The release of Wolbachia-infected mosquitoes has emerged as a promising biocontrol strategy to suppress mosquito-borne diseases. However, the success of such interventions is highly sensitive to ecological conditions, release timing, and seasonal climate variability. In this study, we develop a climate-driven compartmental model that explores the interplay between mosquito life cycle dynamics, human disease transmission, and Wolbachia-based biocontrol. Analysis of the autonomous version of the system shows that the Wolbachia-free subsystem exhibits a backward bifurcation. In contrast, the Wolbachia-invaded subsystem shows a forward bifurcation, which supports disease elimination. In the non-autonomous framework, we establish that disease-free equilibrium is globally attractive for R_0 < 1 , whereas the system admits at least one positive periodic solution for R_0 > 1 , implying disease persistence. Numerical simulations validate the analytical results using real climate data from Niterói, Brazil. We find that the consistent presence of Wolbachia-carrying mosquitoes successfully invades the Wolbachia-free population and effectively reduces disease prevalence. To evaluate the efficacy of Wolbachia deployment, we analyze five release strategies, namely: three-pulse, four-pulse, five-pulse, continuous exponentially decreasing, and quadratically increasing releases. In addition to comparing release modes, we examine release timing and gaps, showing that aligning strategies with seasonal climate patterns ensures sustained Wolbachia establishment and disease control. Additionally, a global sensitivity analysis using partial rank correlation coefficients identifies key parameters of infection dynamics. Parameters such as mosquito biting rate and egg maturation rate positively influence infection, while recruitment of humans, probability of cytoplasmic incompatibility, and recovery of humans exert strong negative effects. These insights help prioritize key biological parameters for an accurate intervention strategy. The model, supported by sensitivity analysis, provides a practical tool for designing adaptive Wolbachia release strategies personalized for local temperature and rainfall conditions.

Mass media reporting during epidemics can significantly influence public behavior and shape societal attitudes. In this study, we construct a Filippov system to examine how media coverage, vaccination, and treatment interact to affect disease transmission. The model incorporates a threshold policy in which interventions are triggered when the number of infected individuals exceeds a critical value. At the same time, a constant rate of media awareness is maintained throughout the epidemic. When the basic reproduction number (R2) is below one, these measures may facilitate disease eradication; however, the presence of backward bifurcation cannot prevent guaranteed elimination. Our analysis explores the global stability of the Filippov model for varying infected thresholds (Ic), emphasizing the existence and stability of equilibria and the nature of sliding-mode dynamics. Depending on the chosen threshold, the system may converge to the endemic equilibrium of the continuous subsystem, a pseudo-equilibrium on the switching manifold, or another endemic equilibrium. Appropriate threshold selection can thus constrain infection levels to remain below a critical or predetermined value. By investigating sliding-mode behavior, boundary equilibrium bifurcations, and the system's global properties, we find that the pseudo-equilibrium can sometimes be globally stable. Alternatively, the system may exhibit bistability, involving combinations of pseudo-equilibrium, endemic, and disease-free equilibrium (DFE). The resulting long-term infection levels depend on whether parameter choices and initial conditions determine a zero value, a fixed intermediate value, or a higher endemic value. Global sensitivity analysis using the PRCC method identifies the key parameters that influence both the threshold values and the overall both system. These results highlight the potential of threshold-based intervention strategies and reveal the inherent difficulties in controlling infectious diseases, even under ideal conditions with highly effective vaccines, sustained media engagement, and adequate treatment capacity.

This study examines a susceptible-infected-temporary-permanent-recovered (SITHR) epidemic model incorporating the Holling type II incidence rate to prevent and control the disease with optimal use of hospital beds. Initially, the well-posedness and feasibility of the model are analyzed, and then valid biological equilibrium points are calculated. Subsequently, the stability of these equilibrium points is assessed and the basic reproduction number (R_0) is calculated as a threshold value that controls the dynamics of the disease. The proposed model undergoes several bifurcations, including transcritical (backward and forward), saddle-node, Hopf, and Bogdanov–Takens bifurcations. The normal form is derived to demonstrate the presence of a Bogdanov–Takens bifurcation. Furthermore, parameter estimation is conducted using COVID-19 data from Italy to refine the model’s accuracy and boost the reliability of the study’s predictions. Using the normalized forward sensitivity index (NFSI), a sensitivity analysis of parameters associated with the basic reproduction number is performed, and the partial rank correlation coefficient (PRCC) is calculated to locate the key parameters affecting disease transmission dynamics. Moreover, the system is expanded to incorporate time-dependent control variables to reduce the infected population and the cost associated with implementing these controls. The developed optimal control system is employed to build the Hamiltonian function, which is solved using Pontryagin’s maximum principle. Also, a cost-effectiveness analysis is performed to evaluate the economic efficiency of various intervention strategies. Beyond the deterministic framework, the study includes formulations for continuous-time Markov chains and stochastic differential equations to assess the impact of environmental noise on the system. Moreover, the Galton–Watson branching process determines the extinction threshold for the stochastic model and sets the parameters that govern disease extinction or persistence. Finally, numerical simulations are demonstrated to illustrate the impact of changes in system parameters on the dynamic behavior of the model. These findings will enhance preparedness and enable more efficient responses to health emergencies, leading to better patient care and less pressure on healthcare systems.

This study introduces an epidemic model with a Beddington-DeAngelis-type incidence rate and Holling type II treatment rate. The Beddington-DeAngelis incidence rate is used to evaluate the effectiveness of inhibitory measures implemented by susceptible and infected individuals. Moreover, the choice of Holling type II treatment rate in our model aims to assess the impact of limited treatment facilities in the context of disease outbreaks. First, the well-posed nature of the model is analyzed, and then, we further investigated the local and global stability analysis along with bifurcation of co-dimensions 1 (transcritical, Hopf, saddle-node) and 2 (Bogdanov-Takens, generalized Hopf) for the system. Moreover, we incorporate a time-delayed model to investigate the effect of incubation delay on disease transmission. We provide a rigorous demonstration of the existence of chaos and establish the conditions that lead to chaotic dynamics and chaos control. Additionally, sensitivity analysis is performed using partial rank correlation coefficient and extended Fourier amplitude sensitivity test methods. Furthermore, we delve into optimal control strategies using Pontryagin's maximum principle and assess the influence of delays in state and control parameters on model dynamics. Again, a stochastic epidemic model is formulated and analyzed using a continuous-time Markov chain model for infectious propagation. Analytical estimation of the likelihood of disease extinction and the occurrence of an epidemic is conducted using the branching process approximation. The spatial system presents a comprehensive stability analysis and yielding criteria for Turing instability. Moreover, we have generated the noise-induced pattern to assess the effect of white noise in the populations. Additionally, a case study has been conducted to estimate the model parameters, utilizing COVID-19 data from Poland and HIV/AIDS data from India. Finally, all theoretical results are validated through numerical simulations. This article extensively explores various modeling techniques, like deterministic, stochastic, statistical, pattern formation(noise-induced), model fitting, and other modeling perspectives, highlighting the significance of the inhibitory effects exerted by susceptible and infected populations.

This study explores the interaction between cancer cells, helper T cells, cytotoxic T cells, and tumour necrosis factors in chemotherapy and immunotherapy treatment microenvironment. The goal is to analyse the connection of helper and cytotoxic T-cell levels with the anti-tumour immune response and the impact of various dosing regimens when combined with immunotherapy and chemotherapy. We discussed the positivity and boundedness of the model. Further, we analyse the biologically valid equilibria and investigate their local stability properties, examining transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations numerically and analytically. Furthermore, direction and stability conditions for periodic solutions are determined. Since cancer treatments are administered in phases or cycles, the choice of therapy, whether sequential or simultaneous, often depends on the patient's critical condition. Thus, the model examines the impact of periodic treatment fluctuations. This study demonstrates that administering chemotherapy before immunotherapy yields better outcomes than starting with immunotherapy followed by chemotherapy. A theoretical analysis is conducted on periodic solutions, global stability, and the persistence of the non-autonomous system. Chaos is extensively demonstrated, and chaotic attractors are depicted through the periodicity of parameters S1 (TNF immunotherapy), S2 (Cytotoxic T immunotherapy), and σ (chemotherapy concentration). When periodicity is introduced in S1 and S2, the non-autonomous system also displays bursting oscillations, indicating both the expeditious growth of the tumour (relapse) and the swift elimination of tumours through treatment remission. Additionally, a sensitivity analysis follows a Latin hypercube sampling-based uncertainty analysis and an eFAST sensitivity analysis to evaluate how parameter uncertainties influence tumour growth. Numerical simulations illustrate how the model's dynamic behaviour changes with system parameter alterations. The findings highlight the critical role of helper, cytotoxic T cells and immunotherapy in tumour elimination. Additionally, the study explores the efficacy of cycle-specific drug administration at lower doses between treatment courses to prevent tumour relapse, suggesting this approach may be superior to shortening treatment intervals. Periodic treatment results indicate that the concurrent use of chemo-immunotherapy rapidly reduces tumour cells and maintains patients' overall health more effectively than sequential use (immunotherapy first, then chemotherapy). However, starting with chemotherapy followed by immunotherapy produces the best results compared to other treatment combinations.

Oncologists and applied mathematicians are interested in understanding the dynamics of cancer-immune interactions, mainly due to the unpredictable nature of tumour cell proliferation. In this regard, mathematical modelling offers a promising approach to comprehend this potentially harmful aspect of cancer biology. This paper presents a novel dynamical model that incorporates the interactions between tumour cells, healthy tissue cells, and immune-stimulated cells when subjected to simultaneous chemotherapy and radiotherapy for treatment. We analysed the equilibria and investigated their local stability behaviour. We also study transcritical, saddle-node, and Hopf bifurcations analytically and numerically. We derive the stability and direction conditions for periodic solutions. We identify conditions that lead to chaotic dynamics and rigorously demonstrate the existence of chaos. Furthermore, we formulated an optimal control problem that describes the dynamics of tumour-immune interactions, considering treatments such as radiotherapy and chemotherapy as control parameters. Our goal is to utilize optimal control theory to reduce the cost of radiotherapy and chemotherapy, minimize the harmful effects of medications on the body, and mitigate the burden of cancer cells by maintaining a sufficient population of healthy cells. Cost-effectiveness analysis is employed to identify the most economical strategy for reducing the disease burden. Additionally, we conduct a Latin hypercube sampling-based uncertainty analysis to observe the impact of parameter uncertainties on tumour growth, followed by a sensitivity analysis. Numerical simulations are presented to elucidate how dynamic behaviour of model is influenced by changes in system parameters. The numerical results validate the analytical findings and illustrate that a multi-therapeutic treatment plan can effectively reduce tumour burden within a given time frame of therapeutic intervention.

The dissemination of public health information through television and social media posts is essential for informing the public about the transmission of contagious diseases, which is crucial in preventing the spread of various infectious diseases. In this paper, we propose a non-linear mathematical model to assess the effect of advertisements through social media in creating awareness and limiting treatment on spreading infectious diseases. These initiatives may alter population behaviour and divide the susceptible population into subgroups. In addition, to comprehend these dynamics better, we use half-saturation constant rates for media coverage and treatment. The model’s well-posedness and feasibility are evaluated. The possible biological equilibrium points are calculated. Local and global stability are carried out. The objective of our study is to produce the model’s bifurcation. Transcritical, Saddle–node, Hopf bifurcation of codimension 1 and Cusp, Generalized-Hopf (Bautin), and Bogdanov–Takens (BT) bifurcation of codimension 2 are studied for this purpose. Due to the limited medical resources and supply efficiency, the model exhibits backward bifurcation, resulting in bistability. Moreover, the occurrence condition for stability and direction of Hopf bifurcation is discussed. This model study demonstrates that the system is significantly influenced by the pace with which awareness programmes are implemented and that raising this value above a threshold may result in continuous oscillation. Sensitivity analysis employs the normalized forward sensitivity index of the basic reproduction number to provide a comprehensive understanding of the effect of various parameters on accelerating and limiting disease spread. Further, the minimum possible cost is determined by formulating an optimal control system based on sensitivity analysis and applying Pontryagin’s maximum principle. Methods of cost-effectiveness, such as ACER and ICER, are used to determine the most cost-effective control intervention strategy among all the strategies. Numerical simulations have been done to support all theoretical findings.

In the current study, we introduce a model for the temporal and spatial interactions between prey and predator. The model incorporates a nonlinear refuge mechanism for prey, along with linear harvesting of prey and nonlinear harvesting for predators. Initially, we examined the well-posed nature of the model by analyzing the presence of all feasible equilibria and investigating the corresponding dynamics. Following that, we delve into the dynamics of the temporal model, focusing specifically on aspects such as uniform boundedness, permanence, and stability of viable equilibria. We demonstrate analytically that the proposed model experiences transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations. It shows a variety of intricate dynamics involving Generalized Hopf and double Hopf. Then, discrete-time delay effects arising from the gestation of predator species have been incorporated into the temporal system. Hopf bifurcation for the delay parameter was detected in this investigation. Subsequently, we established conditions for self-diffusion instability and Turing instability in the spatiotemporal model, both with and without delay, employing the homogeneous Neumann boundary condition. Moreover, the discussion of sensitivity analysis (PRCC) serves to illustrate how crucial parameters influence the dynamics of the system. In addition, we conduct numerical simulations aiming to corroborate and validate the analytical results obtained.

Immunization plays a vital role in eradicating infectious diseases, typically requiring multiple doses at specific time intervals. This study focuses on developing and analyzing an infectious disease model governed by a six-dimensional system of ordinary differential equations, considering the impact of first and second vaccination doses along with hospital beds and treatment. The model’s qualitative behavior is analyzed, including conditions for positive solutions, the invariant region of the solution, equilibrium points, and their stability. When the basic reproduction number ( R_0 ) is less than one ( R_0 < 1 ), the disease will be eradicated; conversely, an epidemic occurs when R_0>1 . Moreover, the transcritical bifurcation of the system is examined using the center manifold theory. Interestingly, backward bifurcation is discovered, and it indicates that the disease is not entirely eradicated even when R_0 < 1 . We have investigated different bifurcations like saddle-node, transcritical, and Hopf bifurcations of codimension 1, as well as Generalized-Hopf (GH), Cusp (CP), and Bogdanov–Takens (BT) bifurcations of codimension 2. Additionally, a delayed epidemiological model is explored, assuming a lag in vaccination among the susceptible population. A Hopf-bifurcation is observed near the endemic equilibrium point, linked to critical parameter values during the latent period. Moreover, the model is calibrated using the least-squares technique, incorporating coronavirus-infected case data and vaccination information from India and Italy’s mass vaccination program between March 1, 2021, and May 30, 2021. Global sensitivity analysis, utilizing the Partial Rank Correlation Coefficient (PRCC), identifies crucial parameters affecting threshold quantities after fitting the model. The study highlights the significance of critical parameters such as the effective transmission rate, rates of first and second-dose vaccinations, and recovery rate due to double-dose vaccination. Further, delayed optimal control measures are determined using Pontryagin’s maximal principle to mitigate infection, prevention, and treatment burdens. Numerical simulations are conducted to understand the effect of these delayed control measures on disease progression and demonstrate the insights obtained through analytical investigations. The study indicates that implementing all control strategies effectively reduces the disease burden among the population. Accurate estimation of vaccine efficacy is crucial for disease prevention, underlining the importance of well-planned vaccination strategies. Moreover, the numerical simulations validate all the theoretical findings, emphasizing the validity of this model in a real-world situation. Relying solely on vaccination might not swiftly or completely control the disease. Complementary pharmaceutical and non-pharmaceutical measures are necessary to combat the infection effectively. Further limitations on medical resources could lead to a backward bifurcation. Simulation results suggest that delaying the implementation of control measures could exacerbate epidemic situations.

This paper deals with the non-linear Susceptible–Infected–Hospitalized–Recovered model with Holling type II incidence rate, treatment with saturated type functional response for the prevention and control of disease with limited healthcare facilities. The well-posedness of the model is ensured with the help of the non-negativity and boundedness of the solution of the system. The feasibility of the model with DFE (Disease-free equilibrium) and EE (endemic equilibrium) is analysed. The local and global stability are discussed with the help of the computed basic reproduction number R0. At R0=1, we use the Centre manifold theory to analyse the transcritical bifurcation exhibited by the system. It is found that the disease is not eradicated even if R0<1 due to the occurrence of backward bifurcation. The occurrence condition of Hopf bifurcation is obtained. The optimal control theory is used to analyse the effects of the minimum possible medical facilities, hospital beds, and awareness creation on the population dynamics. The Hamiltonian function is constructed with the extended optimal control model and solved by Pontryagin’s maximum principle to get the minimum possible expenditure. Different types of control strategies are shown by numerical simulation. The sensitivity analysis is discussed with the help of a crucial parameter that depends on the reproduction number. Further, the model is simulated numerically to support the theoretical studies. This paper emphasizes the significance of treatment intensity, the total number of hospital bed available and their occupancy rate as vital parameters for prevention of disease prevalence.