Due to fear of predation, prey populations adopt various anti-predator defense strategies, which reduce their birth rate. Here, we formulate both autonomous and non-autonomous prey–predator systems with the assumptions that the prey’s per capita birth rate decreases as a consequence of fear of predators, and that predator species waste time searching for prey species due to encounters with other predators. For the autonomous system, we establish the criteria under which the system is uniformly persistent and those under which the coexistence equilibrium point is globally asymptotically stable. Moreover, the existence of local bifurcations, one- and two-parameter bifurcation analyses, and the influence of each ecological parameter on system dynamics are rigorously examined. For the non-autonomous system, we establish the conditions for uniform persistence and prove the existence of a unique globally asymptotically stable periodic solution under certain conditions.
This paper considers a chemostat system with a consumer and a resource. The consumer moves between multiple sink-source patches with both resource and toxicant. By using graph-theoretic method and Lyapunov theory, we show global stability of positive equilibria in the system. Rigorous analysis on the equilibria demonstrates that diffusion could make the consumer persist in sinks and even make the total population abundance larger than that without diffusion. Asymmetry in diffusion, toxicant distribution, and spatial pattern of patches play crucial roles in the persistence and increase, while two formulas are deduced for total abundances with/without diffusion. On the contrary, diffusion could also lead to decrease of the total abundance, even lead to extinction of the consumer in sources. Our results are consistent with empirical observations, and provide new insights. This work is important for both conservation of endangered species and control of pests.
A stochastic predator-prey system with Markov switching and colored noise is studied, where Markov switching is used to represent parameter changes caused by environmental mutations. The joint impact of switching and colored noise on the stationary response is explored by the stochastic averaging method, the stochastic Runge-Kutta algorithm, and the most probable trajectory theory. Findings show that a wider bandwidth tends to make the stationary response more concentrated and drives the most probable response closer to the deterministic one, whereas a narrower bandwidth is associated with stronger population oscillations. A larger stationary probability tends to drive the system response toward that of the corresponding substate, while a faster transition rate generally stabilizes the most probable response. In addition, both the bandwidth parameter and switching may induce stochastic P-bifurcation, accompanied by changes in the concentration and shape of the stationary distribution. The results also indicate that the overall stationary response is influenced not only by stationary probabilities but also by the intrinsic stability of the substates.
This research focuses on the transmission dynamics of Mycoplasma pneumoniae (MP). A mathematical model has been developed, which explicitly incorporates infectivity during the latent period and distinguishes between the spontaneous recovery rate of latent individuals and the treatment rate of symptomatic patients. The key innovation involves decomposing the basic reproduction number R-0 into two components: R-01, representing transmission from latent individuals, and R-02 from infected patients. Notably, the spontaneous recovery rate of latent individuals influences both R-01 and R-02. Additionally, a secondary threshold R-03, related to waning immunity, plays a significant role in disease persistence. It is noted that R-0>1 does not ensure a stable endemic equilibrium unless R03 < 1. Conversely, R-0 < 1 may still allow an unstable endemic equilibrium if R-03>1. Sensitivity analysis reveals that the transmission of MP is influenced by key parameters, with spontaneous recovery rate and transmission rates being critical drivers. It is worth noting that the transmission rate of latently infected persons has a more substantial impact on disease transmission than the transmission rate of infected people. Optimal control strategies, including preventive measures, targeted treatment, and interventions for latent infections, are developed to minimize the disease burden. This work improves the understanding of MP spread and designs effective control strategies.
After being cured of tuberculosis (TB), individuals do not acquire lifelong immunity and may become reinfected. Based on this context, we propose an epidemiological model incorporating treatment and recurrent TB. By calculating the basic reproduction number R0 using the next-generation matrix method, we decouple the transmission thresholds for fast-progressing and slow-progressing TB. Key findings: backward bifurcation where the disease may persist even when R0 < 1, and bistability where multiple endemic equilibria coexist when forward bifurcation occurs at R0 = 1 under certain conditions. We confirm that exogenous reinfection parameter delta induces backward bifurcation, and rigorously prove the existence of a critical threshold sigma c for recurrent reinfection parameter sigma, demonstrating that sigma > sigma c alone can trigger backward bifurcation. The model innovatively incorporates differentiated treatment mechanisms for both successfully treated and treatment-failure cases, better reflecting clinical reality. We employ Bayesian MCMC methods with Guangdong TB surveillance data to estimate parameter posterior distributions and conduct global PRCC analysis. Numerical simulations demonstrate that optimized treatment strategies can reduce TB mortality by 93.33% by 2035, however, reaching the 2050 goals requires integrated measures including latent infection screening. This study provides quantitative evidence for precision TB control strategies.
Considering the complexity of host-vector diseases transmission and the phenomenon of periodic outbreaks, this paper investigates the non-autonomous and time-periodic reaction-diffusion transmission models of host-vector disease with general incidence rates and seasonality. The basic reproduction number R-0 and the critical wave speed c(& lowast;) are defined and the existence and non-existence of traveling wave solutions are studied. Specifically, when R-0 > 1 and c > c(& lowast;), the existence of periodic traveling wave solutions satisfying some boundary conditions is proved by using the methods of the linear parabolic equation, the fixed point theorem and some limit techniques. The non-existence of traveling waves is proved by using Poincare transformation when R-0 > 1 for any 0 < c < c(& lowast;) or R-0 <= 1 for any c > 0. Finally, the numerical simulations are given to verify the correctness of the theoretical results and to explore the spatial spread behavior.
In this paper, we analyze the complex dynamics of a discrete prey-predator model with prey refuge and Holling type-II functional response. Our focus is on investigating the stability, chaotic behavior, and codimension-one bifurcation properties of the model. We begin by establishing the nonnegativity conditions required for the model's biological relevance, followed by the derivation of equilibrium solutions. Specifically, we show that the model always admits trivial and semitrivial equilibria, and an interior equilibrium exists under specific parametric conditions. Next, we apply the linear stability theory to examine the local dynamics at these biologically meaningful equilibria. This analysis reveals the conditions under which the equilibria are stable or unstable, providing a clear understanding of the system's behavior at these points. Furthermore, we conduct a comprehensive bifurcation analysis, identifying the codimension-one bifurcation sets and exploring the occurrence of bifurcations as system's parameter, namely the bifurcation parameter, varies. Our results highlight the key parameters that drive transitions in system behavior, leading to phenomena such as stability loss, periodic solutions, and chaotic dynamics. We also address the chaotic behavior observed in the model by implementing control strategies, including the Ott-Grebogi-Yorke (OGY) method and a hybrid control approach. These methods effectively mitigate chaos, restoring stability to the system under certain conditions. Importantly, each theoretical result is supplemented with a biological interpretation, elucidating the implications of our findings in the context of ecological systems. Finally, we illustrate our theoretical results with numerical simulations, demonstrating the model's complex dynamics, including bifurcations, maximum Lypunov function, phase portraits, and chaotic regimes. These simulations not only validate our analytical results but also provide deeper insights into the rich dynamics exhibited by discrete prey-predator interactions with Holling type-II functional response and prey refuge.
In this work, we have constructed a mathematical model of the neuromuscular activity of a motor unit. It is described in terms of ordinary differential equations and couples the model of Izhikevich of neural activity, the Williams model of calcium activity in the muscle fiber, and a Hill-type model of the resultant muscle force. We have introduced an important novelty in the coupling of the latter two, by introducing a sigmoid activation function, which significantly improves the descriptive capabilities of the model from a quantitative point of view. We have validated the model by fitting with a high degree of accuracy experimental data for twitches of nine different motor units (slow, fast fatigable, fast fatigue-resistant) - a result, which is not known in the scientific literature with a descriptive model (i.e., one, which describes the main underlying mechanisms in the process). We also study numerically and show how the model parameters affect the model solution. As a result, this work provides a computational framework, under which one can perform computer simulations, or process and analyze real experimental data of neuromuscular activity and in the same time relate the findings to main characteristics of the underlying processes. We believe that at present such a framework is missing, but very much needed, in order to computationally study and/or obtain valuable hypotheses for many aspects of the motor unit activity, such as, e.g., neuromuscular disorders.
This work examines a predator-prey model that features multiple ecological mechanisms, along with Allee effect, fear effect, and time delay, with a focus on how time delay influences the model's dynamics behaviors. To begin, we analyze the Hopf bifurcation with non-delay model. Subsequently, by choosing time delay as the bifurcation parameter, we obtain the existence of Hopf bifurcation. Moreover, the direction, stability and periodicity of Hopf bifurcation are determined by the normal form theorem and the center mainfold theorem. Finally, numerical simulations are offered to further confirm these theoretical findings.
In this work, we are interested in the impact of climatic conditions and hibernation on the dynamics of insect pest populations. For this, we use an age-structured population dynamics model describing the biological cycle of the insect at its four stages of development (egg, larva, cocoon, adult). This model allows us to closely monitor the evolution of the population at the cocoon stage as a function of temperature, taking into account the fact that a part of the cocoon population always remains dormant. In the autonomous case, we present qualitative results on the asymptotic behavior of the model and illustrate these results with numerical simulations. We will use the developed model to simulate the dynamics of this pest population over a season with three different hibernation strategies, in order to highlight the evolution of the population as a function of climatic conditions and the different hibernation strategies of the insect.
Leptospirosis is a bacterial infection transmitted through contact with water contaminated by the urine of infected animals, often affecting both humans and animals, particularly in tropical and subtropical regions. This paper illustrates a stochastic model for the transmission dynamics of leptospirosis infections in both human and vector populations. The model enhances key factors such as rainfall impact, vector behavior, and human interactions to capture the transmission of the disease. Through a detailed mathematical investigation, the paper explains the criteria required for the existence and uniqueness of a global solution for the stochastic model. Nonlinear analysis notions are utilized to investigate the ergodic aspects of the stochastic model. To validate the model's effectiveness, numerical simulations are conducted, and the results are compared with deterministic behavior. The comparison highlights the importance of considering stochasticity in accurately capturing the dynamics of leptospirosis. The paper also illustrates the relevance of the proposed system by comparing the model's dynamics with real-world leptospirosis cases globally. This validation displays the model's ability to capture the epidemiological characteristics and trends observed in different geographical regions. Moreover, the impact of the rate of transmission of leptospirosis from an infected vector to a susceptible human on the evolution of infected individuals within the population is visualized, delivering insightful information. Major outputs of the study include the identification of critical parameters influencing disease spread and the demonstration of the stochastic model's superior accuracy over deterministic approaches in reflecting real-world scenarios.
The SARS-CoV-2 epidemic posed devastating effects on public health, and there is growing concern about the proliferation of misinformation and the role awareness plays in altering the dynamics of an epidemic. We formulate an SEIR-type model of SARS-CoV-2 with three susceptible compartments, consisting of individuals who are unaware, aware and misinformed. The media-dependent reproduction number of our model is computed, and the global stability of the disease-free equilibrium and the uniform persistence of SARS-CoV-2 are established. Unknown parameter values of our model are estimated, using data on the cumulative number of cases of symptomatic infectious humans. Numerical simulations suggest a decrease in prevalence when susceptible humans transition from misinformed to aware humans, and when aware susceptible humans send media-related messages on the spread and control of SARS-CoV-2. Furthermore, there is an increase in the media-dependent reproduction number with an increase in misinformation. Finally, an optimal control of awareness and misinformation problem is formulated and analyzed. The optimal control results suggest that awareness and adherence to control measures, as well as the enhancement of media intensity, are vital in reducing disease prevalence within the population.These results highlight the role of awareness and misinformation in disease dynamics.
In this paper, we consider a discrete predator-prey model with fear effects, prey refuge, harvesting and Crowley-Martin functional responses. First, we analyze the existence of the model's fixed points. Subsequently, we apply bifurcation theory to establish the conditions for the occurrence of codimension-two bifurcations, including fold-flip bifurcation, 1:1 strong resonance and 1:2 strong resonance. Finally, we verify our theoretical analysis through numerical simulations, presenting bifurcation diagrams, maximum Lyapunov exponents diagrams, phase portraits and the two-parameter spaces under the specified initial conditions.
Epidemic models are essential tools for understanding the spread of infectious diseases and evaluating containment measures. The COVID-19 pandemic highlighted the critical role of population movement in shaping epidemic dynamics, emphasizing the need for models that incorporate mobility effects. In this work, we study disease transmission between two interconnected populations using a stochastic framework. Building on a deterministic model, we introduce a continuous-time Markov chain stochastic model and compare it with three approximations. While continuous-time Markov chains provide a natural stochastic counterpart to deterministic models, they pose challenges in scalability and implementation for large systems. To address these issues, we explore simplified approximations that retain key stochastic features while reducing computational complexity. Our analysis focuses on the impact of movement on disease persistence, particularly in source-sink scenarios where one population serves as a reservoir of infection. We show that stochastic effects can lead to extinction events absent in deterministic models, underscoring the importance of randomness in epidemic forecasting. Numerical simulations illustrate each approach, providing insights into the interplay between mobility and epidemic spread.
This paper presents a comprehensive analysis of a time-delayed predator-prey model with dispersal among multiple patches, considering the critical threshold parameter, R-0, as the net reproduction number. Our research highlights the significance of R-0 in determining the global behavior of the system. We demonstrate that when R-0 < 1, the predator-free equilibrium is proven to be globally attractive, while for R-0 > 1, this equilibrium becomes unstable. Furthermore, we reveal that, under specific additional conditions, the predator-free equilibrium can exhibit local asymptotic stability when R-0 < 1. In cases where R-0 > 1, the model exhibits uniform persistence and supports the existence of at least one coexistence equilibrium. Our practical application in a two-patch environment with linear release rates of natural enemies yields several noteworthy insights. First, we observe that R-0 exhibits a gradual decrease with an increase in maturation delay, indicating the impact of this delay on the system's stability. Second, we highlight that for cases where R-0> 1, an increase in maturation delay can lead to system destabilization, posing challenges for effective pest control. Finally, we emphasize that alterations in the dispersal rates of both prey and predator species can have a profound influence on the survival or extinction of the predator population. Notably, our findings indicate that increasing the release rates of natural enemies may not always be the optimal strategy for pest control due to the complex interplay of dispersal effects. Our research provides valuable insights into the management of predator-prey interactions in multi-patch environments, offering guidance for more effective ecological pest control strategies.
Objective: This study aims to investigate the mechanisms underlying the progression of hypertension and to evaluate the effectiveness of various intervention strategies. Methods: A four-state dynamic model was developed to describe the natural progression of hypertension, considering clinical variations in blood pressure, disease trajectory, and the patient's life course. Hypertension prevalence data in China from 2002 to 2018 were utilized to calibrate the model and estimate parameters using a Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm. The equilibrium and stability of the system were analyzed mathematically. The model was then employed to forecast future epidemiological trends in hypertension prevalence in China. Sensitivity analyses and numerical simulations were conducted to evaluate the effectiveness of diverse intervention strategies by examining changes in prevalence and disease states. Results: The model demonstrated a strong fit to the observed data (R-2 = 99%). Projections indicate a concerning upward trend, with hypertension prevalence in China forecast to rise from 30.2% in 2022 to over 40% by 2045. Simulation of interventions, however, shows that reducing disease progression and enhancing regression could lower prevalence to approximately 23-29% by 2030. An integrated strategy, featuring a 60% reduction in both prehypertension and hypertension progression rates coupled with a 45% increase in prehypertension regression, would align with WHO management targets. Conclusion: The increasing prevalence of hypertension in China can be effectively mitigated by reducing disease progression rates and strengthening prehypertension management. Attaining the WHO's 2010-2030 control objectives, however, necessitates a comprehensive and coordinated set of interventions.
Understanding the dynamics of COVID-19 requires models that go beyond classical assumptions. While various mathematical frameworks, including those based on evolutionary invasion analysis and multi-strain epidemic models, have the common limitation of treating virulence as a constant. In this study, we develop an advanced SIRD model that integrates time-dependent virulence to bridge the gap between evolutionary theory and epidemiological dynamics. We employ a finite difference method to compute virulence dynamically, using real-time data on daily infections and deaths to estimate mortality rates. This allows us to capture the evolving nature of virulence more realistically than traditional models. The basic reproduction number is derived and analyzed as a function of virulence, offering a nuanced perspective on transmission dynamics. Focusing on the United States data, we investigate the interplay and coexistence of multiple SARS-CoV-2 variants namely the original strain, Alpha, Delta, and Omicron. The model further quantifies the synergistic and antagonistic interactions among these strains. Their evolutionary fitness and invasion potential are assessed using invasion fitness functions and selection gradients, determining the intensity of invasion. Contour plots are used to visualize thresholds of the reproduction number, providing valuable insights into the evolutionary trajectories of viral strains. Our results emphasize the critical role of time-varying virulence and mutation-driven dynamics in shaping epidemic outcomes. The strains Alpha, Delta and resident strain interaction shows the synergism and Omicron shows the antagonism behavior. This study offers a robust modeling framework that can inform future research and support evidence-based public health strategies.
Leptospirosis is a zoonotic disease caused by Leptospira bacteria, that occurs mainly in tropical regions such as Thailand. This study presents a mathematical model that captures the dynamics of Leptospira transmission. The model incorporates direct and environmental transmission pathways and notably includes an exposed human compartment, an often-neglected element in existing leptospirosis models. We combined several control interventions, including environmental management, reservoir control, and human treatment to explore strategies for mitigating disease spread. The equilibrium points of the system are identified, and their stability properties are analyzed. Using real data from Thailand, we estimate key parameters and perform a global sensitivity analysis to identify the dominant factors driving leptospirosis transmission. Finally, we evaluate optimal control theory and conduct a comparative cost-effectiveness analysis of the proposed interventions. Our findings suggest that environmental management is the most effective and potentially cost-effective strategy to reduce leptospirosis transmission.
In this paper, we study transmission dynamics of viral infectious diseases co-infection model and evolution dynamics of different strains virulence. First, original strain transmission model is developed by considering immune loss and reinfection mechanisms, and stability of equilibrium in the model is obtained. When basic reproduction number [Formula: see text], a mutant strain is introduced at endemic equilibrium of original strain transmission model, and two strains co-infection model is established. Second, using invasion analysis method, we obtain evolutionary singularity strategy properties through fitness function and evolutionary dynamics of average virulence and average transmission rate using population genetic method. Finally, numerical simulations are performed with COVID-19 data in the United States to obtain transmission rate as a function of virulence, contour plots of fitness function and time series diagrams of average virulence and average transmission rate, which provide theoretical guidance for controlling development of COVID-19 outbreaks.
In the realm of ecology, the interplay between cooperative behaviors and the Allee effect represents a crucial nexus in understanding natural dynamics. These elements elucidate various mechanisms governing prey–predator intraspecific interactions. In this study, we introduce a two-dimensional prey–predator model that incorporates a strong Allee effect, impacting prey growth function and facilitating hunting cooperation among predators. Our proposed model innovatively integrates a density-dependent death rate term referred to as predator intraspecific competition, denoted as [Formula: see text], and employs a Holling type III functional response to enhance our analysis. The primary focus of this research is to explore how hunting cooperation influences the dynamics of the model, especially whenever the intraspecific predation affects the predator populations. Furthermore, we meticulously document the system’s dynamical behaviors through an extensive bifurcation analysis, using both one-parameter and two-parameter approaches. Through the rigorous local and global bifurcation analysis, we uncover a spectrum of significant bifurcations, including saddle-node bifurcation, Hopf bifurcation, BT bifurcation, and cusp bifurcation points (CP), alongside generalized Hopf (GH) bifurcations as local bifurcations, and a homoclinic bifurcation as a global bifurcation. This comprehensive examination not only contributes to the theoretical understanding of ecological interactions but also underscores the profound implications of cooperation and density-dependent effects in shaping ecosystem dynamics.
