Search Results (200)

Measles remains a significant public health threat despite widespread vaccination, with recent resurgences driven by vaccine hesitancy and coverage gaps. Existing mathematical models often fail to capture the substantial temporal heterogeneity in incubation periods, vaccine-induced protection, and recovery processes that characterize measles transmission. We develop and analyze an SVEIR epidemic model incorporating four independent distributed time delays with exponential survival factors, capturing the realistic variability in these epidemiological processes. The model features compartment-specific mortality rates, disease-induced mortality, and imperfect vaccination with failure probability $\theta$. Using next-generation matrix methods adapted for delay kernels, we derive the delay-dependent reproduction number ${\mathcal{R}}_0^d$ and prove, via systematic construction of Volterra-type Lyapunov functionals, that it constitutes a sharp threshold: the disease-free equilibrium is globally asymptotically stable when ${\mathcal{R}}_0^d\le 1$, while a unique endemic equilibrium emerges and is globally stable when ${\mathcal{R}}_0^d>1$. Normalized forward sensitivity analysis reveals that the transmission rate $\beta$ and recruitment rate $\Lambda$ exhibit maximal positive elasticity, while the vaccination rate p, vaccine failure probability $\theta$, and incubation delay ${\tau }_3$ possess the largest negative elasticities. Critically, ${\tau }_3$ exerts exponential influence via $e^{-n_3{\tau }_3}$, making interventions that delay infectiousness—such as post-exposure prophylaxis—unusually potent. We derive an explicit expression for the critical delay ${\tau }_3^{\mathrm{cr}}$ at which ${\mathcal{R}}_0^d=1$, demonstrating that prolonging the effective incubation period sufficiently can shift the system from endemic persistence to extinction. Numerical simulations using Dirac delta kernels confirm all theoretical predictions. These findings provide three actionable insights for public health: (1) maintaining high vaccination coverage among new birth cohorts remains paramount; (2) improving vaccine quality (reducing $\theta$) yields substantial returns; and (3) the incubation delay represents a quantifiable, measurable target for evaluating the population-level impact of time-sensitive interventions. The framework is broadly applicable to infectious diseases characterized by significant temporal heterogeneity. Full article

This study develops an integrated mathematical model to investigate the interaction between tuberculosis (TB) transmission and childhood stunting, which is aligned with the United Nations Sustainable Development Goals (SDG 3). The population is structured into two age groups (0–5 years and ≥5 years), with stunting explicitly incorporated into the pediatric population to capture its potential influence on TB dynamics. The model is formulated as a system of ordinary differential equations and analyzed using equilibrium and stability analysis, with the basic reproduction number, $R_0$. The disease-free equilibrium is locally asymptotically stable when $R_0<1$, while an endemic equilibrium exists when $R_0>1$. Sensitivity analysis indicates that the transmission rate ($\beta$), progression rate from latent to active infection ($\sigma$), and recovery rate ($\gamma$) are the most influential parameters affecting $R_0$. These parameters are therefore selected as control variables in an optimal control framework to design effective intervention strategies. Numerical simulations show that the combined control strategy significantly reduces TB transmission, resulting in a reduction of more than $80\%$ in active TB cases within a relatively short intervention period. The results suggest that integrated interventions targeting transmission, disease progression, and recovery are substantially more effective than single-measure strategies. This study provides a quantitative framework to support integrated public health policies addressing TB and childhood stunting simultaneously. Full article

In this study, we examine the effectiveness of combining interleukin-2 (IL-2) with highly active antiretroviral therapy (HAART) in controlling HIV replication. A mathematical model of the immune system is developed to analyze immune recovery when IL-2 is administered alongside HAART. We investigate the stability of the endemic equilibrium and Hopf bifurcation and determine the direction and stability of periodic solutions using center manifold theory. Numerical simulations are conducted to support the theoretical findings. The results show that the disease-free equilibrium is stable when the basic reproduction number $R_0<1$, while the endemic equilibrium exists when $R_0>1$. Our results also reveal the presence of a subcritical Hopf bifurcation in the system. An optimal control problem is also studied, showing that the combined therapy of IL-2 and HAART improves treatment outcomes, reduces side effects, and has a unique optimal control pair. Sensitivity analysis further highlights the importance of system parameters in influencing treatment effectiveness. Full article

Tuberculosis (TB) remains a major public health challenge in high-burden regions, where reinfection and seasonal variation play important roles in disease transmission. In this paper, we study a tuberculosis transmission model with reinfection based on the SIRI framework, with particular emphasis on the intrinsic relationship between the averaged system and the periodic system. The averaged system is shown to characterize the long-term epidemiological behavior, whereas the periodic system captures short-term seasonal fluctuations. From a theoretical perspective, we prove that the periodic system and its corresponding averaged system share the same basic reproduction number. We analyze the threshold dynamics of the seasonal model and investigate the dynamical properties of the averaged system, including the existence and stability of equilibria and the occurrence of backward bifurcation. In particular, we show that disease persistence may occur even when the basic reproduction number ($R_0$) is less than one, and we examine the stability of equilibrium points at the critical threshold ($R_0=1$). These results reveal how transmission and reinfection jointly determine the disease burden and equilibrium structure. To validate the theoretical findings, numerical simulations are performed using tuberculosis incidence data from Yunnan Province, China, covering the period from 2005 to 2020. The numerical simulations suggest that the seasonal model provides a better fit to the data, while the averaged model may overestimate the transmission potential of the disease. Under the condition that the two models share the same basic reproduction number, a constrained numerical simulation is performed. The results show that, under certain parameter settings, the endemic equilibrium of the averaged system can approximate the mean prevalence of the periodic solution. However, such an approximation cannot be guaranteed in general. Full article

This paper presents a novel mathematical framework for anthrax transmission by integrating Caputo fractional derivatives, distributed delays, and a Beddington–DeAngelis incidence function. The proposed model captures memory effects in disease progression, temporal heterogeneities in pathogen release, and saturation phenomena in host–pathogen interactions. We establish the well-posedness of the system and derive the basic reproduction number ${\mathcal{R}}_0$, which serves as a sharp threshold for disease dynamics: when ${\mathcal{R}}_0\le 1$, the disease-free equilibrium is globally asymptotically stable; when ${\mathcal{R}}_0>1$, a unique endemic equilibrium emerges and is globally stable. Theoretical analysis demonstrates that the fractional order modulates convergence rates through memory effects, while distributed delays influence oscillatory behaviors and time to equilibrium. Numerical simulations validate these findings and illustrate the impacts of key parameters on disease transmission. The results provide a scientific foundation for designing targeted public health interventions in anthrax control. Full article

This study formulates a deterministic model to assess the effect of a family-based control and management (FBCM) strategy against the transmission of Helicobacter pylori infection and its consequent development of gastric ulcers and gastric cancer. The model includes nine epidemiological compartments to model disease transmission and contact epidemiology between susceptible and infected individuals. In the model analysis, we compute positivity, the invariant region, equilibria, stabilities, and bifurcation analysis. We calculate the control reproduction number ${\mathcal{R}}_0$ and demonstrate that the model has a unique disease-free equilibrium (DFE) and an endemic equilibrium point (EEP) that are locally and globally stable for ${\mathcal{R}}_0<1$ and ${\mathcal{R}}_0>1$, respectively. We perform a thorough mathematical analysis and validate the model by fitting it to real data on gastric cancer cases recorded at Meru Teaching and Referral Hospital, Kenya. The best numerical results are achieved when we combine both preventive measures (sensitization and a family-based approach) and curative measures (prompt treatment and adherence), resulting in the greatest decrease in gastric ulcer and gastric cancer cases compared with a single intervention. This study shows that integrated household-level interventions can reduce transmission and prevent mild-to-severe disease progression through effective sensitization campaigns, high FBCM efficacy, effective gastric ulcer treatment, and adherence to drug protocols. The use of such strategies offers an effective means of reducing Helicobacter pylori-related gastric ulcers and gastric cancer outcomes, with important implications for public health control program design. Full article

Powdery Mildew is a global plant disease caused by fungal species, causing powdery growth on various parts of plants. This study aims to develop, evaluate and simulate the transmission dynamics of Powdery Mildew in cashew plants using a stochastic differential equation with Lévy noise. After providing some preliminary definitions of stochastic differential equations, we first consider the model without noise. We prove positivity, compute the basic reproduction number, ${\mathcal{R}}_0$, the PMD-free equilibrium, and the existence of a unique endemic equilibrium point whenever ${\mathcal{R}}_0>1$. After that, we formulate the stochastic model under Lévy noise. For this model, we also prove the positivity of the solutions and show that it is possible to extend the disease when ${\mathbb{S}}_s<1$. We also found the condition that ensures the persistence of the disease if ${\mathbb{S}}_0^s>1$. To simulate the model, we build a stochastic model numerical scheme and do a number of numerical simulations to support the theoretical findings we have gotten. Full article

This article introduces and thoroughly examines a novel deterministic compartmental model of COVID-19 dynamics. The model uniquely incorporates compartments for symptomatic and asymptomatic individuals alongside an environmental reservoir for the pathogen. It also accounts for a steady inflow of infected visitors and a steady outflow from the removed class. The mathematical soundness of the model is established by identifying the invariant region and ensuring positivity of solutions. Notably, during surges of infected visitors, certain classes maintain positive minimum values. We analytically determine endemic equilibrium points and prove the global stability of the disease-free equilibrium. Sensitivity analysis highlights the significant roles of transmission rates and asymptomatic individuals in disease spread. Simulation results corroborate the theoretical findings and provide additional insights into the model’s predictive capabilities. Full article

Yellow fever (YF) and malaria co-infections are real public health concerns in Africa, especially in countries such as Nigeria, where mosquitoes carrying both pathogens (Aedes for YF, Anopheles for malaria) coexist. A mathematical model that considers the critical factors influencing the transmission dynamics and control interventions of YF and malaria co-infections is formulated and used to analyse the problem. The essential dynamical features of the model, such as the basic reproduction number and disease-free equilibrium, are determined and analysed. The qualitative analysis of the model illustrates the conditions under which the disease can be eradicated or persists. Further analysis, supported by numerical simulations, reveals the intrinsic dynamics of the model and the impact of control interventions such as yellow fever vaccination, use of insecticide-treated mosquito nets, treatment of malaria-infected humans, and use of insecticides. The results of the analysis demonstrate the impact of interventions; specifically, effective implementations of interventions such as yellow fever vaccination, use of insecticide-treated mosquito nets, and use of insecticides appear to have a significant impact in eradicating YF and malaria co-infections in endemic areas. Effective treatment of malaria-infected humans may lead to a decrease in infections but might not necessarily lead to eradicating infections in endemic areas. These findings are expected to aid in improving the management of YF and malaria co-infections in endemic regions for expeditious disease eradication. Full article

Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of public health testing. We extend the standard SIR model to include compartments for ‘Confirmed’ (C) and ‘Monitored’ (M) individuals, resulting in a new SICMR model. By fitting the model to U.S. COVID-19 pandemic data (specifically the Omicron wave of late 2021), we demonstrate that capacity constraints in testing destabilize the testing-free endemic equilibrium ($E_1$). This equilibrium becomes an unstable saddle-focus. The instability is driven by a sociological feedback loop, where the rise in confirmed cases drive testing effort, modeled by a nonlinear Holling Type II functional response. We explicitly verify that the eigenvalues for the best-fit model satisfy the Shilnikov condition (${\lambda }_u>{\lambda }_s$), demonstrating the system possesses the necessary ingredients for complex, chaotic-like dynamics. Furthermore, we employ Stochastic Differential Equations (SDEs) to show that intrinsic noise interacts with this instability to generate ’noise-induced bursting,’ replicating the complex wave-like patterns observed in empirical data. Our results suggest that public health interventions, such as testing, are not merely passive controls but active dynamical variables that can fundamentally alter the qualitative stability of an epidemic. Full article

This paper presents a comprehensive mathematical analysis of a novel compartmental model describing the dynamics of dispersed water pollutants and their interaction with two distinct host populations. The model is formulated as a system of integro-differential equations that incorporates multiple distributed delays to realistically account for time lags in the infection process and pollutant transport. We rigorously establish the biological well-posedness of the model by proving the non-negativity and ultimate boundedness of solutions, confirming the existence of a positively invariant feasible region. The analysis characterizes the long-term behavior of the system through the derivation of the basic reproduction number ${\mathcal{R}}_0^d$, which serves as a sharp threshold determining the system’s fate. For the model without delays, we prove the global asymptotic stability of the infection-free equilibrium (IFE) when ${\mathcal{R}}_0\le 1$ and of the endemic equilibrium (EE) when ${\mathcal{R}}_0>1$. These stability results are extended to the distributed-delay model by using sophisticated Lyapunov functionals, demonstrating that ${\mathcal{R}}_0^d$ universally governs the global dynamics: the IFE (${\mathcal{E}}_0^d$) is globally asymptotically stable (GAS) if ${\mathcal{R}}_0^d\le 1$, while the EE (${\mathcal{E}}_d^{\ast }$) is GAS if ${\mathcal{R}}_0^d>1$. Numerical simulations validate the theoretical findings and provide further insights. Sensitivity analysis identifies the most influential parameters on ${\mathcal{R}}_0^d$, highlighting the recruitment rate of susceptible individuals, exposure rate, and pollutant shedding rate as key intervention targets. Furthermore, we investigate the impact of control measures, showing that treatment efficacy exceeding a critical value is sufficient for disease eradication. The analysis also reveals the inherent mitigating effect of the maturation delay, demonstrating that a delay longer than a critical duration can naturally suppress the outbreak. This work provides a robust mathematical framework for understanding and managing dispersed water pollution, emphasizing the critical roles of multi-source contributions, time delays, and targeted interventions for environmental sustainability. Full article

We present a comprehensive mathematical analysis of a within-host dual-target HIV dynamics model, which explicitly incorporates the virus’s interactions with its two primary cellular targets: CD4⁺ T cells and macrophages. The model is formulated as a system of five nonlinear delay differential equations, integrating three distinct discrete time delays to account for critical intracellular processes such as the development of productively infected cells and the maturation of new virions. We first establish the model’s biological well-posedness by proving the non-negativity and boundedness of solutions, ensuring all trajectories remain within a feasible region. The basic reproduction number, ${\mathcal{R}}_0^d$, is derived using the next-generation matrix method and serves as a sharp threshold for disease dynamics. Analytical results demonstrate that the infection-free equilibrium is globally asymptotically stable (GAS) when ${\mathcal{R}}_0^d\le 1$, guaranteeing viral eradication from any initial state. Conversely, when ${\mathcal{R}}_0^d>1$, a unique endemic equilibrium emerges and is proven to be GAS, representing a state of chronic infection. These global stability properties are rigorously established for both the non-delayed and delayed systems using carefully constructed Lyapunov functions and functionals, coupled with LaSalle’s invariance principle. A sensitivity analysis identifies viral production rates ($p_1,p_2$) and infection rates (${\beta }_1,{\beta }_2$) as the most influential parameters on ${\mathcal{R}}_0^d$, while the viral clearance rate (m) and maturation delay (${\tau }_3$) have a suppressive effect. The model is extended to evaluate antiretroviral therapy (ART), revealing a critical treatment efficacy threshold ${ϵ}^{cr}$ required to suppress the virus. Numerical simulations validate all theoretical findings and further investigate the dynamics under varying treatment efficacies and maturation delays, highlighting how these factors can shift the system from persistence to clearance. This study provides a rigorous mathematical framework for understanding HIV dynamics, with actionable insights for designing targeted treatment protocols aimed at achieving viral suppression. Full article

This paper is dedicated to the analytical investigation of the global dynamics of an SEIR epidemiological model that incorporates latency age (the time spent by an individual in the exposed class before becoming infectious) and a general nonlinear incidence rate. In this model, to reflect the dependence of disease progress on the latency age, the exposed class is structured by the latency age, and the rate at which the latent individual becomes infected, and the removal rate are assumed to depend on the latency age. By analyzing the characteristic equations associated with each equilibrium, we study the local stability of both the disease-free and endemic steady states of the model. Moreover, it is proven that the semiflow generated by this system is asymptotically smooth, and if the basic reproduction number is greater than unity, the system is uniformly persistent. Furthermore, based on Lyapunov functional and LaSalle’s invariance principle, the global dynamics of the model are established. It is obtained that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable and hence the disease dies out; however, if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable, and the disease persists. Numerical simulations are carried out to illustrate the main analytic results. Full article

Tomato Yellow Leaf Curl Virus (TYLCV) has recently caused severe economic losses in global tomato production. According to the International Plant Protection Convention (IPPC), yield reductions of 50–60% have been reported in several regions, including the Caribbean, Central America, and South Asia, with losses in sensitive cultivars reaching up to 90–100%. In developing countries, TYLCV and mixed infections affect more than seven million hectares of tomato-growing land annually. In this study, we construct and analyze a nonlinear dynamic model describing the transmission of TYLCV, incorporating the Caputo fractional-order derivative operator. The existence and uniqueness of solutions to the proposed model are rigorously established. Equilibrium points are identified, and the Jacobian determinant approach is applied to compute the basic reproduction number, $R_0$. Suitable Lyapunov functions are formulated to analyze the global asymptotic stability of both the disease-free and endemic equilibria. The model is numerically solved using the Grünwald–Letnikov-based nonstandard finite difference method, and simulations assess how the memory index and preventive strategies influence disease propagation. The results reveal critical factors governing TYLCV transmission and suggest effective intervention measures to guide sustainable crop protection policies. Full article

This study proposes an SVEIR with a reinfection model of tuberculosis disease spread to examine the impact of saturated infection and imperfect vaccination. Vaccinated individuals are considered vulnerable, as they are still likely to be reinfected. As the recovered individuals still have bacteria in their bodies, they are likely to return to their latent class. The dynamic behavior of the proposed model was analyzed to understand both the local and global stability equilibrium points. To analyze the disease-free and endemic equilibrium stability, the Routh–Hurwitz Criterion and Center Manifold theorems were used, respectively. The local and global stability equilibrium state is entirely dependent on the effective reproduction number. If the effective reproduction number is less than one, the disease-free equilibrium point is locally and globally asymptotically stable, whereas if it is greater than one, the endemic equilibrium point is locally asymptotically stable. Numerical simulations show the time series of the solution of the model, phase-plane trajectory, elasticity indices, bifurcation diagram, partial rank correlation coefficients, and the sensitivity of the infected class to variations in the transmission rate represented both in the peak value and a heatmap. Furthermore, the contour plot illustrates that the disease transmission rate affects the effective reproduction number and the stability of equilibrium points. Full article