed towards the equilibrium points.

A delay differential equation model on covid-19 with vaccination strategy

In this paper, we have extended SEIR model of COVID-19. The model incorporates two vital aspects in the form of vaccine compartment and constant time delay. The vaccination and time delay provide the information about immune protection and actual existence of the infection among the individuals, respectively. The model is analysed numerically and numerical simulation are executed for three different initial histories and constant time delays which affirm the biological relevance of the system. The analysis includes disease-free equilibrium (DFE), endemic equilibrium, and the basic reproduction number. The stability analysis is performed which reveal the asymptotic stability of the DFE when the basic reproduction number R0 &lt; 1. The study addresses the boundedness and positivity of the solution as the time delay approaches zero. In addition, sensitivity analysis and contour plots for R0 with different parameters offer deeper insights into the model. The impact of vaccination and vaccine inefficacy on the model dynamics is explored.

Modeling the transmission dynamics of banana bunch top disease in banana plants

Banana Bunch Top Disease (BBTD) is a severe viral infection that poses a significant threat to banana production in tropical regions. Caused by the Banana Bunchy Top Virus (BBTV) and transmitted primarily by the banana aphid (Pentalonia nigronervosa), BBTD results in stunted growth and poor fruit production, leading to substantial economic losses. This paper presents a mathematical model to analyze the transmission dynamics of BBTD in banana plants and evaluate various control strategies using the Next Generation Matrix approach to compute the basic reproduction number R0. The study reveals that effective control of aphid populations, timely removal of infected plants, and the use of virus-free planting material significantly reduce the spread of BBTV and improve plant health. Increasing the removal rate γ and managing the transmission rate β are crucial strategies for controlling the disease, while elevated aphid infection rates λ and progression rates σ exacerbate its spread. Numerical simulations highlight the effectiveness of targeted interventions, including the application of pesticides and the introduction of resistant banana varieties. The study recommends that, in addition to current control measures, more focus should be placed on comprehensive monitoring programs, early detection of symptoms, and community education on BBTD management strategies. Enhanced efforts in these areas are crucial to reducing the incidence of BBTD and improving the resilience of banana crops.

Impact of awareness and time-delayed saturated treatment on the transmission of infectious diseases

We have proposed a mathematical model with saturated incidence and treatment rates along with awareness and delay in treatment. We analyze the model and find the equilibrium points and their stability. We also find the basic reproduction number R0 to understand the disease dynamics. For R0=1, we get transcritical backward bifurcation which means that the disease persists in the population even if R0<1. We performed the sensitivity analysis and found that treatment along with awareness plays a significant role in controlling infectious disease. We deduce that awareness about the disease affects the transmission rate of infection. As people become aware of aspects of the infectious disease, they amend their behavior so that they fend themselves from catching the disease. We have introduced a time lag in treatment and found the threshold value of the time delay. It is observed that when the value of the time delay crosses the threshold value, we get a Hopf bifurcation i.e. endemic steady state becomes unstable above the threshold value and it may become difficult to control the disease beyond the threshold value of delay in treatment.

Global stability for age-structured heroin model with general nonlinear incidence rate and time delay

In this article, we consider a mathematical model on the consumption of Heroin, it is an epidemiological model describing the interactions between heroin consumers and the rest of the population. The model is a system of differential equations with delay, integro-differential equations and partial differential equations. The first equation describes the evolution of healthy individuals, who do not consume Heroin. The second equation describes the evolution of consumer individuals, and in the last equation we find the evolution of individuals under treatment, so that they become healthy. In this work, we are interested in the qualitative study of the system considered. We examine the effects of a nonlinear incidence function with lag on the dynamics of an age-of-infection structured Heroin consumption model in the framework of mathematical epidemiology. We prove that incorporating the lag into the model does not change the asymptotic behavior of its equilibrium states. By defining the basic reproduction number R0 as a critical threshold parameter, we show that if R0 &lt; 1, the disease-free equilibrium (DFE) is globally stable regardless of the lag τ. Conversely, if R0 &lt; 1, the model exhibits an endemic equilibrium (EE) that remains locally and globally asymptotically stable for any lag τ. Our results highlight the robustness of the equilibrium stability in response to variations in the lag and provide important insights into the long-term dynamics of Heroin consumption, providing a clearer understanding of the conditions under which epidemic outcomes are stable or unstable.

Modeling and analysis of a two-strain immuno-epidemiological model with reinfection

In this paper, we formulate a two-strain model with reinfection that combines immunological and epidemiological dynamics across scales, using the COVID-19 pandemic as a case study. Firstly, we conduct a qualitative analysis of both within-host and between-host models. For the within-host model, we prove the existence and stability of equilibria, and Hopf bifurcation occurs from the infection equilibrium with immune response. This implies that, under specific immune states, the virus within an infected individual may persist, and its concentration may also oscillate periodically. For the between-host model, the disease-free equilibrium always exists and is locally asymptotically stable when the epidemiological basic reproduction number ℜ0<1. In addition, the model can have boundary equilibria of strain 1 or strain 2, which are locally asymptotically stable under specific conditions. However, the co-existence equilibrium does not exist. Secondly, to explore the infection and transmission mechanisms of two strain models and obtain reliable parameter values, we utilize statistical data to fit the immuno-epidemiological model. Simultaneously, we conduct an identifiability analysis of the immuno-epidemiological model to ensure the robustness of the fitted parameters. The results demonstrate the reliable estimation of parameter ranges for structurally unidentifiable parameters with minor measurement errors using the affine invariant ensemble Markov Chain Monte Carlo algorithm (GWMCMC). Moreover, simulations illustrate that enhancing treatment of patients infected with BA.2 strains to inhibit the number of viruses released by infected cells can significantly reduce disease spread.

Estimating the prevalence of terrorism under control policies. A statistical modelling approach

Measuring the intensity of the prevalence of radicalization plays a vital role in countering violent extremism. Hence, several mathematical models have been proposed aiming to describe the complex dynamics of this social phenomenon. It should be noted that existing literature mainly includes deterministic compartmental models, while the measuring of the extremism's intensity is based on the basic reproduction number R0. However, deterministic approaches can only provide an estimation of the mean tendency of the phenomenon, while the R0 index is accompanied by several drawbacks. In this paper, we propose a new compartmental scheme which considers control policies, containing susceptible, resistant, extremist and under-rehabilitation cases. The aforementioned model is combined with the methodology of continuous time Markov processes, leading to the quantification of not just the mean trend but the uncertainty of the phenomenon, too. Moreover, we introduce two new stochastic descriptors for the assessment of the radicalization's severity, namely the exact (Rε) and aggregate (Ra) reproduction numbers that address the limitation of the widely used R0 index. Emphasis is placed on the presentation of theorems and algorithms that describe the computation of their complete distribution functions. We derive formulas for the calculation of elasticities, and perform a perturbation analysis, shedding light on the overall impact of system's parameters on the selected stochastic descriptors. A series of illustrative examples is presented for the illustration of the derived theoretical properties, along with numerical comparisons between the three reproduction numbers. Finally, based on the proposed measures, we pinpoint the most impactful factors, and discuss control policies that facilitate the effective combating of the negative repercussions of terrorism.

Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with Examples

The fact that the famous basic reproduction number R0, i.e., the largest eigenvalue of the next generation matrix FV−1, sometimes has a probabilistic interpretation is not as well known as it deserves to be. It is well understood that half of this formula, −V, is a Markovian generating matrix of a continuous-time Markov chain (CTMC) modeling the evolution of one individual on the compartments. It has also been noted that the not well-enough-known rank-one formula for R0 of Arino et al. (2007) may be interpreted as an expected final reward of a CTMC, whose initial distribution is specified by the rank-one factorization of F. Here, we show that for a large class of ODE epidemic models introduced in Avram et al. (2023), besides the rank-one formula, we may also provide an integral renewal representation of R0 with respect to explicit “age kernels” a(t), which have a matrix exponential form.This latter formula may be also interpreted as an expected reward of a probabilistic continuous Markov chain (CTMC) model. Besides the rather extensively studied rank one case, we also provide an extension to a case with several susceptible classes.

Modeling hepatocyte apoptosis in chronic HCV infection with impulsive drug control

Chronic hepatitis C virus (HCV) infection is one of the leading life-threatening diseases owing to its non-cytopathic nature and scarcity of vaccines to combat. Hepatocyte apoptosis casts extensive impact in regulating chronic HCV infection. We propose a four dimensional deterministic model scheming the role of hepatocyte apoptosis and significance of HCV-specific CTL response in the intermediate stages of chronic HCV infection. Numerical simulation indicates that unregulated behavior of hepatocyte apoptosis triggers hepatocellular carcinoma and prolonged continuation of apoptosis induces liver damages. The system undergoes backward bifurcation and the value of the basic reproduction number R0=0.98<1 shows recurrence of infection. Local sensitivity analysis has been performed to reveal the most sensitive parameters responsible for the progression and mitigation of infection. In order to control of chronic HCV infection through pan-genotypic direct acting antiviral drugs (DAAs), impulsive differential equations are considered to study the effect of perfect drug adherence to both fixed and non-fixed dosing. For possible eradication of the chronic HCV infection, impulsive drug control suggests significant viral load reduction. The analytical results obtained for both non-impulsive and impulsive models are quantified biologically.

Schistosomiasis mathematical model in a spatially heterogeneous environment

Schistosomiasis is classified by WHO as a neglected tropical disease. Recent research works have shown that large-scale development projects involving massive population displacement and water irrigation, such as the construction of dams, lakes, and the development of agricultural areas, favour the proliferation of bilharzia. These observations motivate us to propose a reaction–diffusion model to assess the role of the displacements of humans, snails, cercaria, miracidia in the transmission dynamics of Schistosomiasis. The model incorporates a general non-linear contact functions and density-dependent parameters. The aim is to better understanding the role of spatial interactions on the spread of Schistosomiasis, in order to propose appropriate recommendations for the control of that silent threat. We characterize the basic reproduction number R0 of the model. The uniform persistence theory, the maximum principle are used to conduct an in-depth analysis of both the homogeneous and heterogeneous models. Theoretical results are illustrated through numerical simulations.

Global dynamics and threshold behavior of an SEIR epidemic model with nonlocal diffusion

This paper studies the global dynamics of an SEIR (Susceptible–Exposed–Infectious–Recovered) model with nonlocal diffusion. We show the model’s well-posedness, proving the solutions’ existence, uniqueness, and positivity, along with a disease-free equilibrium. Next, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number R0, defined as the spectral radius of the next-generation operator. We show that the solution map has a global compact attractor, offering insights into long-term dynamics. In particular, the analysis shows that for R0<1, the disease-free equilibrium is globally stable. Using the persistence theory, we show that there is an endemic equilibrium point for R0>1. Moreover, by constructing an appropriate Lyapunov function, we establish the global stability of the unique endemic equilibrium in two distinct scenarios.

Optimal vaccination strategies for imperfect vaccines and variable host susceptibility

I present a method to allocate a given number of vaccines to members of a population who differ in their susceptibility to the disease so that the final size of the epidemic is minimised. I consider an arbitrary distribution of protection that the vaccine confers, including the extreme cases of leaky and all-or-none vaccines. The optimal vaccination policy depends on the distribution of protection. While for low values of the basic reproduction number R0 the optimal policy prioritises the most susceptible hosts, I show that for almost any distribution the order of priority reverses and the least susceptible hosts should be vaccinated when R0 is high. The exception where this does not happen is the all-or-none vaccine. However, even a small deviation from the ideal all-or-none distribution can imply that the limited number of vaccines should be given to less susceptible hosts already at realistic values of R0.