Endemic Equilibrium Research Articles

Competitive networked bi-virus spread: Existence of coexistence equilibria

The paper studies multi-competitive continuous-time epidemic processes. We consider the setting where two viruses are simultaneously prevalent, and the spread occurs due to individual-to-individual interaction. In such a setting, an individual is either not affected by any of the viruses, or infected by one and exactly one of the two viruses. One of the equilibrium points is the coexistence equilibrium, i.e., multiple viruses simultaneously infect separate fractions of the population. We provide a sufficient condition for the existence of a coexistence equilibrium. We identify a condition such that for certain pairs of spread matrices either every coexistence equilibrium lies on a line that is locally exponentially attractive, or there is no coexistence equilibrium. We then provide a condition that, for certain pairs of spread matrices, rules out the possibility of the existence of a coexistence equilibrium, and, as a consequence, establishes global asymptotic convergence to the endemic equilibrium of the dominant virus. Finally, we provide a mitigation strategy that employs one virus to ensure that the other virus is eradicated. The theoretical results are illustrated using simulations.

Optimal control strategies for infectious diseases with consideration of behavioral dynamics

We present a simple SVIR (susceptible, vaccinated, infected, recovered) model to analyze the spread of COVID‐19, incorporating the influence of the population's caution on the transmission rate, which is considered nonlinear in current number of infected. Demonstrating a positive bound solution confirms the model's biological relevance. Through a formula for the basic reproduction number, we explore the local asymptotic stability of the disease‐free equilibrium (DFE) and endemic equilibrium (EE), showing that the existence of the EE relies on the basic reproduction number. Furthermore, we establish the global stability of the DFE by constructing a Lyapunov function. We present an optimal control problem for vaccination, demonstrating the existence and uniqueness of the optimal strategy. Our simulations indicate that optimal vaccination is effective in reducing infections and costs. We also investigate the effect of integrating education into the model to underscore its importance in decreasing disease transmission rates and reducing the necessity for vaccine uptake.

Global dynamic analyzes of the discrete SIS models with application to daily reported cases

Emerging infectious diseases, such as COVID-19, manifest in outbreaks of varying magnitudes. For large-scale epidemics, continuous models are often employed for forecasting, while discrete models are preferred for smaller outbreaks. We propose a discrete susceptible-infected-susceptible model that integrates interaction between parasitism and hosts, as well as saturation recovery mechanisms, and undertake a thorough theoretical and numerical exploration of this model. Theoretically, the model incorporating nonlinear recovery demonstrates complex behavior, including backward bifurcations and the coexistence of dual equilibria. And the sufficient conditions that guarantee the global asymptotic stability of a disease-free equilibrium have been obtained. Considering the challenges posed by saturation recovery in theoretical analysis, we then consider the case of linear recovery. Bifurcation analysis for of the linear recovery model displays a variety of bifurcations at the endemic equilibrium, such as transcritical, flip, and Neimark–Sacker bifurcations. Numerical simulations reveal complex dynamic behavior, including backward and fold bifurcations, periodic windows, period-doubling cascades, and multistability. Moreover, the proposed model could be used to fit the daily COVID-19 reported cases for various regions, not only revealing the significant advantages of discrete models in fitting, evaluating, and predicting small-scale epidemics, but also playing an important role in evaluating the effectiveness of prevention and control strategies. Furthermore, sensitivity analyses for the key parameters underscore their significant impact on the effective reproduction number during the initial months of an outbreak, advocating for better medical resource allocation and the enforcement of social distancing measures to curb disease transmission.

Investigation and Analysis on the Interactions Between Mathematics Literacy Skills Using Mathematical Modeling

This study investigated and analyzed the interactions between mathematics literacy skills using mathematical modeling. The study used an ASMD model to represent the population of individuals who have skills on addition (A), subtraction (S), multiplication (M) and division (D). The objectives achieved were that the recruitment parameter and coefficient of leaving any compartments significantly influence the system based on the free-equilibrium analysis of mathematics literacy skills. The study showed that system has direct and indirect dynamics in the three states: subtraction, multiplication and division. It also revealed that addition skill is easier to learn than others. Subtraction and multiplication do not interact and have no inter coefficient. The system showcased that good number of individuals are not efficient and effective in the utilization of these skills from the endemic equilibrium of the model showed that E<sub>n</sub>>1 (asymptotically unstable). Finally, this study discovered that these elementary skills in mathematics is fundamental to learners, educated and uneducated, support continued inclusive, workable economic growth, full creative employment, decent work and improve academic performance for all at all levels and in the world at large. The study recommends that curriculum planners should give more time to study these skills thoroughly say one academic session, discourage the use of calculator at early stage, student and teacher factors should be taken into considerations and so on.

Stability and Hopf Bifurcation of a Cytokine-Enhanced HIV Infection Model with Saturation Incidence and Three Delays

To discuss the influence of cytokine-enhanced rate and antibody delay on viral infection, in this paper, we propose a cytokine-enhanced HIV model with saturation incidence, antibody immune response and time delays. Multiple time delays, namely intracellular delay, virus replication delay and antibody delay, are included. First, we prove that the model is well-posed. Then, we obtain two basic reproductive numbers of the model. Next, the stability of the equilibria is investigated by using Lyapunov functions and LaSalle’s invariance principle. The results show that intracellular delay and virus replication delay do not impact on the stability of the three equilibria. However, if antibody immune response delay is positive, we determine some conditions for stability switches of the endemic equilibrium by using antibody immune response delay as a bifurcation parameter. It is concluded that with the increase of antibody immune response delay, the endemic equilibrium loses its stability and the system generates Hopf bifurcation. Finally, the numerical simulation results also show the correctness of the theoretical results.

Analysis of SIS epidemic model in bi-uniform hypernetworks

To describe the dynamics of epidemic spread with multiple individuals interacting with each other, we develop a susceptible-infected-susceptible (SIS) spread model with collective and individual contagion in general hypernetworks with higher-order interactions. The constructed model is applied to a bi-uniform hypernetwork to obtain a mean-field model for the SIS model. The threshold value at which an epidemic can spread in the bi-uniform hypernetwork is obtained and analyzed dynamically. By analysis, the model leads to bistability, in which a disease-free equilibrium and an endemic equilibrium coexist. Finally, numerical simulations of the developed model are carried out to give the effect of the proportion of individual contagion hyperedges on the spread of an epidemic.

Mathematical Model for Describing the Transmission of Animal African Trypanosomiasis in Cattle

In this paper, we have formulated a mathematical model for the transmission dynamics of African Animal Trypanosomiasis (AAT) by incorporating spraying of the tsetse y population(vector) and treating the cattle population (host) as control strategies. It has been shown that the disease free equilibrium point is globally asymptotically stable and the endemic equilibrium point is locally asymptotically stable. We have also shown that treatment of the host population reduces significantly infection by AAT as compared to spraying of the vector population.

Analysis and Simulation of SEIR Mathematical Model of Stunting Case in Indonesia

Stunting is one of the focuses of the target for improving nutrition in the world until 2025. The government of Indonesia targets the prevalence of stunting to decrease to 14% in 2024. There are some factor affect stunting, some of them are sanitation, health care and parenting of child. In this research, we constructed a SEIR mathematical model of stunting case in Indonesia. There are four compartements in this model : the population of newborn are likely to be exposed to stunting (S), the population of children were having an early symptom stunting (E), the population of affected child stunting permanent (I) and the population of children showing symptom stunting but no caught stunting (free stunting) (R). The disease free and endemic equilibrium point are the equilibrium of the formed model. The local stability of equilibrium points have been analyzed, determined the basic reproduction number, and conducted a sensitivity analysis. If then is asymptotically stable If then is asymptotically stable. Sensitivity analysis revealed that the parameters and significantly contribute to the increase in the basic reproduction number . It means that the good sanitation, good healthcare, good parenting of child and decrease the rate of transmission indirectly from the person affected stunting are the most important thing to reduce the rate of stunting in Indonesia.

Evaluating the Risk of Malaria Transmission within the Central African Republic with the Goal of Stabilising and Eliminating the Infection

Abstract In this paper, we formulate a multi compartmental mathematical model for human and mosquito. We construct the system of differential equations for an SEITVR for human compartment and SEI for mosquito compartment. We investigate the outbreak of malaria and effect on Central African Republic. The analysis of the compartmental model is carried out using stability analysis. Our model exhibits two equilibrium points, disease free equilibrium points and endemic equilibrium points. The next generation matrix is used to determine the basic reproduction number R0. Numerical example are provided to validate our results for both the disease free state and endemic state of each model. We believe that this investigation will be more effective on reducing malaria infection and stop spreading.

Investigation of sliding mode dynamics and near-optimal controls for a reaction–diffusion population model in a polluted environment

In this paper, we present a reaction–diffusion population model developed for a polluted environment, and investigate the impact of different control measures on population-toxicant dynamics. Toxicant concentrations is taken as the reference to determine whether to implement control measures, and sliding mode dynamics are studied. The dynamical behaviors of each subsystem are discussed, and results show that model solutions ultimately approach either two endemic equilibria or the sliding equilibrium on a surface of discontinuity. Furthermore, we formulate the near-optimal control problem by minimizing the control cost while reducing the toxicant concentration. The sufficient and necessary conditions for the near-optimality are established using Ekeland’s variational principle and Hamiltonian function. The theoretical results are illustrated by numerical simulations.

Mathematical Model for Mycobacterium Tuberculosis

To demonstrate the dynamics of the Mycobacterium tuberculosis disease population, a mathematical model was presented. The model has five compartments, and the resulting equations were resolved. While multiple cases of illness transmission were simulated using the compartmental model of infectious disease spread for a structured population model, the fundamental reproduction number was found using the next-generation matrix. Based on the results of the simulations, the system’s disease-free and endemic equilibrium was created by presenting, analyzing, and graphing the various subpopulations across time. The Homotopy Perturbation Method (HPM) analytical technique was then used to resolve the model.

Optimal control with the effects of ivermectin and live stock availability on malaria transmission

Malaria remains a global threat and the conventional methods used for combating the disease leave out mosquitoes that feed outdoors. This study addresses the challenge posed by such mosquitoes based on a tool called ivermectin drug which is lethal to mosquitoes that ingest bloodmeal containing a concentration of it. We formulated a mathematical model with three control tools (insecticide treated nets, treatment of infective individuals and ivermectin drug on livestock and humans) for the transmission and control of malaria under optimal condition. The model’s basic reproduction number, R0 was estimated and the local and global stability analyses of the disease-free and endemic equilibrium points of the model were carried out. Sensitivity analysis carried out showed that R0 is most sensitive to the mosquito biting rate and to the proportion of blood meal on human with cattle availability in such a way that any percent increase in the value of any of these parameters will lead to an equal percent increase in the value of R0. The result of an optimal control analysis based on three time dependent controls suggests that the combination of all three controls gives the best result followed by the strategy that combines the use of ivermectin drug and the treatment of infective human. Depending on available resources, any of these is recommended to be adopted in malaria intervention programmes because of their effectiveness on both the infective human and mosquito populations with the potential of contributing significantly to the disease elimination within a minimal time frame.

Stability analysis of a [formula omitted] epidemic compartmental model with saturated incidence rate, vaccination and elimination strategies

In this study, we developed and analyzed a compartmental epidemic model and a system model that incorporate vaccination, elimination, and quarantine techniques, with a saturated incidence rate, by theoretical and numerical means. The model is described by a system of four nonlinear differential equations. Our analysis included determining the reproduction number Rq and examining equilibrium solutions. The outcomes of the disease are identified through the threshold Rq. When Rq<1, the disease-free equilibrium is globally asymptotically stable, as proved by the LaSalle invariance principle and disease extinction analysis. However, using the Routh–Hurwitz criterion, we have proved that the disease-free equilibrium is not stable, and when Rq>1, the unique endemic equilibrium is asymptotically stable locally. The stability of the endemic and disease-free equilibria globally has been examined using the Routh–Hurwitz and Dulac criteria. Subsequently, numerical simulations were utilized to illustrate the theoretical findings effectively.

Dynamics and control of typhoid fever in Sheno town, Ethiopia: A comprehensive nonlinear model for transmission analysis and effective intervention strategies.

This study presents a reliable mathematical model to explain the spread of typhoid fever, covering stages of susceptibility, infection, carrying, and recovery, specifically in the Sheno town community. A detailed analysis is done to ensure the solutions are positive, stay within certain limits, and are stable for both situations where the disease is absent and where it is consistently present. The Routh-Hurwitz stability criterion has been used and applied for the purpose of stability analysis. Using the next-generation matrix, we determined the intrinsic potential for disease transmission. It showing that typhoid fever is spreading actively in Sheno town, with cases above a critical level. Our findings reveal the instability of the disease-free equilibrium point alongside the stability of the endemic equilibrium point. We identified two pivotal factors for transmission of the disease: the infectious rate, representing the speed of disease transmission, and the recruitment rate, indicating the rate at which new individuals enter the susceptible population. These parameters are indispensable for devising effective control measures. It is imperative to keep these parameters below specific thresholds to maintain a basic reproduction number favorable for disease control. Additionally, the study carefully examines how different factors affect the spread of typhoid fever, giving a detailed understanding of its dynamics. At the end, this study provides valuable insights and specific strategies for managing the disease in the Sheno town community.

Epidemiological Modelling of Yellow Fever Dynamics

Aims: Yellow fever is a severe and often fatal viral illness caused by the yellow fever virus Despite being largely overlooked, yellow fever continues to silently claim lives in many parts of the world. The study focuses on the epidemiological modelling of yellow fever dynamics between a host (human) and vector (mosquito) populations The human population was divided into five main compartments: Susceptible, Exposed, Infected, Isolated, and Recovered. The vector population was also divided into two compartments: Susceptible and Infected. Nonlinear differential equations describing these compartments were formulated. Stability analysis and numerical simulations were then performed based on the formulated equations. From the stability analysis, it was observed that the disease-free equilibrium is both locally and globally asymptotically stable. Similarly, the endemic equilibrium was found to be locally and globally asymptotically stable. The simulation also revealed a direct correlation between the transmission rate and disease spread.

Machine learning investigation of tuberculosis with medicine immunity impact

Tuberculosis (T.B.) remains a prominent global cause of health challenges and death, exacerbated by drug-resistant strains such as multidrug-resistant tuberculosis MDR-TB and extensively drug-resistant tuberculosis XDR-TB. For an effective disease management strategy, it is crucial to understand the dynamics of T.B. infection and the impacts of treatment. In the present article, we employ AI-based machine learning techniques to investigate the immunity impact of medications. SEIPR epidemiological model is incorporated with MDR-TB for compartments susceptible to disease, exposed to risk, infected ones, preventive or resistant to initial treatment, and recovered or healed population. These masses' natural trends, effects, and interactions are formulated and described in the present study. Computations and stability analysis are conducted upon endemic and disease-free equilibria in the present model for their global scenario. Both numerical and AI-based nonlinear autoregressive exogenous NARX analyses are presented with incorporating immediate treatment and delay in treatment. This study shows that the active patients and MDR-TB, both strains, exist because of the absence of permanent immunity to T.B. Furthermore, patients who have recovered from tuberculosis may become susceptible again by losing their immunity and contributing to transmission again. This article aims to identify patterns and predictors of treatment success. The findings from this research can contribute to developing more effective tuberculosis interventions.

Effect of homophily on coupled behavior-disease dynamics near a tipping point

Understanding the interplay between social activities and disease dynamics is crucial for effective public health interventions. Recent studies using coupled behavior-disease models assumed homogeneous populations. However, heterogeneity in population, such as different social groups, cannot be ignored. In this study, we divided the population into social media users and non-users, and investigated the impact of homophily (the tendency for individuals to associate with others similar to themselves) and online events on disease dynamics. Our results reveal that homophily hinders the adoption of vaccinating strategies, hastening the approach to a tipping point after which the population converges to an endemic equilibrium with no vaccine uptake. Furthermore, we find that online events can significantly influence disease dynamics, with early discussions on social media platforms serving as an early warning signal of potential disease outbreaks. Our model provides insights into the mechanisms underlying these phenomena and underscores the importance of considering homophily in disease modeling and public health strategies.

SUSCEPTIBLE VACCINATED INFECTED RECOVERED SUSCEPTIBLE MODEL: EQUILIBRIA POINTS AND APPLICATION ON COVID-19 CASE DATA IN INDONESIA

Severe Respiratory Syndrome Coronavirus-2 is the infectious agent that causes COVID-19. A vaccine program is an effort to stop the spread of COVID-19 infections in Indonesia. The susceptible vaccinated infected recovered susceptible (SVIRS) model can be used to represent the spread of infectious diseases. This study aims to construct the SVIRS model, identify the equilibria points thus apply it to COVID-19 case data in Indonesia, and determine transmission patterns, model accuracy, and interpretation. Literature and applications are the research methodologies employed. First-order nonlinear differential equations form the obtained SVIRS model. The model has two equilibrium points: a disease-free equilibrium point, and the other is endemic equilibrium point. The SVIRS model on the spread of COVID-19 in Indonesia was obtained using daily secondary data from January 11 to November 30, 2022. The model is solved by the fourth-order Runge-Kutta method. The model’s accuracy is accurate enough to explain the spread of COVID-19 in Indonesia with a mean average percentage error (MAPE) value of 43%. According to the transmission pattern, the number of COVID-19 cases in Indonesia peaked on July 27, 2022, then decreased to zero, obtaining an equilibrium point when no more cases of the disease were present.

Untangling the memory and inhibitory effects on SIS-epidemic model with Beddington–DeAngelis infection rate

The dynamical behaviors of an epidemic model based on the susceptible–infected–susceptible (SIS) model are investigated. The Beddington–DeAngelis functional response is used for the infection rate to present the dependence of the transmission of the infection on the ratio of both susceptible and infected populations. A Caputo fractional derivative is applied to show the existence of memory in nature affects population dynamics. The disease-free and endemic equilibrium points are obtained as the equilibrium points that describe the condition when the population is free from the disease or the disease exists in the population throughout time. The existence, uniqueness, non-negativity, and boundedness are proven which state the biological validity of the mathematical model. The local and global stability of each equilibrium point is studied including the basic reproduction number and its influence on the dynamical behaviors. Some numerical simulations are portrayed to explore more about the dynamics of the model which are relevant to the analytical findings. The partial rank correlation coefficient is presented to investigate the dominant parameter with respect to the basic reproduction number and the density of susceptible and infected populations. The parameter continuations are demonstrated to show the impact of infection rate and the inhibitory effect which lead to the occurrence of forward bifurcations. The memory effect is also demonstrated numerically to show the changes in convergence rate due to the changes in memory strength.

Unraveling the importance of early awareness strategy on the dynamics of drug addiction using mathematical modeling approach.

A drug is any substance capable of altering the functioning of a person's body and mind. In this paper, a deterministic nonlinear model was adapted to investigate the behavior of drug abuse and addiction that incorporates intervention in the form of awareness and rehabilitation. In the mathematical analysis part, the positivity and boundedness of the solution and the existence of drug equilibria have been ascertained, which shows that the model consists of two equilibria: a drug-free equilibrium and a drug endemic equilibrium point. The drug-free equilibrium was found to be both globally and locally asymptotically stable if the effective reproduction number is less than or equal to one (Rc≤1). Furthermore, we were able to show the existence of a unique drug endemic equilibrium whenever Rc>1. Global asymptotic stability of a drug endemic equilibrium point has been ascertained using a nonlinear Lyapunov function of Go-Volterra type, which reveals that the drug endemic equilibrium point is globally asymptotically stable if an effective reproduction number is greater than unity and if there is an absence of a reversion rate of mended individuals (i.e., ω=0). In addition, an optimal control problem was formulated to investigate the optimal strategy for curtailing the spread of the behavior using control variables. The control variables are massive awareness and rehabilitation intervention of both public and secret addicted individuals. The optimal control simulation shows that massive awareness control is the best to control drug addiction in a society. In sensitivity analysis section, the proportion of those who are exposed publicly shows to be a must sensitive parameter that can reduce the reproduction number, and the effective contact rate shows to be a must sensitive parameter to increase the reproduction number. Numerical simulations reveal that the awareness rate of exposed publicly and the rehabilitation rate of addicted publicly are very important parameters to control drug addiction in a society.