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With the advancement of technology, social media has become an integral part of people's daily lives. This has resulted in the emergence of a new group of individuals known as "professional operation people". These individuals actively engage with social media platforms, taking on roles as content creators, influencers, or professionals utilizing social media for marketing and networking purposes. Therefore, in this article, we designed a six-dimensional fractional-order social media addiction model (FOSMA) in the sense of Caputo, which took into account the professional operations population. Initially, we established the positivity and boundedness of the FOSMA model. After that, the basic regeneration number and the equilibrium points (no addiction equilibrium point and addiction equilibrium point) were computed. Then, the local asymptotic stability of the equilibrium points were proved. In order to investigate the bifurcation behavior of the model when R 0 = 1 , we extended the Sotomayor theorem from integer-order to fractional-order systems. Next, by the frequency analysis method, we converted the fractional order model into an equivalent partial differential system. The tanh function was introduced into the scheme of sliding mode surface. The elimination of addiction was achieved by the action of the fractional order sliding mode control law. Finally, simulation results showed that fractional order values, nonlinear transmission rates, and specialized operating populations had a significant impact on predicting and controlling addiction. The fractional-order sliding mode control we designed played an important role in eliminating chatter, controlling addiction, and ensuring long-term effectiveness. The results of this paper have far-reaching implications for future work on modeling and control of fractional-order systems in different scenarios, such as epidemic spread, ecosystem stabilization, and game addiction. With the advancement of technology, social media has become an integral part of people's daily lives. This has resulted in the emergence of a new group of individuals known as "professional operation people". These individuals actively engage with social media platforms, taking on roles as content creators, influencers, or professionals utilizing social media for marketing and networking purposes. Therefore, in this article, we designed a six-dimensional fractional-order social media addiction model (FOSMA) in the sense of Caputo, which took into account the professional operations population. Initially, we established the positivity and boundedness of the FOSMA model. After that, the basic regeneration number and the equilibrium points (no addiction equilibrium point and addiction equilibrium point) were computed. Then, the local asymptotic stability of the equilibrium points were proved. In order to investigate the bifurcation behavior of the model when we extended the Sotomayor theorem from integer-order to fractional-order systems. Next, by the frequency analysis method, we converted the fractional order model into an equivalent partial differential system. The tanh function was introduced into the scheme of sliding mode surface. The elimination of addiction was achieved by the action of the fractional order sliding mode control law. Finally, simulation results showed that fractional order values, nonlinear transmission rates, and specialized operating populations had a significant impact on predicting and controlling addiction. The fractional-order sliding mode control we designed played an important role in eliminating chatter, controlling addiction, and ensuring long-term effectiveness. The results of this paper have far-reaching implications for future work on modeling and control of fractional-order systems in different scenarios, such as epidemic spread, ecosystem stabilization, and game addiction. [ABSTRACT FROM AUTHOR]

Vaccination has been used as strategy to eradicate the spread of COVID-19. But imperfect vaccine has been reported to induce backward bifurcation and hysteresis in mathematical models of disease transmission. Backward bifurcation is a phenomenon whereby a stable endemic equilibriumexists contemporaneously with a stable disease-free equilibriumwhen the basic reproduction number is less than 1. This situation can cause difficulty in controlling an epidemic because the basic reproduction is no longer the only means of eradicating the disease. In this paper, we propose a mathematical model for the transmission of disease which includes imperfect vaccination. We show that our model is capable of capturing backward bifurcation under certain conditions. By using parameters that are relevant to COVID-19 transmission in Malaysia, our numerical analysis shows that low vaccine efficacy can trigger backward bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, the author proposes and studies a mathematical model for the spread of coronavirus 2019. The effect of preventive programs on the control of disease is also considered. The basic reproduction number R0$$ {R}_0 $$ was computed and the results showed that forward bifurcation occured. Local and global stability of the equilibrium points of the model was studied by geometric stability method and Lyapunov function. Finally, the particular data of China which was reported for Covid‐19 were used for numerical simulation. [ABSTRACT FROM AUTHOR]

Infectious diseases continue to be a significant threat to human health and civilization, and finding effective methods to combat them is crucial. In this paper, we investigate the impact of awareness campaigns and optimal control techniques on infectious diseases without proper vaccines. Specifically, we develop an SIRS-type mathematical model that incorporates awareness campaigns through media and treatment for disease transmission dynamics and control. The model displays two equilibria, a disease-free equilibrium and an endemic equilibrium, and exhibits Hopf bifurcation when the bifurcation parameter exceeds its critical value, causing a switch in the stability of the system. We also propose an optimal control problem that minimizes the cost of control measures while achieving a desired level of disease control. By applying the minimum principle to the optimal control problem, we obtain analytical and numerical results that show how the infection rate of the disease affects the stability of the system and how awareness campaigns and treatment can maintain the stability of the system. This study highlights the importance of awareness campaigns in controlling infectious diseases and demonstrates the effectiveness of optimal control theory in achieving disease control with minimal cost. [ABSTRACT FROM AUTHOR]

The red palm weevil is one of the most dangerous pests affecting date palm trees in the Gulf region and the Middle East. This paper investigates the control of the red palm weevil (RPW) in date palms using several techniques, namely stem injection, sex pheromone traps, and the Sterile Insect Technique. A mathematical model describing the spread of the red palm weevil is presented, and the boundedness of the solution for the RPW system is studied. The conditions for local stability, forward bifurcation, and Hopf bifurcation are also obtained. Additionally, this research highlights the significance of optimal control in managing RPW and simulates the model using the Forward-Backward Sweep numerical method. It is observed that the parameters of the sterile insect technique, the pheromone trap, and the mechanical injection play important roles in controlling the RPW. Mechanical Injection may be more effective in controlling the palm weevil compared to the other two methods. [ABSTRACT FROM AUTHOR]

This paper studies the global dynamics of a delayed air pollution dynamic model with saturated functional response. This model exhibits forward=backward bifurcation. Through some iterative analysis techniques and constructing appropriate Lyapunov functionals, the global stability of the equilibria and the permanence of the model are obtained. For the case of forward bifurcation, it is shown that the boundary equilibrium is globally asymptotically stable (globally attractive) if R0 < 1 (R0 = 1), the positive equilibrium is globally asymptotically stable if R0 > 1, ab < c, or R0 > 1, ab > c, n < n', or R0 > 1, ab > c, n ≥ n' and the delay is small (the delay can change the stability of the positive equilibrium if R0 > 1, ab > c, n > n'). For the case of backward bifurcation, it is shown that the boundary equilibrium is globally asymptotically stable if R0 < !, the model has bistable equilibria (the boundary equilibrium and one positive equilibrium are locally asymptotically stable, and the other positive equilibrium is unstable) if ! < R0 < 1, the positive equilibrium is globally asymptotically stable if R0 > 1. Our results largely improve existing results. [ABSTRACT FROM AUTHOR]

Recent studies have demonstrated that immune impairment is an essential factor in viral infection for disease development and treatment. In this paper, we formulate an age-structured viral infection model with a nonmonotonic immune response and perform dynamical analysis to explore the effects of both immune impairment and virus control. The basic infection reproduction number is derived for a general viral production rate, which determines the global stability of the infection-free equilibrium. For the immune intensity, we get two important thresholds, the post-treatment control threshold and the elite control threshold. The interval between the two thresholds is a bistable interval, where there are two immune-present infected equilibria. When the immune intensity is greater than the elite control threshold, only one immune-present infected equilibrium exists and it is stable. By assuming the death rate and virus production rate of infected cells to be constants, with the immune intensity as a bifurcation parameter, the system exhibits saddle-node bifurcation, transcritical bifurcation, and backward/forward bifurcation. [ABSTRACT FROM AUTHOR]

Recently, bistable viral infection systems have attracted increased attention. In this paper, we study bistability and robustness for virus infection models with nonmonotonic immune responses in viral infection systems. The results show that the existing transcritical bifurcation undergoes backward or forward bifurcation in viral infection models with nonmonotonic immune responses. Our investigation demonstrates that the backward bifurcation threshold is the elite control threshold. When the immune intensity is greater than the elite control threshold, the virus will be under elite control; when the immune intensity is less than the elite control threshold, the virus may rebound. We also give a new definition of robustness to characterize bistable systems. [ABSTRACT FROM AUTHOR]

Given A, B, C, and D, block Toeplitz matrices, we will prove the necessary and sufficient condition for AB - CD = 0, and AB - CD to be a block Toeplitz matrix. In addition, with respect to change of basis, the characterization of normal block Toeplitz matrices with entries from the algebra of diagonal matrices is also obtained. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Guangxi Normal University - Natural Science Edition is the property of Gai Kan Bian Wei Hui and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper deals with the qualitative behavior of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals. Firstly, the positivity and boundedness of solutions for integer-order model are proved. The basic reproduction number R 0 is driven and it is shown that the disease-free equilibrium is globally asymptotically stable if R 0 < 1 in integer-order model. Using the methods of bifurcations theory, it is proved that the integer-order model exhibits forward bifurcation and Hopf bifurcation. Next, with the aim of the stability theory of fractional-order systems, some conditions, which can guarantee the local stability of the fractional-order model, are developed and occurrence of forward and Hopf bifurcations in this model are studied. Lastly, numerical simulations are illustrated to support the theoretical results and a comparison between the integer and fractional-order systems is presented. [ABSTRACT FROM AUTHOR]

In this paper, a discrete-time fractional-order SIR epidemic model with saturated treatment function is investigated. The local asymptotic stability of the equilibrium points is analyzed and the threshold condition basic reproduction number is derived. Backward bifurcation is shown when the model possesses a stable disease-free equilibrium point and a stable endemic point coexisting together when the basic reproduction number is less than unity. It is also shown that when the treatment is partially effective, a transcritical bifurcation occurs at ℜ 0 = 1 $\Re_{0}=1$ and reappears again when the effect of delayed treatment is getting stronger at ℜ 0 < 1 $\Re_{0}<1$. The analysis of backward and forward bifurcations associated with the transcritical, saddle-node, period-doubling and Neimark–Sacker bifurcations are discussed. Numerical simulations are carried out to illustrate the complex dynamical behaviors of the model. By carrying out bifurcation analysis, it is shown that the delayed treatment parameter ε should be less than two critical values ε1 and ε2 so as to avoid ℜ 0 $\Re_{0}$ belonging to the dangerous range ℜ 0 , 1 $\left[ \Re_{0},1\right]$. The results of the numerical simulations support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number R 0 is obtained such that R 0 ≤ 1 indicates the possibility of dysentery eradication in the community while R 0 > 1 represents uniform persistence of the disease. The Lyapunov–LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for R 0 > 1. Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation. [ABSTRACT FROM AUTHOR]

In this paper, we incorporate immune systems into an HIV model, which considers both logistic target-cell proliferation and viral cell-to-cell transmission. We study the dynamics of this model including the existence and stability of equilibria. Based on the existence of equilibria, we focus on the backward bifurcation and forward bifurcation. Considering the stability of equilibria, Hopf bifurcation is discussed by identifying the basic reproduction number R 0 as bifurcation parameter. The direction and stability of Hopf bifurcation are investigated by computing the first Lyapunov exponent. Specially, the effects of immune response on the basic reproduction number R 0 and viral dynamics are addressed by deriving the sensitivity analysis. As a result, we find that the removal rate of infected cells by cytotoxic T lymphocytes (CTLs), γ 1 , is the predominant factor of R 0 . However, we conclude from numerical results that it is unfeasible to decrease R 0 by increasing the value of γ 1 constantly. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions. These dynamics are investigated by the proposed model to point out the importance and complexity of immune responses in fighting HIV replication. [ABSTRACT FROM AUTHOR]

TV and radio advertisements are widely acknowledged as important interventions in raising issues of public health care and play promising role to control the infection through propagating awareness among the individuals. In this paper, a nonlinear susceptible‐infected‐susceptible (SIS) model is proposed and analyzed to see the impacts of TV and radio advertisements on the spread of influenza epidemic. In the model formulation, it is assumed that the susceptible individuals contract infection through the direct contact with infected individuals. The information regarding the protection against the disease is propagated via TV and radio advertisements, and their growth rates are assumed to be proportional to the fraction of infected individuals. However, the growth rate of TV advertisements decreases with the increase in number of aware individuals. The information broadcasted through TV and radio advertisements induces behavioral changes among the susceptible individuals, and they form an isolated aware class. The epidemiological feasible equilibria, their stability properties, and direction of bifurcation are discussed. The expression for modified basic reproduction number is obtained. The model analysis shows that the dissemination rate of awareness among susceptible individuals due to TV and radio advertisements and baseline number of TV and radio advertisements have potential to reduce the epidemic peak and, thus, control the spread of infection. Further, the analytical findings are well supported through numerical simulation. [ABSTRACT FROM AUTHOR]

The red palm weevil is one of the most dangerous pests affecting date palm trees in the Gulf region and the Middle East. This paper investigates the control of the red palm weevil (RPW) in date palms using several techniques, namely stem injection, sex pheromone traps, and the Sterile Insect Technique. A mathematical model describing the spread of the red palm weevil is presented, and the boundedness of the solution for the RPW system is studied. The conditions for local stability, forward bifurcation, and Hopf bifurcation are also obtained. Additionally, this research highlights the significance of optimal control in managing RPW and simulates the model using the Forward-Backward Sweep numerical method. It is observed that the parameters of the sterile insect technique, the pheromone trap, and the mechanical injection play important roles in controlling the RPW. Mechanical Injection may be more effective in controlling the palm weevil compared to the other two methods.

Measles, a highly contagious airborne disease, remains endemic in many developing countries with low vaccination coverage. In this paper, we present a deterministic mathematical compartmental model to analyze the dynamics of measles. We establish global stability conditions for both disease-free and endemic equilibria using the Lyapunov functional stability method. By using arbitrary parameters, we find that the proposed model exhibits forward bifurcation. To simulate the solution of the model for the forward problem, we perform numerical integration using MATLAB software. Moreover, we calibrate the model with real data from Ethiopia and estimate the parameters along with a 95 percent confidence interval (CI) by formulating an inverse problem. It is noteworthy that our model fits well with the actual data from Ethiopia. The estimated basic reproduction number (R0) is determined to be R0=1.3973, demonstrating the endemic status of the disease. Additionally, our local sensitivity analysis indicates that reducing the transmission rate and increasing vaccination coverage can effectively minimize R0.

Pandemic is an unprecedented public health situation, especially for human beings with comorbidity. Vaccination and non-pharmaceutical interventions only remain extensive measures carrying a significant socioeconomic impact to defeating pandemic. Here, we formulate a mathematical model with comorbidity to study the transmission dynamics as well as an optimal control-based framework to diminish COVID-19. This encompasses modeling the dynamics of invaded population, parameter estimation of the model, study of qualitative dynamics, and optimal control problem for non-pharmaceutical interventions (NPIs) and vaccination events such that the cost of the combined measure is minimized. The investigation reveals that disease persists with the increase in exposed individuals having comorbidity in society. The extensive computational efforts show that mean fluctuations in the force of infection increase with corresponding entropy. This is a piece of evidence that the outbreak has reached a significant portion of the population. However, optimal control strategies with combined measures provide an assurance of effectively protecting our population from COVID-19 by minimizing social and economic costs. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyzed a mathematical model that describes the dynamics of visceral leishmaniasis in a population with immigration of infective humans under mass vaccination strategy. Our result shows that in order for the vaccine to play a role on disease control, it must be very effective. Results also show that vaccination coverage does not have any impact on disease control when the immigration rate is small, and it does not affect the long-term behavior when the immigration rate is high. In the case of no immigration of infective, our system has disease-free equilibrium, and it is globally asymptotically stable when [ABSTRACT FROM AUTHOR]

In this paper, an insect-parasite-host model with logistic growth of triatomine bugs is formulated to study the transmission between hosts and vectors of the Chagas disease by using dynamical system approach. We derive the basic reproduction numbers for triatomine bugs and Trypanosoma rangeli as two thresholds. The local and global stability of the vector-free equilibrium, parasite-free equilibrium and parasite-positive equilibrium is investigated through the derived two thresholds. Forward bifurcation, saddle-node bifurcation and Hopf bifurcation are proved analytically and illustrated numerically. We show that the model can lose the stability of the vector-free equilibrium and exhibit a supercritical Hopf bifurcation, indicating the occurrence of a stable limit cycle. We also find it unlikely to have backward bifurcation and Bogdanov-Takens bifurcation of the parasite-positive equilibrium. However, the sustained oscillations of infected vector population suggest that Trypanosoma rangeli will persist in all the populations, posing a significant challenge for the prevention and control of Chagas disease.

In this paper, we study a dengue model governed by an eight-dimensional autonomous system of ordinary differential equations using dynamical system theory. Appropriate Lyapunov functions are used to carry out an extensive investigation of the global asymptotic dynamics of the model around the dengue-free and dengue-present equilibria. The model is shown to exhibit a forward bifurcation phenomenon using Center Manifold Theory. Sensitivity analysis is carried out to determine the relative importance of the model parameters to the spread of the disease. Using optimal control theory, the model is further extended to a nonlinear optimal control model to explore the impact of four time-dependent control variables, namely, personal protection, treatment drug therapy for latently infected individuals, treatment control for symptomatic individuals and insecticide control for mosquito reduction, on dengue disease dynamics in a population. Cost-effectiveness analysis is conducted on various strategies with combinations of at least three optimal controls to determine the least costly and most effective strategy that can be implemented to contain the spread of dengue in a population. [ABSTRACT FROM AUTHOR]

Infectious diseases continue to be a significant threat to human health and civilization, and finding effective methods to combat them is crucial. In this paper, we investigate the impact of awareness campaigns and optimal control techniques on infectious diseases without proper vaccines. Specifically, we develop an SIRS-type mathematical model that incorporates awareness campaigns through media and treatment for disease transmission dynamics and control. The model displays two equilibria, a disease-free equilibrium and an endemic equilibrium, and exhibits Hopf bifurcation when the bifurcation parameter exceeds its critical value, causing a switch in the stability of the system. We also propose an optimal control problem that minimizes the cost of control measures while achieving a desired level of disease control. By applying the minimum principle to the optimal control problem, we obtain analytical and numerical results that show how the infection rate of the disease affects the stability of the system and how awareness campaigns and treatment can maintain the stability of the system. This study highlights the importance of awareness campaigns in controlling infectious diseases and demonstrates the effectiveness of optimal control theory in achieving disease control with minimal cost.

A deterministic model which describes measles' dynamic using newborns and adults first and second dose of vaccination and medical treatment is constructed in this paper. Mathematical analysis about existence of equilibrium points, basic reproduction number, and bifurcation analysis conducted to understand qualitative behaviour of the model. For numerical purposes, we estimated the parameters' values of the model using monthly measles data from Jakarta, Indonesia. Optimal control theory was applied to investigate the optimal strategy in handling measles spread. The results show that all controls succeeded in reducing the number of infected individuals. The cost-effective analysis was conducted to determine the best strategy to reduce number of infected individuals with the lowest cost of intervention. Our result indicates that the use of the first dose measles vaccine with medical treatment is the most optimal strategy to control measles transmission.

Copyright of Revista Integración is the property of Universidad Industrial de Santander and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Recently, bistable viral infection systems have attracted increased attention. In this paper, we study bistability and robustness for virus infection models with nonmonotonic immune responses in viral infection systems. The results show that the existing transcritical bifurcation undergoes backward or forward bifurcation in viral infection models with nonmonotonic immune responses. Our investigation demonstrates that the backward bifurcation threshold is the elite control threshold. When the immune intensity is greater than the elite control threshold, the virus will be under elite control; when the immune intensity is less than the elite control threshold, the virus may rebound. We also give a new definition of robustness to characterize bistable systems.

In this paper, we investigate an SIRS epidemic model with chronological age structure in a demographic steady state. Although the age-structured SIRS model is a simple extension of the well-known age-structured SIR epidemic model, we have to develop new technique to deal with problems due to the reversion of susceptibility for recovered individuals. First we give a standard proof for the well-posedness of the normalized age-structured SIRS model. Next we examine existence of endemic steady states by fixed point arguments and bifurcation method, where we introduce the next generation operator and the basic reproduction number $R_0$ to formulate endemic threshold results. Thirdly we investigate stability of steady states by the bifurcation calculation and the comparison method, and we show existence of a compact attractor and discuss the global behavior based on the population persistence theory. Finally we give some numerical examples and discuss the effect of mass-vaccination on $R_0$ and the critical coverage of immunization based on the reinfection threshold.

In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number $ R_0 $ is obtained such that $ R_0 \leq ~1 $ indicates the possibility of dysentery eradication in the community while $ R_0 \gt 1 $ represents uniform persistence of the disease. The Lyapunov–LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for $ R_0 \gt 1 $ . Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation.

Kawasaki disease, also referred to as mucocutaneous lymph node syndrome (MCLS), is an autoimmune disease. In this paper, we consider an ordinary differential equation model of Kawasaki disease pathogenesis. In this model, there exhibits forward ∕ backward bifurcation. By constructing suitable Lyapunov functions and using Lyapunov–LaSalle invariance principle, we establish some practical sufficient conditions for the global stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibrium. [ABSTRACT FROM AUTHOR]

Copyright of Revista Integración is the property of Universidad Industrial de Santander and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We propose a mathematical model describing the dynamics of hepatitis B or C virus infection. The method of characteristics reduces the age-structured model to a system of differential equations with distributed delay. The model is rigorously analyzed, and a detailed stability analysis is performed. The model, consisting of two mutually exclusive compartments representing the interaction of the virus in the liver and the blood, has a locally asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number R0, is less than unity. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained. Further, under biologically realistic hypotheses, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. This means that reducing the basic reproduction number R0 below 1 is not always sufficient to control HBV/HCV infection when the virus interacts with cells in both the liver and the blood. Numerical analysis confirms the results and stresses the role of some parameters involved in the elimination and persistence of the infection. [ABSTRACT FROM AUTHOR]