In this paper, a model of computer virus propagation with social media coverage and saturated treatment is established to obtain an optimal control strategy so that the number of infected computers is minimized while controlling costs as low as possible. Firstly, we show that the virus-free equilibrium is asymptotically stable for R 0 < 1, indicating computer virus disappears eventually. The global stability of the positive equilibrium is investigated through the Li-Muldowney geometric approach. Secondly, sensitivity analysis is performed by applying Latin hypercube sampling(LHS) to understand how each parameter affects virus spread. Based on the sensitivity analysis of parameters, we propose an optimal control system whose objective function is to both minimize the number of infected computers and reduce the costs of implemented control measures. The necessary conditions for optimal control are derived by applying Pontryagin’s maximum principle. Furthermore, the optimal control strategies are characterized by means of the numerical simulations to verify our results. Finally, a brief discussion is presented.
This study develops a comprehensive mathematical framework for analyzing the transmission dynamics of malaria, with particular emphasis on a four-dose vaccination strategy combined with treatment interventions as primary control mechanisms. The model stratifies the human population into nine compartments—susceptible, exposed, infected, under treatment, recovered, and four sequential vaccination classes (V 1 through V 4 )—together with susceptible and infected vector populations, all governed by a system of ordinary differential equations. Analysis of the diseasefree equilibrium demonstrates local asymptotic stability when the basic reproduction number R 0 < 1, while a unique endemic equilibrium exists and is stable when R 0 > 1. Sensitivity analysis identifies the mosquito biting rate, transmission probability, first-dose vaccination rate, and treatment rate as the most influential parameters governing malaria dynamics. Numerical simulations confirm that increasing vaccination coverage and completion rates across all four doses, in conjunction with optimized treatment, substantially reduces malaria incidence. To capture long-term memory effects inherent in malaria transmission—particularly non-exponential immunity waning and heterogeneous parasite development—a Caputo fractional-order version of the model is introduced, solved using the Adams–Bashforth predictor-corrector method. Comparison of the integer-order and fractional-order results reveals that memory effects slow the approach to equilibrium and may lead to higher endemic levels under identical intervention parameters, underscoring the value of the fractional extension. These findings emphasize that achieving high completion rates across all four vaccination doses, sustained treatment access, and integrated vector control are essential for meaningful progress toward malaria elimination in Sub- Saharan Africa.
Dynamic problems involving the mechanical model of respiratory devices prepared based on the working principle of the lung bring a different perspective to the current situation in clinical medicine in case the time scale changes. This paper delves into mathematical models performed by delta derivative and proportional derivative on arbitrary time scale for the instantaneous volume in the single compartment lung. First, these problems are solved with appropriate methods. Finally, results are supported on graphical representations of the equations to assess how well the operators mentioned perform in real-world problems and statistical analysis of selected arbitrary values is performed.
This study presents a nonlinear mathematical model that integrates two crucial nonpharmaceutical interventions: media-driven awareness programs and the availability of hospital beds to halt the spread of infectious diseases. By incorporating the dynamic influence of media awareness programs and the finite capacity of hospital beds, the model captures the complex interplay between public behavior and healthcare resource limitations during an infectious disease outbreak. A comprehensive analytical investigation is conducted, which includes equilibrium analysis, computation of the basic reproduction number, detailed stability and bifurcation analysis. The presented model exhibits a range of rich dynamical behaviors, such as forward and backward transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations. Numerical simulations validate the theoretical findings and reveal that the synergistic impact of media awareness programs and healthcare resources (in terms of hospital beds) are crucial for the effective disease control. This work offers strategic insights for policymakers aiming to design robust intervention frameworks that combine media awareness programs with healthcare resource optimization.
Poliomyelitis, also known as polio, is a contagious viral illness that predominantly impacts young children, leading to paralysis and, in severe instances, death. Despite worldwide initiatives aimed at elimination, the spread of the poliovirus persists in various areas, highlighting the need for robust vaccination strategies. This research employed a mathematical model to explore the dynamics of poliovirus transmission, integrating vaccination as a crucial method for disease control. The model was analyzed to determine the basic reproduction number (R 0 ) and the stability properties of the disease-free and endemic equilibrium. Our findings demonstrated that the system achieves local asymptotic stability when the basic reproduction number is less than one, global asymptotic stability when it is exactly one, and maintains a stable endemic equilibrium when it is greater than one. Sensitivity analysis revealed critical parameters influencing the basic reproduction number, emphasizing the impact of vaccination coverage and disease transmission rates on polio dynamics. More so, numerical simulations showed that vaccination coverage of about 75% would be enough to eradication polio from the population.
This paper incorporates the effects of the stochastic perturbations and multiple time-delays into a generalized predator-prey system. The presented system includes two types of perturbations: one is a nonlinear comprehensive perturbation, and the other is a perturbation on the predator’s attacking rate. Firstly, the unique existence of the global positive solutions and various boundedness properties of the proposed system are considered. Secondly, the mean persistence and extinction of predator and prey populations are investigated. The results demonstrate that stochastic perturbations and multiple time-delays have an important effects on the dynamical behaviors of the considered system. Finally, all theoretical results are verified by numerical simulations.
This paper introduces a novel class of delayed virus-to-cell HIV models incorporating general incidence and Logistic growth, with the objective of elucidating the complex reactivation dynamics of latently infected cells. Analytical results demonstrate that the HIV infection is cleared from the T-cells population when the basic reproduction number [Formula: see text]; if [Formula: see text], the criteria for local and global asymptotic stability of the endemic equilibrium are established under the condition that the death rate [Formula: see text] of target T-cells is greater than or equal to the intrinsic mitosis rate [Formula: see text]. Conversely, when [Formula: see text] is less than [Formula: see text], the system is shown to exhibit rich dynamical behaviors. Furthermore, the characteristic equation at endemic equilibrium with the coefficients dependent on two time delays is analyzed. Hopf bifurcation criteria are derived for both the general case, where both delays vary simultaneously as bifurcation parameters, and the special case involving only a single delay. The analyses reveal that stability switches can be induced, while chaotic phenomena emerge only when the delays become sufficiently large. Theoretical findings are validated through numerical simulations, which further indicate that a higher latency reactivation rate contributes to more effective control of HIV transmission. In contrast, a prolonged latent infection delay is ultimately predicted to precipitate a rapid viral load rebound.
We develop a stochastic diffusion model with two component Allee effects via a continu- um limit of a discrete Markov process. The boundary classification proves that zero is exit and infinity is entrance. Using intrinsic ultracontractivity of the associated semigroup, we establish the existence and uniqueness of a quasi-ergodic distribution, characterizing metastable persistence prior to extinction. Scaling laws for the mean extinction time with respect to key parameters-initial population size, mate-finding efficiency, and predation intensity-are derived. This work provides a rigorous mathematical framework for ana- lyzing transient dynamics and extinction in populations subject to strong Allee effects and environmental stochasticity.
In this paper, we mainly consider the dynamics of a stochastic SIRI epidemic model with media coverage. Firstly, a sufficient condition for the existence and uniqueness of the ergodic stationary distribution of the model is obtained by establishing an appropriate Lyapunov function. Then the approximate expression of probability density function of the solution to this model around quasi-endemic equilibrium is obtained by solving the corresponding Fokker-Planck equation. In addition, the sufficient conditions for the extinction of the disease are given. Finally, some numerical simulations are introduced to verify our theoretical results. The results indicate that as media coverage parameter increases, the time average of infected individuals decreases, which implies that media coverage can promote the extinction of disease.
This study focuses on the dynamics of a diffusive nutrient-microorganism model subject to Neumann boundary conditions. An in-depth analysis is conducted to explore how time delay affects the stability of the constant steady state. Analytical results reveal that time delay can destabilize the system, thereby inducing unstable oscillations in originally stable regions. Specifically, under certain conditions, the equilibrium state undergoes stability switches and gives rise to Hopf bifurcation as the time delay pa- rameter varies. Furthermore, under the synergistic effects of diffusion and time delay, the system exhibits spatially inhomogeneous periodic solutions, which elucidates the dynamical mechanism underlying the alternating distribution of nutrients patches and microbial enrichment zone in sediments. The direction and stability of Hopf bifurcation are determined by applying the center manifold theory and the normal form method for partial functional differential equations. Finally, numerical simulations are carried out to verify the theoretical findings.
Cholera and HIV are two major infectious diseases that often coexist and result in severe health outcomes. In this paper, we propose a co-infection model for the transmission dynamics of cholera and HIV to explore the distinctive correlation between these two diseases. First, we analyze the sub-models at their steady states and calculate the basic reproduction number of the sub-models and co-infection model. Second, the proposed stochastic co-infection model has been investigated for the existence and uniqueness of a global solution, stationary distribution and ergodicity for the co-infection stochastic model are also studied by constructing stochastic Lyapunov candidates. At last, numerical simulations are carried out to validate theoretical results.
This paper introduces a novel stochastic [Formula: see text] epidemic model to investigate the dynamics of two co-circulating pathogen strains in a vaccinated population. The model incorporates both Gaussian white noise and Lévy jump processes to capture a spectrum of random disturbances, ranging from continuous environmental fluctuations to abrupt, large-scale events such as outbreaks or intervention shocks. We rigorously establish the existence, uniqueness and positivity of a global solution using Lyapunov functions and stopping time theory. Stochastic threshold conditions, [Formula: see text] and [Formula: see text], are derived to determine the extinction or persistence of the infections. Our analysis shows that when [Formula: see text], the diseases are eradicated exponentially fast, whereas if [Formula: see text], the infections persist endemically. Numerical simulations validate these theoretical results and reveal that Lévy noise induces more pronounced fluctuations compared to Gaussian noise. Sensitivity analysis indicates that transmission rates are key drivers of both infection prevalence and stochastic variability. This study provides a rigorous analytical framework and valuable insights for understanding and managing concurrent epidemics in complex and uncertain environments, highlighting the critical role of stochasticity in shaping disease outcomes.
Eastern equine encephalitis virus (EEEV) is a deadly arboviral pathogen with 30% severe case fatality. EEEV exhibits pronounced 2-3 year cyclical outbreak patterns in the northeastern United States, linked to shifts in mosquito feeding preferences between hatch-year and adult avians. We developed an age-structured vector-host model incorporating differential feeding patterns of Culiseta melanura mosquitoes on European Starlings and American Robins. Global sensitivity analysis revealed mosquito biting rate (a) as the dominant driver in transmission, with avian infectivity (delta) and exposure (alpha) playing secondary roles. Pairing tree-based machine learning algorithms with SHAP analysis on 100,000 parameter set simulations identified parameter hierarchies that govern cyclic transmission. Adult avian mortality (mu A) was identified as the key parameter underlying unstable cyclic and stable transmission patterns. SHAP further revealed these patterns to be grounded on opposite sides of the same epidemiological mechanism. Stable endemics emerge from demographic stability paired with transmission optimization, while unstable cyclic endemics emerge from demographic instability paired with minimal transmission optimization. Numerical simulations illustrated critical threshold dynamics at 0.2 <= alpha <= 0.4, where heightened hatch-year exposure triggers demographic instability responsible for observed 2-3 year cycles, while balanced exposure (alpha approximate to 0.5) leads to stable endemics. These mechanisms provide a foundation for targeted surveillance and control interventions.
Tuberculosis, one of the oldest infectious diseases in the world, is caused by M. tuberculosis, and currently the infectious disease with the highest single-disease mortality rate. The invention of anti-tuberculosis drugs has aided in controlling tuberculosis, but drug resistance remains a significant challenge in its prevention and control. This study is based on tuberculosis reporting data released by the China CDC from 2004 to 2020. Considering that drug-resistant tuberculosis has both primary and acquired transmission routes, a dynamic model coupled with economic factors is established. Nonlinear least squares fitting is applied to existing reports of new cases to derive model parameters. Sensitivity analysis reveals that transmission coefficients, disease progression rates, and economic parameters significantly influence tuberculosis transmission. Different economic effect parameters exert differentiated intervention effects on the transmission of susceptible and drug-resistant strains, clinical outcomes, and strain conversion. Calculations indicate that under current control measures, the control reproduction numbers for tuberculosis transmission among drug-sensitive and drug-resistant populations are [Formula: see text] =0.5460, [Formula: see text] =0.6419, respectively. Given the current control measures and economic investment, it will be impossible to achieve the WHO’s goal of eliminating tuberculosis in China by 2035. Increased economic investment in high-prevalence areas, coupled with a focus on measures to interrupt human-to-human transmission, can effectively control tuberculosis outbreaks. In low-prevalence areas, increased investment prioritizing the reduction of disease progression yields superior outcomes. Furthermore, enhancing economic investment in adherence monitoring for drug-sensitive infections and improving conversion rates can reduce the number of drug-resistant cases while preventing a significant increase in drug-sensitive infections. This provides a basis for optimizing resource allocation and strategic planning of control measures.
The stochastic dynamics of molecular motors play a pivotal role in comprehending and simulating critical processes in biological systems. This understanding is essential for elucidating how these motors achieve precise navigation and functionality amidst the intricacies of cellular environments. First, we employed the stochastic energy landscape function to analyze the stability of the stochastic molecular motor systems under various conditions. In this paper, we concentrate on the stochastic dynamics of a molecular motor system, particularly examining the mean trajectories and the most probable trajectories under varying levels of noise. In terms of the mean trajectories, we observe that molecular motors, regardless of their initial concentrations, can transition to the same position. Moreover, we discover that an increase in noise intensity correlates with a reduction in the time needed to transition to the mean position. In terms of the most probable trajectories, an escalation in noise intensity results in a decrease in concentration at the most probable position for the molecular motor [Formula: see text], in contrast to an increase in concentration at the most probable position for the external system [Formula: see text]. Our findings, which are dedicated to understanding the behavior of motors under different stochastic noise, contribute significantly to a more profound comprehension of molecular motors.
In this study, a two-dimensional (2D) multi-block lattice Boltzmann method (LBM) is employed to develop a computational model of human blood circulation. Based on actual arteriovenous network data, the vascular model of the blood circulation was simplified into an idealized 2D rigid channel network. Among these structures, the approach of incorporating porous media within the blood flow channels was employed to simulate the capillary network. The model was calibrated based on experimentally measured flow variation patterns, and through grid independence verification, the research validity of the model has been substantiated. As an application, the model is coupled with a three-dimensional (3D) localized radial artery model constructed using the immersed boundary method. The effects of elastic walls on the propagation of flow, displacement, and pressure waves were systematically investigated. Furthermore, through simulations of amputation, vascular plaque formation, and organ pathologies, the impact of abnormal blood flow on circulatory dynamics and pulse wave characteristics was quantitatively assessed. The results demonstrate that localized blood flow patterns significantly influence the systemic circulation process, and abnormalities in regional hemodynamics can alter the pulse wave characteristics of the radial artery. This model offers a novel methodological framework for investigating the mechanisms underlying blood circulation dynamics.
This paper is concerned with a nonlocal dispersal predator–prey system with single predator and multiple preys. We study the invading phenomenon of an alien predator to the habitat of multiple aborigine preys by traveling wave solutions connecting the predator-free state to the co-existence state. The existence of traveling wave solutions that converge to predator-free equilibrium as the moving coordinate goes to [Formula: see text] is proven by constructing suitable upper-lower solutions and using Schauder’s fixed point theorem, when the wave speed [Formula: see text] for some positive number [Formula: see text]. Then by constructing a Lyapunov functional, we show that the traveling wave solutions converge to the co-existence state as the moving coordinate goes to [Formula: see text]. Furthermore, by using a limiting argument, we prove the existence of traveling wave solutions with speed [Formula: see text]. Finally, by using a contradictory approach with the properties of the corresponding characteristic equation to the system, we prove the non-existence of traveling wave solutions when [Formula: see text]. It turns out that [Formula: see text] is the minimum wave speed for traveling wave solutions connecting the predator-free state and the co-existence state.
This paper investigates a delayed stage-structured predator–prey model that incorporates prey refuge, fear effect, and intraspecific protection among predators. Different predation mechanisms are assigned to predator life stages: mature predators follow a Crowley–Martin functional response, while immature predators obey a Holling type II response. The analysis proves positivity, boundedness and global stability of the coexistence equilibrium through a Lyapunov approach. The study reveals two major dynamical findings. First, predator maturation can trigger a transcritical bifurcation, determining whether predators persist or become extinct. Second, the delayed fear effect can induce a Hopf bifurcation, generate persistent population oscillations and destabilize the ecosystem. Numerical simulations further show that prey refuge and protective behavior among predators promote species coexistence and enhance ecosystem resilience, whereas excessive fear delay may lead to instability or extinction. These results demonstrate that behavioral responses and stage structure jointly play a crucial role in maintaining ecological stability, providing new biological insight into predator–prey coexistence and conservation management.
Host individuals with vaccinations for infectious diseases often transition among different states of susceptibility and infectivity. Thus, we derive a class of epidemic models in which susceptible and infected individuals have a discrete set of susceptibility and infectivity states, respectively. Our model is based on double-dose vaccination and accounts for all possible state transitions between states. A comprehensive mathematical analysis of the proposed model is conducted, including assessing the control reproduction number and the global dynamical behaviors of the solutions. Using an improved affine-invariant ensemble Markov Chain Monte Carlo method, we also fit the model with measles case data in the United States. The results indicate that despite high vaccination coverage and efficacy rates, sporadic or small-scale measles outbreaks still occur. This suggests that reliance solely on vaccination policies is insufficient to fully interrupt its transmission chain. Consequently, enhancing the diagnosis rate of measles cases is another crucial measure for controlling measles transmission and outbreaks, building upon existing vaccination efforts.
This paper investigates a free boundary problem focused on the growth dynamics of vascularized tumors, incorporating time delays and the impact of inhibitors. Unlike existing vascularized tumor models with inhibitors, our model contains time delays, which represent the intrinsic delays of cell proliferation. The substance exchange between the vascularized tumor and its surrounding vascular network is represented by the Robin boundary. The problem comprises a system of nonlinear reaction-diffusion equations that characterizes the nutrient concentration u(r, t) and the inhibitor concentration v(r, t), together with an ordinary differential equation representing the tumor radius R(t). Firstly, it is shown that the model possesses at least one steady-state solution under certain sufficient conditions. Next, we demonstrate that the quasi-steady state system possesses a non-negative solution and analyze its stability. Finally, we establish the existence and uniqueness of the global classical solution and further analyze the asymptotic stability of the steady-state solution. Our results demonstrate that time delays in cell proliferation do not change the overall evolution trend of the tumor, but only slow the tumor growth process.
