This paper proposes a within-host Mycobacterium tuberculosis (Mtb) infection model integrating M1/M2 macrophage conversion to explore macrophage polarization regulation in pulmonary infection with cellular immunity. The basic reproduction number R0 is calculated, and criteria on the locally and globally asymptotic stability of infection-free equilibrium E0 are established if R0<1. Criteria on the existence and locally asymptotic stability of immune-free infection equilibrium E¯ and immune-activated infection equilibrium E∗ are further obtained if R0>1. Moreover, the model exhibits forward, backward and saddle-node bifurcations at E¯, as well as Hopf bifurcation and chaotic dynamics at E∗, indicating that Mtb infection is highly sensitive to the dynamic shifts during macrophage polarization. Numerical simulations verify theoretical results, revealing that the effective conversion rates γ1 and γ2 between M1 and M2 macrophages are key regulators of Mtb infection progression, which may provide theoretical insights into tuberculosis latency and relapse, and offer testable predictions for immunotherapeutic strategies.
The threshold interpretation of R0 can fail to describe elimination in epidemic models with imperfect vaccination and reinfection. We study an SEIRV model in which vaccinated individuals remain partially susceptible and recovered individuals may be reinfected. The main novelty is an explicit threshold account of how reinfection can sustain endemic equilibria even when the vaccinated-population reproduction number satisfies R0<1. We derive the disease-free equilibrium, R0, and a sufficient condition for convergence to the disease-free state, showing why local disease-free stability and global elimination are distinct. We reduce the endemic-equilibrium problem to a scalar equation and identify a reinfection threshold β2∗ determining the local direction of bifurcation at R0=1. When β2>β2∗, we prove a lower critical threshold R0C<1 separating elimination from subcritical endemic persistence. We also give an efficient-vaccine approximation and general numerical algorithm for computing R0C. Simulations using an illustrative Omicron-era parameterisation show how these thresholds organise the model dynamics.
Using an appropriate mathematical model, this study aims to examine the transmission dynamics and optimal control of hepatitis B virus spreading, employing the Beddington-DeAngelis incidence function, a hybrid method of 4th order Runge-Kutta method (RK4) and feed-forward neural network (FFNN), as the integration of epidemiological models with neural network is particularly important for representing disease propagation. The well-posedness, local and global stability conditions are obtained using the threshold parameter. Some sensitive epidemic parameters and their relative impacts are quantified. Based on the local and global properties of the model and sensitivity analysis, a control problem is formulated to control the infection by minimizing the HBV-infected population and maximizing the recovered population using three control measures. Finally, a hybrid method of supervised FFNN and RK4 with two hidden layers is used to effectively approximate the temporal dynamics of HBV transmission and verify the theoretical results, as well as the effects of controls.
A mathematical model for Moringa was developed and analyzed in this study to manage the plantation's natural pests through integrated pest management (IPM). Bio-pesticides are expensive, difficult to apply and need a lengthy process. However, this method will be more efficient and less costly if chemical pesticides are added to the farming system in addition to biopesticides. Here, we examine the use of integrated pest management (IPM) combined with chemical and biological pesticides. We determine which parameter can lead to stability switches. By utilizing optimal control theory, the ideal concentration profile for both pesticides has been established to reduce side effects and maximize process economy. Numerical simulations justify the primary results.
During the recent pandemic, a rise in COVID-19 cases was followed by a decline in influenza. In the absence of cross-immunity, a potential explanation for the observed pattern is behavioural: non-pharmaceutical interventions (NPIs) designed and promoted for one disease also reduce the spread of others. We study short-term and long-term dynamics of two pathogens where NPIs targeting one pathogen indirectly influence the spread of another - a phenomenon we term behavioural spillover. We examine how perceived risk of and response to one disease substantially alter the spread of other pathogens, revealing how waves of different pathogens emerge over time as a result of behavioural interdependencies and human response. Our analysis identifies the parameter space where two diseases simultaneously co-exist, and where shifts in prevalence occur. Our findings are consistent with observations from the COVID-19 pandemic, where NPIs contributed to significant declines in infections such as influenza, pneumonia, and Lyme disease.
In this paper, we investigate the propagation speed of bistable traveling waves in a diffusive three-species Lotka-Volterra competition model. By constructing a novel pair of upper and lower solutions, we derive sufficient conditions that determine whether the bistable wave speed is positive or negative. These criteria offer a theoretical foundation for predicting the direction of wave propagation in such systems. The validity and applicability of our theoretical results are further confirmed through several numerical examples.
To study the macroscopic dynamics of susceptible host cells, infected host cells, free virus particles and antibodies after viral infection within-host, this study develops a novel dynamic model that integrates three key mechanisms: a general incidence function capturing the complexity of antibody production, the inhibitory effect of antibodies on viral infectivity and the cytokine-mediated self-cure of infected cells. The basic reproduction numbers for both the virus and immune response are derived, along with sufficient conditions for the stability of equilibria. Bifurcation analysis revealed that a Hopf bifurcation may occur when the basic reproduction number of the immune response exceeds one. Numerical simulations highlight the critical role of saturation effects in viral replication and the immune response for infection control. Antibody immunity, once depleted, may not be replenished and neglecting saturation effects could overestimate both the oscillatory parameter range and the severity of infection.
We study periodic dynamics and error-threshold behavior in a delayed quasispecies model consisting of a master sequence (x0) and two mutant populations (x1,x2). The system, formulated as delay differential equations with time-periodic replication rates, yields new conditions for the existence and absence of T-periodic solutions. Using topological degree arguments, we show that when mutation probabilities (Qji) lie strictly between 0 and 1 and at least one fitness function (fj) is periodic, the system supports nontrivial positive periodic orbits, with or without backward mutations. This shows that fluctuating environments, such as circadian or treatment-induced cycles, can sustain oscillatory genotype distributions. Conversely, if mutations are strictly unidirectional and the master sequence is consistently dominated in fitness, no positive T-periodic orbit arises. In this regime, the master sequence decays monotonically to extinction without time delays, while time delays induce non-monotonic decay, recovering the classical error-threshold phenomenon and linking it to cancer-related quasispecies dynamics.
Hepatitis B transmission is influenced by environmental variability, immune response and vaccination, making its spread inherently stochastic, while the integration of stochastic modeling with machine learning is particularly important for representing disease uncertainty in varied surroundings. We present a hybrid innovative framework that combines a stochastic epidemiological model with a feed-forward neural network to study hepatitis B virus (HBV) dynamics. We show the well-posedness by establishing the existence of solutions with uniqueness, and analyze extinction and persistence of the disease. In addition, a supervised approach of feed-forward neural network (FFNN) having two hidden layers, each consist of 20 neurons will be used to effectively approximate the dynamics of the model. To evaluate robustness of the network while handling the stochastic model, we evaluate the performance by regression and mean squared error (MSE), and to show a strong alliance among the stochastic trajectories and predicted simulations obtained by the neural network.
This work examines the dynamics of a discrete-time plankton interaction model, in which phytoplankton generate toxins and are vulnerable to external contamination. The model includes a Holling Type-II predation response and uses a piecewise constant argument approach to break it up into smaller pieces. This keeps the ecological realism of the continuous system while making it possible to study complex discrete-time behaviors. Our focus is on the formation of Neimark-Sacker bifurcation, a phenomena associated with the initiation of quasi-periodic oscillations in population densities. We show how toxin buildup and outside contamination can make plankton populations unstable, which could cause blooms to happen in an irregular way, using stability analysis and numerical simulations. The results show how useful discrete-time models are for capturing rapid changes in ecosystems, such damaging algal blooms. They also give ideas for managing ecosystems and reducing blooms.
Based on the considerations of round-trip in the treatment process, this paper presents a mathematical model aimed at studying the dynamic behaviour and epidemiological trends of HIV/AIDS. We first calculate the basic reproduction number R̅0 and discuss the stability of equilibrium points and the existence of forward bifurcations, validating the theoretical results through numerical simulations. Subsequently, using cumulative HIV/AIDS case data reported in China, we estimate model parameters using the least squares method, achieving a good fit. Furthermore, sensitivity analyses were performed on the model parameters to explain the dependence of the parameters on the infection variables. Finally, the model is applied to evaluate the control effects of treatment coverage at different stages of infection. The results suggest that reducing HIV/AIDS exposure, improving HIV/AIDS screening, promoting infectious disease treatment and increasing disease prevention awareness are the most effective measures to prevent HIV/AIDS infection.
This study examines competition models based on the Lotka-Volterra form that incorporate starvation-driven diffusions (SDD). Such dispersal assumes that species disperse in response to resource abundance or scarcity in a heterogeneous habitat. The primary objective of this study is to examine how SDD, in combination with diverse interspecific interactions, affects species’ fitness and coexistence states. To this end, the study introduces a refined classification for competing interactions based on a novel metric that quantifies the variability of resource heterogeneity across the environment. This approach contrasts with traditional models that assume uniform diffusion within homogeneous environments. This study investigates the local stability of two semitrivial steady states and establishes the existence and uniqueness of positive steady states by eigenvalue analysis and monotone dynamical systems theory. Through this analytical exploration, the study reveals that the interplay between species’ dispersal strategies and the varying intensities of interspecific competition significantly impacts ecological outcomes.
COVID-19 infection exhibits significant age-related differences. In this paper, we consider an infectious disease model with age-structure in susceptibility and evolutionary game and analyze the impact of mandatory and voluntary vaccination strategies on disease progression. We derive the conditions for the existence of equilibria and confirm that the basic reproduction number R0 serves as a threshold parameter that fully determines the dynamical properties of the model. Theoretical analyses indicate that the persistence of COVID-19 is contingent upon the value of the basic reproduction number. By conducting numerical simulations, we investigate the impacts of various factors, including relative vaccine cost and vaccine effectiveness, on disease dynamics under a voluntary vaccination policy. Our analysis reveals that enhancing vaccine effectiveness does not reduce disease transmission when vaccination rates are extremely low. Under voluntary vaccination policies, it is crucial to keep relative vaccine costs below a certain threshold to promote higher vaccination uptake.
In this paper, we introduce a mathematical simulation that captures the dynamics of lumpy skin disease (LSD) by considering three key transmission paths: vector-borne, direct cattle-to-cattle contact and environmental contamination. Additionally, this model incorporates three control measures, including vector control, environmental management and isolation/treatment of infected cattle. We perform a comprehensive mathematical analysis to demonstrate the model well-posedness, like proving the existence, uniqueness, positivity and boundedness of the solution. The basic reproduction number (R0) is calculated. The local and global stability analysis is presented for the disease-free and endemic equilibrium points. Sensitivity analysis for the model parameters is shown, which reveals that isolation and treatment control measures are the most effective in eliminating disease transmission. We construct an objective function to formulate an optimal control problem (OCP) and derive the optimality necessary conditions. Numerical simulations confirm the theoretical findings, demonstrating that strategic implementation of combined control measures can efficiently suppress LSD.
Human T-lymphotropic virus (HTLV) and human immunodeficiency virus (HIV) are two retroviruses that pose a certain threat to human psychology and physiology. In this paper, we propose a diffusive HTLV and HIV coinfection model with macrophages, two delays, cell-to-cell transmission and three latently infected cells in which latent HIV infected CD4+T cells, latent HIV infected macrophages, and latent HTLV infected CD4+T cells are considered. Four reproduction number and four equilibria, namely, infection-free equilibrium, HIV infection equilibrium, HTLV infection equilibrium and HTLV and HIV coinfection equilibrium, are calculated and proved the global asymptotic stability of the coinfection model. Numerical simulations are executed to showcase the corresponding theoretical outcomes and uncover how macrophages and latently infected cells influence the dynamics of HTLV and HIV coinfection.
We extend the predator-prey model developed by Ackleh et al. [Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics. J. Differ. Equ. Appl. 2019;25:1568-1603] to incorporate the evolution of a predator’s resistance to toxicant effects. We consider three cases: (1) lethal effects, where the toxicant directly influences the predator’s survival; (2) sublethal effects, where the toxicant impacts the predator’s fecundity, and (3) mixed effects, where the toxicant impacts both vital rates. For the first two cases, we derive conditions for existence and stability of model equilibria and for system persistence. These cases are also analyzed numerically to further understand the system dynamics. Overall, we find that evolution of a predator to resist a toxicant may allow for predator survival when otherwise it would have faced extinction. However, evolution in response to lethal effects can generate multiple boundary equilibria, leading to alternative stable states. When this occurs, evolution in response to a toxicant may result in the extinction of the predator while, without evolution, the predator survives.
Determining optimal antibiotic dosing strategies is complex. Clinically, some antibiotics work best in continuous low doses, while others require high repeated pulses. However, a rational understanding of the best approach depending on the specific pairing of antibiotics and bacterial species remains unclear. Using mathematical models, we analyze bacterial populations under two strategies—constant concentration and repeated dosing—with fixed pharmacodynamic and pharmacokinetic properties. Our results reveal that the shape of the dose‒response curve, which measures the bacterial net growth rate against the antibiotic concentration, is crucial. Specifically, its concavity determines the best strategy. In cases where the curve exhibits multiple concavities, additional factors, such as the tolerable dosing range, influence the regimen. These findings challenge the universal application of ‘hit hard and hit early’, as some recommended schedules include lower, constant doses. This work contributes to the literature on rational antibiotic prescription, aiming to minimize antibiotic use and combat antimicrobial resistance.
In this paper, we analyze a deterministic model of malaria and Corona Virus Disease 2019 co-infection within a homogeneous population. We first studied the single infection model of each disease and then the co-infection dynamics. We calculate the basic reproduction number of each model and study the existence and stability of the steady states. Then, we show that under some suitable conditions, both the malaria single infection model and co-infection model exhibit backward bifurcation. Furthermore, we analyze the conditions of extinction, competitive exclusion and co-existence of these two diseases. In addition, a local sensitivity analysis of the basic reproduction numbers is performed to explore the impact of the different parameters’ variability on the dynamics of each disease. Moreover, we apply Pontryagin’s maximum principle to determine optimal strategies to control the two diseases in case of co-infection. Finally, some numerical simulation results are presented to support the theoretical findings.
This paper examines a three-species ecological competition model with two predators and one prey, incorporating food-limited growth and both linear and quadratic harvesting strategies. Using mathematical analysis, we identify equilibrium points and derive conditions for their stability and persistence. The results reveal that quadratic harvesting significantly enhances stability, promotes coexistence, and mitigates extinction risks compared to linear harvesting. Numerical simulations validate the theoretical findings, highlighting the effectiveness of quadratic harvesting in managing population dynamics. These insights contribute to the mathematical understanding of sustainable harvesting strategies in complex ecological systems.
This paper proposes an immunosuppressive infection model with time delay and stochastic perturbation. A stochastic threshold R0s is constructed, and the sufficient conditions for virus extinction and weak persistence are given. Subsequently, we respectively fit the SDDE and ODE models to the real data, and conduct a sensitivity analysis of the equilibrium. The greater the noise intensity, the more obvious the oscillation amplitude of the solution curve around the immune-free equilibrium, the larger noise intensity can cause the originally persistent virus in ODE to go extinct in the SDDE model. The greater the time delay, the longer it takes for the virus and immune cells to reach their first peaks. The viral replication rate significantly affects the virus-immune system, and the reproduction of HIV-1 can be inhibited by modulating it. A relatively high viral inhibition rate will lead to the extinction of immune cells while the virus persists.
