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In this paper, we construct a COVID-19 dynamic model that included both the initial and reinfected population compartments, and conduct a structural identifiability analysis of the model parameters to ensure the robustness of the parameter fitting results. We use some actual statistical data from North Carolina to fit the model and estimate the values of some important parameters. In order to accurately fit the parameters in the model, we improve the physics-informed neural networks (PINNs) method in this paper, so that the fitting results can be reproduced on Matlab. The results of this study show that the transmission capacity of the virus in the reinfected person is only slightly lower than that of the first infected person, and vaccination is not effective in reducing the transmission rate of the virus. The death rate of the reinfected is much higher than that of the first infected person. Finally, we conduct a cost-effectiveness study using optimal control methods and found that, while it is easier to reduce reinfection by combining multiple strategies, the most effective strategy for reducing reinfection is to increase treatment cure rates and reduce direct or indirect contact among those who have recovered.

Giardiasis, caused by the protozoan parasite Giardia intestinalis, poses a significant public health burden worldwide, particularly in regions with limited access to clean water and sanitation infrastructure. Understanding the transmission dynamics of giardiasis and evaluating intervention strategies are crucial for effective disease control. In this study, we apply the theory of optimal control to the giardiasis model. The model includes public health education, treatment, and sanitation as the control measures for giardiasis. The goal is to minimize the infections in the population brought on by interactions with asymptomatic, symptomatic, and contaminated environments while reducing the cost of the control measures. We accomplish this goal by using Pontryagin’s Maximum Principle, where fourth-order Runge–Kutta is used to perform numerical simulations for both forward and backward in-time schemes. We simulate the model under different control scenarios to determine which strategy could produce the greatest results. The results demonstrate that the control strategy combining the three control measures (public health education, treatment, and sanitation) proves to be more effective in curbing the spread of giardiasis. Moreover, the incremental cost-effectiveness ratio (ICER) is used to analyze cost-effectiveness. The cost-effectiveness analysis revealed that the strategy that contains treatment and sanitation is the best strategy to implement in the population in case of budget constraints. Hence, our study recommends that sanitizing the environment and treating the infected individuals immediately could be the best practice for controlling the spread of giardiasis disease on an entire population.

In this paper, we have introduced a software reliability growth model (SRGM) that integrates a Weibull testing effort function (TEF) within the software fault detection process (FDP) and fault correction process (FCP), effectively capturing the intricacies of software testing. Leveraging the least squared estimation method, we estimated both model parameters and the TEF, leading to a notably improved fit with the dataset and enhancing our comprehension of software reliability dynamics. Our approach also involved employing a reliability-based method to identify the optimal time for software release, further reinforcing the robustness of our model. Subsequently, we devised an optimal control model to minimize testing efforts throughout the software development life cycle. This work provides a comprehensive methodological framework for computing the total cost while ensuring debugging costs follow a learning curve pattern. Through numerical simulations involving hypothetical values and the Weibull TEF, we observed that once the software achieves the desired reliability for release, subsequent costs gradually decline to zero. In essence, both testing and future costs converge to zero post-release, reducing instantaneous expenses.

In this paper, we develop a dynamic control to investigate the hospitals’ quality improvement and congestion reduction, along with corresponding knowledge accumulations. The main features of our work are: (i) developing a dynamic analysis model of the hospitals’ quality improvement and congestion reduction, considering knowledge accumulations in health care markets; (ii) the demand function for medical treatment depends on treatment quality, congestion degree, price, and traveling cost; (iii) each hospital’s instantaneous costs of quality improvement and congestion reduction depend on treatment quality, quality-improving investment, corresponding knowledge accumulation as well as congestion degree, congestion-reducing investment and the corresponding knowledge accumulation; (iv) the hospitals’ dynamic competitions are not only in treatment quality and congestion degree but also in treatment prices. Our results show that (i) there exists a unique saddle-stable steady-state equilibrium under hospital optimum and social optimum; (ii) knowledge accumulation and the complementarity or substitutability affect the hospitals’ decision behavior; (iii) whether the price and the hospitals’ investments in quality improvement and congestion reduction are higher or lower under hospital optimum than that under social optimum depends on the parameter regions of complementarity or substitutability.

Optimal control of nonlinear vibration requires precise knowledge of the system and the solution to Hamilton–Jacobi–Bellman (HJB) equation. However, in practical engineering applications, acquiring precise system parameters poses challenges, and the analytical solutions for the HJB equation are difficult to obtain. In this paper, two reinforcement learning algorithms, Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm and Soft Actor-Critic (SAC) algorithm, are employed to train neural network-based optimal controllers for the van der Pol vibration system in the presence of unknown system parameters. To validate their performance, the controllers undergo testing in a series of experiments, including assessments of free vibration, frequency sweep excitation, and Gaussian noise excitation. The results indicate that both the TD3-trained and SAC-trained neural network-based controller are capable of proficiently suppress the vibration of the van der Pol oscillator. Additionally, these two model-free controllers can approximate the optimal control law which solved based on the dynamic model of the nonlinear system.

Understanding the impact of human behavior on the spread of infectious diseases might be the key to developing better control strategies. Tuberculosis (TB) is an infectious disease caused by bacteria that mostly affects the lungs. TB remains a global health issue due to its high mortality. The paper proposes a spatiotemporal discrete tuberculosis model, based on the assumption that individuals can be classified as susceptible, exposed, infected, and recovered (SEIR). The objective of this work is to introduce a strategy of control that will reduce the number of exposed and infected individuals. Three controls are established to accomplish this. The first control is a public awareness campaign that will educate the public on the signs, symptoms, and treatments of tuberculosis, allowing them to seek treatment if they are at risk. The second control initiates chemoprophylaxis efforts for people who are latently infected, and the third control characterizes the treatment effort for people who are actively infected. We have shown the existence of optimal controls to give a characterization of controls in terms of states and adjoint functions by using Pontryagin’s maximum principle. Using numerical simulations, our results indicate that awareness campaigns should be combined with treatment and chemoprophylaxis techniques to reduce transmission. As a result, it demonstrates the efficacy of the suggested control strategies in reducing the impact of the disease.

In this paper, we consider the optimal boundary control problem of a two-species competitive system with time delay and size structure in a polluted environment. First, the well posedness of the system is studied by using the characteristic line method and the fixed point principle. Second, the necessary conditions for optimal boundary control are obtained by conjugate system and normal cone property. Finally, the existence and uniqueness of the optimal strategy are proved by Ekeland variational principle.

This study introduces a novel fractional-order model to investigate the interplay between cancer and obesity and their treatment. Initially, we examine the solution’s existence and uniqueness for the proposed model. Additionally, we establish the boundedness of these solutions. Subsequently, we identify some potential equilibrium points of the cancer–obesity model and investigate their stability. To address the considered model, we propose fractional Euler’s and Adam’s methods. Theoretical and numerical analyses are conducted to assess the error estimates and performance of both methods with varying fractional-order derivatives. Moreover, we formulate an optimal control problem concerning cancer density and drug concentration. We delve into the existence of control and explore the first-order optimality conditions. We validate the analytical findings through numerical computations, demonstrating that administering drugs with control variables enhance immunity levels and reduce the burden of cancer.

In this paper, a delayed fractional-order epidemic model with general incidence rate and incubation period is proposed for the Corona Virus Disease 2019 (COVID-19) pandemic. The corresponding sufficient conditions are established to analyze the existence and stability of disease-free equilibrium and endemic equilibrium of the proposed model. The conditions for the existence of Hopf bifurcation are obtained by selecting the time delay as the bifurcation parameter. The control strategies for the COVID-19 pandemic are designed, and the corresponding delay fractional order optimal control problem (DFOCP) is analyzed based on the generalized Euler–Lagrange equation. The parameters of the model are identified based on the data of multiple types of the COVID-19 pandemic. Further, the effectiveness of the model in describing the trend of the COVID-19 pandemic is verified. Based on the results of parameter identification, the influence of incubation period on the COVID-19 pandemic is discussed. The forward–backward sweep method (FBSM) is adopted to numerically solve DFOCP, and the control effects under different control measures are analyzed.

One of the promising strategies to reduce dengue transmission is to release Wolbachia-infected mosquitoes, which can reduce the reproductive success of wild female mosquitoes. We develop a dengue transmission model coupled with the Wolbachia infection to consider the impact of the increased mortality of Wolbachia-infected immature mosquitoes on Wolbachia invasion and dengue transmission. To begin with, we analyze the infection model of Wolbachia without dengue transmission dynamics. Next, we establish an optimal control model by introducing a control variable to simulate the continuous (daily) releases of Wolbachia-infected male mosquitoes and find the optimal control by using Pontryagin’s Maximum Principle. We determine the optimal release strategy by minimizing the total cost of releasing infected male mosquitoes. Then, the full dengue transmission model is analyzed. The basic reproduction number of dengue transmission is calculated using the next-generation matrix method. The stability of the dengue-free equilibrium is proved by using the method of monotone dynamical systems. Furthermore, we carry out sensitivity analysis to study the barrier effect of Wolbachia and the impact of the increased mortality of immature mosquitoes on dengue transmission. Our results suggest that the increased mortality of immature Wolbachia-infected mosquitoes is not conducive to Wolbachia establishment and dengue control, which also induces more Wolbachia-infected mosquitoes to be released. In particular, we estimate the threshold mortality rate of infected larvae by using bifurcation analysis, which provides a quantitative basis and theoretical support for rational selection of Wolbachia strains and scientific and effective practice of dengue control.

This paper presents a brucellosis disease model with reaction–diffusion and time delay. The model takes into account both the direct and indirect transmission of infected animals and pathogens in the environment. By analyzing the associated characteristic equation, the local stability of the unique positive equilibrium point is established. The existence of Hopf bifurcations at the positive equilibrium point is also examined by considering the discrete time delay as a bifurcation parameter. Additionally, an optimal control analysis is conducted to minimize disease outbreaks and control costs. This includes reducing the exposure of susceptible animals to infected animals, removing infected animals from herds, and reducing emissions of brucella into the environment. By constructing Hamiltonian function and applying Pontryagin’s maximum principle, the necessary conditions for the existence of optimal control are given. Finally, the existence of bifurcation periodic solutions and the effectiveness of control strategies are illustrated through numerical simulations.

In this paper, a stochastic brucellosis model with nonlocal transmission and spatial diffusion is established. Existence, uniqueness and positivity of mild solution to the model are obtained by adopting a truncation method. To ensure the Mean Square Exponential Stability (MSES) of the solution, sufficient conditions are given by employing the inequality techniques and the stability implies that the brucellosis die out in a short period of time. Moreover, we introduce elimination of infected animals and disinfection of brucella to the model as control strategies. Necessary conditions are given to obtain the optimal control which best balance the outcomes and costs of the control by applying Pontryagin’s maximum principle. Finally, the theoretical results are demonstrated by the numerical simulations.

In this paper, we introduce an age-structured quit-smoking model that incorporates the duration of smoking cessation, a crucial parameter related to smoking cessation counseling and disease treatment. We identify the threshold value $R_0$, which determines the existence and stability of the smoking-free steady state. Our findings show that the smoking-free steady state is locally and globally stable if $R_0<1$, whereas it becomes unstable, and the unique steady state with present smokers emerges and is locally and globally stable if $R_0>1$. We propose an optimal control strategy, and our results show its ability to effectively decrease the number of smokers and promote smoking cessation. To validate our findings and to study the influence of varying parameters on the model’s dynamics, numerical simulations are carried out. A concise conclusion is then shared at the end of the paper.

Improving epidemic models to better reflect reality has long been a prominent concern for governments and researchers. This paper presents a novel Susceptible–Infected–Recovered–Susceptible (SIRS) epidemic model for human populations, offering a comprehensive analysis. The proposed model introduces a generalized SIRS epidemics framework encompassing three propagation scenarios. The paper establishes the positivity and boundedness of the system and demonstrates the stability of its equilibrium points. Furthermore, a controlled system is introduced, accompanied by three suggested control strategies to minimize the infected population while optimizing cost. To validate the analytical findings, a numerical example is provided. The paper concludes with a summary and outlines future research directions.

Inability to become pregnant after 12 months of regular, unprotected intercourse is defined as Infertility. Couples who have not conceived after 12 months of unprotected vaginal intercourse should be offered further evaluation. Evaluation includes workup for anovulation (hormonal evaluations), hysterosalpingography, hysteroscopy, Laparoscopy. Treatment options vary from low-cost non-pharmacological therapy, like counseling on the timing of intercourse during the most fertile period, may wait for another year of unprotected intercourse, weight loss (target Body Mass Index $<30$), smoking cessation, limiting alcohol consumption, psychological interventions like cognitive behavioral therapy (CBT) to reduce stress, anxiety and depression related to infertility, to high-cost pharmacological approach including ovulation induction medication, intrauterine insemination (IUI), in vitro fertilization (IVF). Mathematical models are rising as a key factor to add to our knowledge of the fertility process and help us understand the intricacies in the reproductive system to be able to predict the possibilities of pregnancy precisely. We have created a mathematical model with five compartments to understand the success of treatment of infertility in women. We have carried out local stability, global stability at pregnancy-free and pregnancy exist equilibrium points and numerical analysis. We have also tried optimal control by maximizing fertility through non-pharmacological measures and applied cost control to IVF treatment. Our results showed non-pharmacological and pharmacological treatments have a positive impact on the overall success of treatment of infertility however cost is the important determining factor. We recommend maximizing non-pharmacological measures before opting for costly pharmacological measures. We also recommend that the government or other Non-Governmental Organizations (NGOs) help with the cost for women with infertility.

In this work, we propose a stochastic Human Papillomavirus (HPV) epidemic model with two kinds of delays and media influences. These two time delays are the delay time caused by media receiving the disease information and the delay time of public feedback after the media coverage. In addition, media coverage not only has a negative impact on the infection rate, but it also has a positive impact on the vaccination rate of disease. We discuss the existence and uniqueness of the positive solution for the HPV epidemic model, and then put forward a positively invariant set. The sufficient conditions of the extinction and persistence for the HPV epidemic are given. For the optimal control problem of the HPV epidemic, we obtain an optimal strategy. Our numerical simulations validate the theoretical results of this paper, showing that appropriate media coverage can help control the development of the disease.

Car-following is an approach to understand traffic behavior restricted to pairs of cars, identifying a “leader” moving in front of a “follower”, which at the same time, it is assumed that it does not surpass to the first one. From the first attempts to formulate the way in which individual cars are affected in a road through these models, linear differential equations were suggested by author like Pipes or Helly. These expressions represent such phenomena quite well, even though they have been overcome by other more recent and accurate models. However, in this paper, we show that those early formulations have some properties that are not fully reported, presenting the different ways in which they can be expressed, and analyzing them in their stability behaviors. Pipes’ model can be extended to what it is known as Helly’s model, which is viewed as a more precise model to emulate this microscopic approach to traffic. Once established some convenient forms of expression, two control designs are suggested herein. These regulation schemes are also complemented with their respective stability analyses, which reflect some important properties with implications in real driving. It is significant that these linear designs can be very easy to understand and to implement, including those important features related to safety and comfort.

Rumor is an unauthenticated statement that gives significant changes in the social life of the people, financial markets (stocks and trades), etc. By incorporating the dissemination of rumor through groups in social, mobile networks and by considering the people’s cognitive factor (hesitate and forget), a new model on the rumor spreading process is presented in this paper. The spreading dynamics of rumor in homogeneous and heterogeneous networks is analyzed by using mean-field theory. The reproduction number is obtained by using the next-generation matrix. The global stability of the rumor-free equilibrium for the homogeneous and heterogeneous model is proved elaborately. An optimal control problem is developed to minimize the hesitators and infected persons and the existence of optimality is shown using Pontryagin’s Minimum Principle. The hesitating and forgetting mechanism has a great impact on the model and is similar to the real-life. Further, the control parameters work superior in controlling the spreading of rumors. Finally, the numerical results are verified by the analytical results.

In this paper, a model describing the transmission of Human Papilloma Virus (HPV) in a bisexually active human host community is presented. We analyze the model in a feasible region where the model is realistic in the sense of HPV transmission. Since the trivial equilibrium does not exist, we obtain the HPV-free equilibrium solutions and make use of the next generation matrix method to compute the basic reproduction number $R_{hpv}$, which governs HPV extinction and persistence whenever it is less or greater than unity, respectively. We perform the sensitivity analysis of the model parameters of $R_{hpv}$ as to HPV prevalence and found that parameters ${\beta }_1,{\beta }_2,{\gamma }_o$ and ${\gamma }_1$, which are the effective HPV transmission and progression to recovery parameters, are positively sensitive to $R_{hpv}$. In order to minimize the increasing effect of $R_{hpv}$ as regards the positive sensitive parameters, we re-construct the model via optimal control theory to incorporate controls of condom usage $u_1$, vaccination $u_2$ and medical counseling $u_3$ respectively. With these controls, we characterize and discuss the existence and uniqueness of the control model and solve the optimality system using the forward–backward Runge–Kutta fourth-order technique via the Matlab computational software. Simulations show that each of the control strategies is potent in combating HPV but the combination of the three controls proved more efficient in minimizing HPV infection in the human bisexual host community.

The rapid development of social networks makes the rumour, other false news disseminate to the people in a short period. Online users in social networks are dynamically changing the connectivity over time. The effect of dynamic connections results in stochastic variation which is termed as noise. In this paper, a nonlinear rumour propagation model is formulated, the basic regeneration number $R_0$ of the proposed model is computed and the stability for the model is discussed. Further, we extend the model to stochastic rumour propagation for online social networks incorporating noise. The existence and uniqueness of the stochastic rumour propagation for the homogeneous network are investigated. Optimal control strategy of stochastic rumour spreading model in online social network is investigated to control the parameters. A comparison between deterministic and stochastic rumour spreading model in online social network is numerically illustrated.