Disease-free Equilibrium Research Articles

Ebola virus disease model with a nonlinear incidence rate and density-dependent treatment

This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages.

SIHQR model with time delay for worm spread analysis in IIoT-enabled PLC network

Industrial Internet of Things (IIoT) is a product of deep integration between Internet of Things (IoT) and industrial control networks. As an important component of IIoT, Programmable Logic Controller (PLC) has become a springboard for many hackers to disrupt industrial networks by spreading viruses. It is known that worm can cause enormous loss. Therefore, revealing the rules of worm spread in PLC networks and exploring methods to inhibit worm spread are increasingly important. To this end, we propose a new worm spread model named SIHQR with time delay. Additionally, we establish a game model between susceptible and infected nodes in order to express the infection rate by several game parameters. Through analysis, we discuss the conditions for the emergence of three scenarios: disease-free equilibrium, endemic equilibrium and Hopf bifurcation. We study the stability at the equilibrium points and obtain an expression for a threshold value which determines whether the system exhibits endemic equilibrium or Hopf bifurcation. In the experimental section, we analyze the impact of initial values of node states and the magnitude of game parameters on worm spread in the three aforementioned scenarios. Finally, we propose a method to mitigate or even eliminate the Hopf bifurcation. The experimental results show that the proposed model has welldone performance.

Modeling the Inflow of Exposed and Infected Migrants on the Dynamics of Malaria

Malaria is currently a life-threatening vector borne disease which is endemic in most of the developing and underdeveloped countries associated with poor health care systems. In this study, a host-vector mathematical model that takes into account the inflow of human migrants who have been exposed or infected with malaria is formulated and analysed. The reproduction number of the mosquito vector population is derived and used as a threshold quantity for determining the existence of the model trivial and realistic steady states. The Routh-Hurwitz criterion and some stability theorems of Metzler matrices are used to show that the realistic disease free equilibrium is both locally and globally asymptotically stable whenever the disease reproductive number is less than one. We derived an equation for the model endemic condition and used Descartes Rule of Sign Change to established the conditions for the model to admit one or three endemic equilibrium state(s). It is further shown that in the absence of inflow of exposed or infected migrants, the model admits a globally asymptotically unique endemic equilibrium when R0>1 and two endemic equilibria when R0<1. Our local sensitivity analysis revealed that the adults mosquito removal and biting rates were respectively the most significant contributing parameters to the spread of malaria. The numerical simulations results suggested that the exposed and infected immigrants have no significant impact on the dynamical behaviour of the model population sub-classes.

Nonlinear dynamics and optimal control strategies of a novel fractional-order lumpy skin disease model

Fractional order differential equations are often employed to better and accurately model several natural and physical processes with memory characteristics than the integer-order models. In this work, a proposed fractional-order model is presented to simulate one of the most dangerous viral cattle diseases. The present fractional-order Lumpy Skin Disease model considers the interactions between different compartments of cattle and virus vectors. The solution of the model is examined in terms of existence, uniqueness, positivity and boundedness. Then, the stability analysis of the different equilibrium points in the model is carried out. The basic reproduction number, R0, has been obtained too. The dependence of stability regions on the values of key parameters is demonstrated and the sensitivity of R0 to different parameters in the system is also investigated. Finally, a proposed optimal control scheme is adopted to successfully diminish the disease spread. It is found that the stability of disease-free equilibrium point is verified at small values of the infection rate of cattle caused by infected cattle or insects. In addition, the disease-free equilibrium point maintains its stability in cases where the recovery rate of cattle or the vaccination rate of cattle are increased. Moreover, the high-efficiency vaccination program helps in stabilizing the disease-free equilibrium point. Also, the fractional-order parameter is found to provide additional degree of freedom in the model to represent different scenarios of evolution for solution orbits. The optimal control parameters of the model are examined in different scenarios to limit or suppress the spread of the disease. It is demonstrated that two control parameters are sufficient to achieve the goals of disease elimination and stabilization of an infection-free steady state. Numerical simulations are verified to confirm our theoretical findings and estimations.

Modeling the transmission dynamics of a two-strain dengue disease with infection age

In this paper, we introduce a partial differential equation (PDE) model to describe the transmission dynamics of dengue with two viral strains and possible secondary infection for humans. The model features the variable infectiousness during the infectious period, which we call the infection age of the infectious host. We define two thresholds [Formula: see text] and [Formula: see text], and show that the strain [Formula: see text] can not invade the system if [Formula: see text]. Further, the disease-free equilibrium of the system is globally asymptotically stable if [Formula: see text]. When [Formula: see text], strain [Formula: see text] dominance equilibrium [Formula: see text] exists, and is locally asymptotically stable when [Formula: see text], [Formula: see text], [Formula: see text]. Then, by applying Lyapunov–LaSalle techniques, we establish the global asymptotical stability of the dominance equilibrium corresponding to some strain [Formula: see text]. This implies strain [Formula: see text] eliminates the other strain as long as [Formula: see text], where [Formula: see text] denotes the probability of a given susceptible mosquito being transmitted by a primarily infected human with strain [Formula: see text]. Finally, we study the existence of the coexistence equilibria under some conditions.

Analysis of COVID-19 Disease Model: Backward Bifurcation and Impact of Pharmaceutical and Nonpharmaceutical Interventions

The SEIQHR model, introduced in this study, serves as a valuable tool for anticipating the emergence of various infectious diseases, such as COVID-19 and illnesses transmitted by insects. An analysis of the model’s qualitative features was conducted, encompassing the computation of the fundamental reproduction number, R0. It was observed that the disease-free equilibrium point remains singular and locally asymptotically stable when R0<1, while the endemic equilibrium point exhibits uniqueness when R0>1. Additionally, specific conditions were outlined to guarantee the local asymptotic stability of both equilibrium points. Employing numerical simulations, the graphical representation illustrated the influence of model parameters on disease dynamics and the potential for its eradication across different noninteger orders of the Caputo derivative. In essence, the adoption of a fractional epidemic model contributes to a deeper comprehension and enhanced biological insights into the dynamics of diseases.

A novel simulation-based analysis of a stochastic HIV model with the time delay using high order spectral collocation technique

The economic impact of Human Immunodeficiency Virus (HIV) goes beyond individual levels and it has a significant influence on communities and nations worldwide. Studying the transmission patterns in HIV dynamics is crucial for understanding the tracking behavior and informing policymakers about the possible control of this viral infection. Various approaches have been adopted to explore how the virus interacts with the immune system. Models involving differential equations with delays have become prevalent across various scientific and technical domains over the past few decades. In this study, we present a novel mathematical model comprising a system of delay differential equations to describe the dynamics of intramural HIV infection. The model characterizes three distinct cell sub-populations and the HIV virus. By incorporating time delay between the viral entry into target cells and the subsequent production of new virions, our model provides a comprehensive understanding of the infection process. Our study focuses on investigating the stability of two crucial equilibrium states the infection-free and endemic equilibriums. To analyze the infection-free equilibrium, we utilize the LaSalle invariance principle. Further, we prove that if reproduction is less than unity, the disease free equilibrium is locally and globally asymptotically stable. To ensure numerical accuracy and preservation of essential properties from the continuous mathematical model, we use a spectral scheme having a higher-order accuracy. This scheme effectively captures the underlying dynamics and enables efficient numerical simulations.

Optimal treatment and stochastic stability on a fractional-order epidemic model incorporating media awareness

Infectious diseases have become a severe concern in our society in recent times. In this respect, conducting a qualitative investigation of infectious disease systems is very crucial. Nowadays, developing a mathematical model and its analysis for disease control has been very popular. Our present study has formulated a SIR-type epidemic model by considering a saturated incidence function with media impact. To account for the influence of memory, we have modified the model by employing a system of fractional-order differential equations of Caputo’s notion. We have determined all the system equilibrium points and discussed their stability criteria based on the basic reproduction number (R0) as the threshold parameter. It is observed that for the threshold R0<1, disease-free equilibrium becomes both locally and globally asymptotically stable. A transcritical bifurcation occurs at the threshold R0=1. Further, as the threshold R0 exceeds unity, the endemic equilibrium becomes asymptotically stable. A fractional-order optimal control problem with treatment as the control parameter is developed to reduce the impact of disease. Using Pontryagin’s Maximum Principle yields a theoretical and numerical solution to the fractional order optimal control problem. It is observed that both the media and memory effect dramatically reduces infection rates. Considering environmental variations, the stochastic stability of the endemic equilibrium point has also been studied. Sensitivity analysis is carried out to comprehend how the parameters impact the threshold value (R0). Finally, extensive numerical simulations are performed to demonstrate our theoretical examination.

Global analysis of an age-structured tuberculosis model with an application to Jiangsu, China.

Diagnostic delay for TB infected individuals and the lack of TB vaccines for adults are the main challenges to achieve the goals of WHO by 2050. In order to evaluate the impacts of diagnostic delay and vaccination for adults on prevalence of TB, we propose an age-structured model with latent age and infection age, and we incorporate Mycobacterium TB in the environment and vaccination into the model. Diagnostic delay is indicated by the age of infection before receiving treatment. The threshold dynamics are established in terms of the basic reproduction number . When , the disease-free equilibrium is globally asymptotically stable, which means that TB epidemic will die out; When , the disease-free equilibrium is globally attractive; there exists a unique endemic equilibrium and the endemic equilibrium is globally attractive when . We estimate that the basic reproduction number (95% CI (0.3060, 0.7556)) in Jiangsu Province, which means that TB epidemic will die out. However, we find that the annual number of new TB cases by 2050 is 1,151 (95%CI: (138, 8,014)), which means that it is challenging to achieve the goal of WHO by 2050. To this end, we evaluate the possibility of achieving the goals of WHO if we start vaccinating adults and reduce diagnostic delay in 2025. Our results demonstrate that when the diagnostic delay is reduced from longer than four months to four months, or 20% adults are vaccinated, the goal of WHO in 2050 can be achieved, and 73,137 (95%CI: (23,906, 234,086)) and 54,828 (95%CI: (15,811, 206,468)) individuals will be prevented from being infected from 2025 to 2050, respectively. The modeling approaches and simulation results used in this work can help policymakers design control measures to reduce the prevalence of TB.

SIR FRACTIONAL ORDER OF COVID-19 by ADAMS BASHFORTH-MOULTON METHOD

This study addresses a research gap by introducing fractional order derivatives into the SIR model for tracking COVID-19 in Malaysia. The Caputo sense fractional derivative and the Adams Bashforth Moulton method are employed to analyse the COVID-19 behavior and stability. By manipulating fractional order derivative values, this study investigates their impact on key SIR parameters, observing that lower values accelerate the attainment of asymptotic behavior in populations. The stability analysis reveals two equilibrium points: an unstable disease-free equilibrium and a stable endemic equilibrium within the system. This pioneering exploration of fractional order derivatives in the context of Malaysia's COVID-19 modeling contributes valuable insights, enhancing our understanding the behavior of the disease.

Delay induced stability switch in a mathematical model of CD8 T-cell response to SARS-CoV-2 mediated by receptor ACE2.

The pathogen SARS-CoV-2 binds to the receptor angiotensin-converting enzyme 2 (ACE2) of the target cells and then replicates itself through the host, eventually releasing free virus particles. After infection, the CD8 T-cell response is triggered and appears to play a critical role in the defense against virus infections. Infected cells and their activated CD8 T-cells can cause tissue damage. Here, we established a mathematical model of within-host SARS-CoV-2 infection that incorporates the receptor ACE2, the CD8 T-cell response, and the damaged tissues. According to this model, we can get the basic reproduction number R0 and the immune reproduction number R1. We provide the theoretical proof for the stability of the disease-free equilibrium, immune-inactivated equilibrium, and immune-activated equilibrium. Finally, our numerical simulations show that the time delay in CD8 T-cell production can induce complex dynamics such as stability switching. These results provide insights into the mechanisms of SARS-CoV-2 infection and may help in the development of effective drugs against COVID-19.

ANALYSIS OF MEASLES DISEASE IN INDIVIDUAL USING BASIC SIR MODEL

In this paper, we examine the measles pandemic using basic SIR model. We investigate endemic equilibrium points, disease-free equilibrium points, reproduction number as well as basic reproduction numbers in order to examine the effects of various partition transitions. It is proposed that the measles model is both locally and globally asymptotically stable at the disease-free equilibrium point. Additionally we have analysis the spreading of disease inside human body and its recovery time. The illustration is carried out by Euler method

Understanding the spread of typhoid fever: Combining vaccination and sanitation methods for better public health policies

Typhoid fever remains a major public health concern, affecting over nine million individuals globally each year. Mathematical modeling approaches can provide valuable insights into typhoid transmission dynamics and inform preventive strategies. In this study, we developed a compartmental model incorporating key features of typhoid epidemiology and two crucial interventions: vaccination and sanitation practices. The model stratifies the population into susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered compartments and tracks the bacterial load in the environment. We established a disease-free equilibrium and basic reproduction number R0. We also identified the endemic equilibrium and analyzed its existence. Numerical simulations demonstrated the critical impact of enhanced sanitation and vaccination in curtailing infections. Our model underscores the need for multifaceted control measures that encompass vaccine coverage, sanitation enforcement, and healthcare capacity building to mitigate typhoid in high-risk regions. This study provides a comprehensive framework to model the intricate transmission dynamics of typhoid fever, supporting informed public health policies and decision-making.

A Study of a Class Continuous SIR Epidemic Model with History

The SIR model, an epidemiological model, divides a population into three compartments: Susceptible (S), Infected (I), and Recovered (R). It is widely used to understand the spread of infectious diseases and predict epidemic outcomes based on factors such as transmission rates and population dynamics. A deterministic epidemic mathematical model to describe the transmission dynamics of an infectious disease was constructed and analyzed by incorporating memory term which provides information on the current and past disease states. The model revealed two key equilibria: a disease-free equilibrium and an endemic equilibrium. The calculated basic reproductive number R0, was employed to establish that when R0&lt;1, the disease-free equilibrium is locally asymptotically stable, while the endemic equilibrium is locally asymptotically stable when R0&gt;1. Additionally, we explored the global stability of these equilibria using Lyapunov functions and Dulac's method, respectively. To validate our analytical findings, we conducted numerical simulations of the model which show the importance of history in the dynamic spread and elimination of 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&lt;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.

A study of growth of COVID-19 with super-spreaders using the modified SIR model including iceberg phenomenon

This paper explores the dynamics of COVID-19 transmission, particularly focusing on super-spreaders, through the lens of the SIR model. The model comprises six compartments representing susceptible, exposed, symptomatic infected, super-spreader, asymptomatic infected, and recovered individuals. Utilizing a set of non-linear, interdependent differential equations, we numerically solve for model parameters to examine the influence of super-spreaders on infection spread within the population. We calculate the basic reproduction number (R0) and discuss the stability of disease-free equilibrium. Our findings underscore the significant role played by super-spreaders and asymptomatic individuals in disease dissemination. Drawing on the epidemiological concept of the iceberg phenomenon, we offer insights into super-spreader events (SSEs) in India and their ramifications.

A node-based SIRS epidemic model on two-layer interconnected networks: Dynamical analysis of interplay between layers

From the micro-level perspective of individuals, this paper proposes a node-based SIRS epidemic model on two-layer networks, and investigates the stability of the equilibrium in the model as well as the influence of network structures and infected or recovered rate on disease propagation. By using the stability theory, we give the accurate sufficient condition for the stability as well as the threshold of disease prevalence, showing the threshold of such two-layer network is less than that of isolated single-layer network. Also, an easy-to-use approximate criterion for stability of disease-free equilibrium is provided for the two-layer network with general intra-layer structures when the infected rate across layers is far less than that within layers. Further, extensive numerical simulations on several classical network structure models verify the theoretical results, and show some new interesting results: (1) for the duplex network with equivalent number of connections each layer, the infected rate across layers has more significant effect on epidemic spreading than that within layers; (2) when fixing the whole network edge density, the same number of connections each layer always be the most conductive to propagate of the disease for the duplex network of ER structures, but it is the most/worst conductive to propagate at large/small infected rate for the duplex network of small-world structures.

A Co-infection Model for Monkeypox and HIV/AIDS: Sensitivity and Bifurcation Analyses

Monkeypox can make people very sick. The skin becomes infected with bacteria, thus causing severe skin damage. This can lead to corneal infection with loss of vision, pneumonia, difficulty swallowing, diarrhoea and vomiting leading to harsh malnutrition or dehydration, several organs inflammation or death. HIV-AIDS is a life-threatening and chronic condition. In HIV-infected individuals whose immune systems have been compromised, monkeypox mortality alone may be much higher. The co-infection of monkeypox and HIV/AIDS infections has been studied from a mathematical perspective by constructing a 13-compartment deterministic model. Basic mathematical analyses were performed on the co-infection model and the sub-models. The disease equilibrium points, the non-negativity of solutions, the basic reproduction numbers, the invariant region and the stability patterns. When the basic reproduction number is less than unity, the disease-free equilibrium points of each sub-model are globally asymptomatically stable. Certain calculations were done using the maple 18 programming language. The sensitivity analysis reveals that the parameters of the basic reproduction of the monkeypox sub-model with positive sensitivity indices are the probability of catching the monkeypox virus, the rate of effective contact, the compartment Im ’s coefficient of infection and the monkeypox vaccine’s waning rate, while the parameters of the basic reproduction of the HIV/AIDS sub-model with positive sensitivity indices are the probability of catching HIV virus, the rate effective contact, the compartment Ih ’s coefficient of infection and the compartment Ah ’s coefficient of infection. Via the centre manifold theorem, the bifurcation analysis reveals a forward bifurcation pattern for the monkeypox sub-model and the HIV/AIDS sub-model, and under a certain condition, a critical value of the monkeypox basic reproduction exists such that an effective management and possible elimination of the monkeypox infection would require that the monkeypox basic reproduction number should be kept below unity and above the critical value.

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

Mathematical Model for the Dynamics of Paragonimiasis Disease in Human, Snail, and Crustacean Populations

This paper proposes a mathematical model for the dynamics of Paragonimiasis transmission in human, snail, and crustacean populations. We proved that the solutions to the differential equations are both positive and bounded. In addition, we calculated the equilibrium states of Paragonimiasis in both its free and present forms. We computed the basic reproduction number using the next-generation matrix. Furthermore, we used the Routh-Hurwitz criteria to determine local stability, which revealed that the system is locally asymptotically stable (LAS). Similarly, we assessed the global stability of the disease-free equilibrium and found that the system is globally asymptotically stable (GAS). To better understand the dynamics of Paragonimiasis transmission, we carried out sensitivity index analyses and computational simulations. Our data show varied indices of rise and decline as parameters are changed. Asymptptically stable, Dynamics, Endemic equilibrium, Reproduction number, Transmission