Search Results (306)

Monkeypox is a viral disease belonging to the smallpox family. Although it has milder symptoms than smallpox in humans, it has become a global threat in recent years, especially in African countries. Initially, incidental immunity against monkeypox was provided by smallpox vaccines. However, the eradication of smallpox over time and thus the lack of vaccination has led to the widespread and clinical importance of monkeypox. Although mathematical epidemiology research on the disease is complementary to clinical studies, it has attracted attention in the last few years. The present study aims to discuss the indispensable effects of three control strategies such as vaccination, treatment, and quarantine to prevent the monkeypox epidemic modeled via the Atangana–Baleanu operator. The main purpose is to determine optimal control measures planned to reduce the rates of exposed and infected individuals at the minimum costs. For the controlled model, the existence-uniqueness of the solutions, stability, and sensitivity analysis, and numerical optimal solutions are exhibited. The optimal system is numerically solved using the Adams-type predictor–corrector method. In the numerical simulations, the efficacy of the vaccination, treatment, and quarantine controls is evaluated in separate analyzes as single-, double-, and triple-control strategies. The results demonstrate that the most effective strategy for achieving the aimed outcome is the simultaneous application of vaccination, treatment, and quarantine controls. Full article

Motivation: We aim to study the boundary stability and persistence of positive odes in mathematical epidemiology models by importing structural tools from chemical reaction networks. This is largely a review work, which attempts to congregate the fields of mathematical epidemiology (ME), and chemical reaction networks (CRNs), based on several observations. We started by observing that epidemiologic strains, defined as disjoint blocks in either the Jacobian on the infected variables, or as blocks in the next generating matrix (NGM), coincide in most of the examples we studied, with either the set of critical minimal siphons or with the set of minimal autocatalytic sets (cores) in an underlying CRN. We leveraged this to provide a definition of the disease-free equilibrium (DFE) face/infected set as the union of either all minimal siphons, or of all cores (they always coincide in our examples). Next, we provide a proposed definition of ME models, as models which have a unique boundary fixed point on the DFE face, and for which the Jacobian of the infected subnetwork admits a regular splitting, which allows defining the famous next generating matrix. We then define the interaction graph on minimal siphons (IGMS), whose vertices are minimal siphons, and whose edges indicate the existence of reactions producing species in one siphon from species in another. When this graph is acyclic, we say the model exhibits an Acyclic Minimal Siphon Decomposition (AMSD). For AMSD models whose minimal siphons partition the infection species, we show that the NGM is block triangular after permutation, which implies the classical max structure of the reproduction number $R_0$ for multi-strain models. In conclusion, using irreversible reaction networks, minimal siphons and acyclic siphon decompositions, we provide a natural bridge from CRN to ME. We implement algorithms to compute IGMS and detect AMSD in our Epid-CRN Mathematica package (which already contain modules to identify minimal siphons, criticality, drainability, self-replicability, etc.). Finally, we illustrate on several multi-strain ME examples how the block structure induced by AMSD, and the ME reproduction functions, allow expressing boundary stability and persistence conditions by comparing growth numbers to 1, as customary in ME. Note that while not addressing the general Persistence Conjecture mentioned in the title, our work provides a systematic method for deriving boundary instability conditions for a significant class of structured models. 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

In this article, mathematical modeling and stability analysis of Lyme disease and its transmission dynamics using Caputo fractional-order derivatives is presented. The compartmental model has been formulated to analyze the spread of Borrelia burgdorferi virus through tick vectors and mammalian hosts. The feasible region is established, and the boundedness of the model is verified. Analytically, the disease-free equilibrium and the basic reproduction number $\left({\Re }_0\right)$ has been determined to assess outbreak potential. By virtue of the fixed-point theory, the existence and uniqueness of solutions has been established. The numerical simulations are obtained via the Runge–Kutta 4 method, demonstrating the model’s ability to capture realistic disease progression. Finally, sensitivity analysis and control strategies (tick population reduction, host vaccination, public awareness, and early treatment) are evaluated, revealing that integrated control measures significantly reduce infection rates and enhance recovery. Full article

Rabies remains a persistent zoonotic threat, particularly in regions where domestic dogs are the main source of human and animal infections. This mathematical model studies the dynamics of rabies transmission between canine populations (dog-to-dog) and from canines to humans (dog-to-human). The model incorporates susceptible, infected, and vaccinated compartments for both species, with pre-exposure vaccination as the key control strategy. Processes such as encapsulation, stability enhancement, and controlled release are modelled as parameters influencing vaccination rates in both dogs and humans. Specifically, the model introduces processing-dependent vaccination functions that reflect improved bioavailability, immunogenicity, and delivery efficiency due to advanced formulation techniques. This interdisciplinary approach bridges mathematical epidemiology and pharmaceutical technology. Earlier rabies models focus on transmission and static vaccination, often ignoring vaccine formulation and delivery. Our current work fills this gap by incorporating pharmaceutical and particle engineering parameters into the vaccination terms of the model, thereby providing a more comprehensive framework for optimizing rabies control strategies in endemic regions. Positivity and boundedness analyses confirm that all model variables remain biologically feasible and bounded over time. Stability analysis identifies thresholds for disease elimination or persistence. Numerical simulations show that enhancing pharmaceutical parameters increases vaccination impact, reducing peak infection prevalence in dogs from 18% to 5% and in humans from 4% to 0.8%, and shortening elimination time from 8 years to 3 years. Formulations with controlled release and improved stability maintain over 90% reduction in transmission for more than 5 years, compared to 60% over 3 years for conventional vaccines. This will ensure that the model’s predictions are validated against realistic conditions and can effectively guide rabies control strategies. Full article

Lassa fever remains a significant zoonotic threat in West Africa, characterized by complex human-to-human and rodent-to-human transmission pathways and prolonged immune responses. Existing integer-order models often neglect the long-term memory and delayed recovery effects inherent to the disease. In this study, we develop and analyze a fractional-order Caputo model for Lassa fever transmission incorporating disability feedback among recovered individuals. The model captures memory-dependent infection and recovery dynamics, offering a more realistic description of epidemic persistence. The model is symmetric when the fractional approach to unity where it recovers its classical ODE counterpart. Analytical results establish the positivity, boundedness, existence, and uniqueness of solutions, while Picard stability and contraction mapping confirm well-posedness within the fractional framework. A Grünwald–Letnikov discretization scheme is constructed for numerical simulation, validated under varying fractional orders ($\lambda \in [0.7,1]$). The results reveal that decreasing the fractional order slows the infection decay rate and prolongs epidemic duration, highlighting the biological significance of memory effects. A global sensitivity analysis based on Latin Hypercube Sampling and Partial Rank Correlation Coefficients (LHS–PRCC) identifies the rodent-to-human transmission rate (${\kappa }_1$), human-to-human transmission rate (${\eta }_1$), and rodent interaction rate (${\xi }_r$) as the most influential parameters. These findings provide critical insight into the control and management of Lassa fever through rodent population control, improved recovery rates, and early human intervention. The fractional-order formulation thus extends existing models both mathematically and epidemiologically by capturing delayed dynamics and disability-induced feedback mechanisms. Full article

Most existing fractional models of COVID-19 describe only the infection process without explicitly accounting for the role of vaccination. In this study, a refined Caputo fractional model is proposed that incorporates a vaccinated class to better understand how immunization influences disease progression. The mathematical formulation guarantees the existence, uniqueness, and positivity of solutions, ensuring that all system trajectories remain biologically valid. The equilibrium points are obtained, and the reproduction number is derived to identify the conditions for disease control. The stability investigation covers local behavior alongside Ulam–Hyers and its extended variants, ensuring the system remains stable under small perturbations. Numerical experiments performed with the Adams–Bashforth–Moulton algorithm illustrate that vaccination reduces infection peaks and shortens the epidemic duration. Overall, the proposed framework enriches fractional epidemiological modeling by providing deeper insight into the combined effects of memory and vaccination in controlling infectious diseases. Full article

Background/Objectives: Machine learning is an extremely important issue, considering the potential to prevent the onset of long-term complications from coronavirus disease or to ensure timely detection and effective treatment. The aim of our study was to develop an algorithm and mathematical model to predict the risk of developing long COVID in children who have had acute SARS-CoV-2 viral infection, taking into account a wide range of demographic, clinical, and laboratory parameters. Methods: We conducted a cross-sectional study involving 305 pediatric patients aged from 1 month to 18 years who had recovered from acute SARS-CoV-2 infection. To perform a detailed analysis of the factors influencing the development of long-term consequences of coronavirus disease in children, two models were created. The first model included basic demographic and clinical characteristics of the acute SARS-CoV-2 infection, as well as serum levels of vitamin D and zinc for all patients from both groups. The second model, in addition to the aforementioned parameters, also incorporated laboratory test results and included only hospitalized patients. Results: Among 265 children, 138 patients (52.0%) developed long COVID, and the remaining 127 (48.0%) fully recovered. We included 36 risk factors of developing long COVID in children (DLCC) in model 1, including non-hospitalized patients, and 58 predictors in model 2, excluding them. These included demographic characteristics of the children, major comorbid conditions, main symptoms and course of acute SARS-CoV-2 infection, and main parameters of complete blood count and coagulation profile. In the first model, which accounted for non-hospitalized patients, multivariate regression analysis identified obesity, a history of allergic disorders, and serum vitamin D deficiency as significant predictors of long COVID development. In the second model, limited to hospitalized patients, significant risk factors for long-term sequelae of acute SARS-CoV-2 infection included fever and the presence of ≥3 symptoms during the acute phase, a history of allergic conditions, thrombocytosis, neutrophilia, and altered prothrombin time, as determined by multivariate regression analysis. To assess the acceptability of the model as a whole, an ANOVA analysis was performed. Based on this method, it can be concluded that the model for predicting the risk of developing long COVID in children is highly acceptable, since the significance level is p < 0.001, and the model itself will perform better than a simple prediction using average values. Conclusions: The results of multivariate regression analysis demonstrated that the presence of a burdened comorbid background—specifically obesity and allergic pathology—fever during the acute phase of the disease or the presence of three or more symptoms, as well as laboratory abnormalities including thrombocytosis, neutrophilia, alterations in prothrombin time (either shortened or prolonged), and reduced serum vitamin D levels, are predictors of long COVID development among pediatric patients. Full article

Sandflies spread the neglected vector-borne disease anthroponotic cutaneous leishmaniasis (ACL), which only affects humans. Despite decades of control, asymptomatic carriers, vector pesticide resistance, and low public awareness prevent eradication. This study proposes a fractional-order optimal control model that integrates biological and behavioral aspects of ACL transmission to better understand its complex dynamics and intervention responses. We model asymptomatic human illnesses, insecticide-resistant sandflies, and a dynamic awareness function under public health campaigns and collective behavioral memory. Four time-dependent control variables—symptomatic treatment, pesticide spraying, bed net use, and awareness promotion—are introduced under a shared budget constraint to reflect public health resource constraints. In addition, Caputo fractional derivatives incorporate memory-dependent processes and hereditary effects, allowing for epidemic and behavioral states to depend on prior infections and interventions; on the other hand, standard integer-order frameworks miss temporal smoothness, delayed responses, and persistence effects from this memory feature, which affect optimal control trajectories. Next, we determine the optimality conditions for fractional-order systems using a generalized Pontryagin’s maximum principle, then solve the state–adjoint equations numerically with an efficient forward–backward sweep approach. Simulations show that fractional (memory-based) dynamics capture behavioral inertia and cumulative public response, improving awareness and treatment efforts. Furthermore, sensitivity tests indicate that integer-order models do not predict the optimal allocation of limited resources, highlighting memory effects in epidemiological decision-making. Consequently, the proposed method provides a realistic and flexible mathematical basis for cost-effective and sustainable ACL control plans in endemic settings, revealing how memory-dependent dynamics may affect disease development and intervention efficiency. Full article

Polycystic Ovary Syndrome (PCOS) is a widespread hormonal disorder affecting women of reproductive age, often leading to infertility and associated complications. This study presents a comprehensive stochastic mathematical framework to analyze the dynamics of PCOS with a particular focus on infertility and treatment outcomes. Here, the transitions between compartments represent progression of women through clinical states of PCOS (risk, diagnosis, treatment, recovery) rather than infection or transmission, since PCOS is a non-communicable disorder. The model incorporates probabilistic elements to break the symmetric and predictable assumptions inherent in deterministic approaches. This allows it to reflect the randomness and asymmetry in hormonal regulation and ovulation cycles, enabling a more realistic representation of disease progression. By utilizing stochastic differential equations, the study evaluates the impact of treatment adherence on fertility restoration. We establish the conditions for disease extinction versus the existence of an ergodic stationary distribution, which represents a form of long-term statistical symmetry. The results emphasize the importance of early diagnosis and consistent treatment. Furthermore, the proposed approach provides a valuable tool for clinicians to predict patient-specific trajectories and optimize individualized treatment plans, accounting for the asymmetric nature of patient responses. Full article

The recent resurgence of COVID-19 in a Hepatitis B virus some endemic countries could lead to adverse outcomes. In this article, we formulate and analyse a mathematical model to explains the co-infection dynamics of Hepatitis B virus and COVID-19. Our aim is to investigate the effect of Hepatitis B virus prevention, COVID-19 prevention, COVID-19 vaccination, and environmental factors on transmission dynamics, and formulate conditions for extinction and persistence of the diseases. First, we derive the basic reproduction number for HBV only, COVID-19 only, and co-infection stochastic models using the next-generation matrix method. Next, we establish the conditions for stability in the stochastic sense for HBV only, COVID-19 only sub-models, and the co-infection model using suitable Lyapunov functions. Furthermore, we devote our attention to finding sufficient conditions for extinction and persistence. Finally, motivated by Ghana data, we applied the Euler–Murayama scheme to illustrate the dynamics of the co-infection, COVID-19, HBV, and the effect of some parameters on disease transmission dynamics by means of numerical simulations. Full article

Grapevineleafroll disease (GLD) is one of the more severe and persistent diseases in grapevines worldwide and is caused by several species of grape leafroll-associated viruses (GLRaVs). GLRaVs enter vines mainly by infected plant material or insect vectors. Mealybugs (Hemiptera: Pseudococcidae) are important vectors of GLRaVs and, among them, Pseudococcus viburni is the primary key vector in many regions. To reduce GLRaV spread, acquiring vines from virus-free certified nurseries, removing infected vines, and controlling insect vectors are crucial control tools. Sustainable mealybug control relies on eco-friendly products, cultural practices that limit mealybug population growth, and biological control by natural enemies. For P. viburni, biological control is primarily based on the action of predators and parasitoids, such as Cryptolaemus montrouzieri Mulsant and Acerophagus flavidulus Brethes, respectively, which will obviously have a different mode of action than chemical insecticides. However, the long-term effect of biological control on GLRaV spread within vineyards has rarely been studied. With the aim of better predicting the impact of biological control on insect vectors, such as mealybugs, we developed a mathematical model to predict the GLRaV spread. The results highlight the importance of establishing vineyards with virus-free material and having a pest management program that reduces the vector population to reduce the economic loss from GLRaVs. Full article

According to the World Health Organization (WHO), globally, cervical cancer ranks as the fourth most common cancer in women, with around 660,000 new cases in 2022. In the same year, about 94 percent of the 350,000 deaths caused by cervical cancer occurred in low- and middle-income countries. This paper focuses on the dynamics of HPV by modeling the interactions between four compartments, as follows: S(t), the number of susceptible females; I(t), females infected with HPV; X(t), females infected with HPV but not yet affected by cervical cancer (CCE); and V(t), females infected with HPV and affected by CCE. A compartmental model is formulated to analyze the progression of HPV, ensuring all key mathematical properties, such as existence, uniqueness, positivity, and boundedness of the solution. The equilibria of the model, such as the HPV-free equilibrium and HPV-present equilibrium, are analyzed, and the basic reproduction number, $R_0$, is computed using the next-generation matrix method. Local and global stability of these equilibria are rigorously established to understand the conditions for disease eradication or persistence. Sensitivity analysis around the reproduction number is carried out using partial derivatives to identify critical parameters influencing $R_0$, which gives insights into effective intervention strategies. With appropriate positivity, boundedness, and numerical stability, a new stochastic non-standard finite difference (NSFD) scheme is developed for the proposed model. A comparison analysis of solutions shows that the NSFD scheme is the most consistent and reliable method for a stochastic fractional delay model. Graphical simulations are presented to provide visual insights into the development of the disease and lend the results to a more mature discourse. This research is crucial in highlighting the mathematical rigor and practical applicability of the proposed model, contributing to the understanding and control of HPV progression. Full article

H1N1 influenza, also known as swine flu, is a subtype of the influenza A virus that can infect humans, pigs, and birds. Sensitivity analysis and optimal control studies play a crucial role in understanding the dynamics of H1N1 influenza. In this study, we have derived a mathematical model incorporating both symptomatic and asymptomatic infections, as well as vaccination, to assess the impact of key parameters on disease transmission. Also, we have assumed a density-dependent infection transmission in the modeling process of H1N1 dynamics. We determine the basic reproduction number using the next-generation matrix method and found that the disease-free equilibrium is stable when the basic reproduction number $R_0<1$ and the endemic equilibrium exists and is stable globally when $R_0>1$. By performing sensitivity analysis, the most influential factors affecting infection spread are identified, aiding in targeted intervention strategies. Optimal control techniques are then applied to determine the best approaches to minimize infections while considering resource constraints. The findings provide valuable insights for public health policies, offering effective strategies for mitigating H1N1 outbreaks and enhancing disease management efforts using optimal vaccination. Full article