Forward-backward Sweep Method Research Articles

Modeling environmental-born melioidosis dynamics with recurrence: An application of optimal control

Melioidosis is a significant health problem in tropical and subtropical regions, especially in Southeast Asia and Northern Australia. Recurrent melioidosis is a major obstacle to eliminating the disease from the community in these nations. This work aims to propose and analyze a human melioidosis model with recurrent phenomena and an optimal control model by incorporating time-dependent control functions. The basic reproduction number (R0) of the uncontrolled model is derived using the method of the next-generation matrix. Using the construction of a Lyapunov functional, we present the global asymptotic dynamics of the autonomous model in the presence of recurrent for both disease-free and endemic equilibria. The global asymptotic stability of the model’s equilibria shows the absence of a backward bifurcation for the model in both cases, whether in the absence or presence of relapse. The sensitivity analysis aims to identify the parameters that have the most significant impact on the model’s dynamics. Furthermore, qualitative analysis of the model’s global dynamics and the changing effect of the most influential parameters on R0 are supported by numerical experiments, with the results being illustrated graphically. The model with time-dependent controls is analyzed using optimal control theory to assess the impact of various intervention strategies on the spread of the epidemic. The numerical results of the optimality system are carried out using the Forward–Backward Sweep method in Matlab. We also conducted a cost-effectiveness analysis using two approaches: the average cost-effectiveness ratio and the incremental cost-effectiveness ratio.

Fast power flow calculation for distribution networks based on graph models and hierarchical forward-backward sweep parallel algorithm

IntroductionIn response to the issues of complexity and low efficiency in line loss calculations for actual distribution networks, this paper proposes a fast power flow calculation method for distribution networks based on Neo4j graph models and a hierarchical forward-backward sweep parallel algorithm.MethodsFirstly, Neo4j is used to describe the distribution network structure as a simple graph model composed of nodes and edges. Secondly, a hierarchical forward-backward sweep method is adopted to perform power flow calculations on the graph model network. Finally, during the computation of distribution network subgraphs, the method is combined with the Bulk Synchronous Parallel (BSP) computing model to quickly complete the line loss analysis.Results and DiscussionResults from the IEEE 33-node test system demonstrate that the proposed method can calculate network losses quickly and accurately, with a computation time of only 0.175s, which is lower than the MySQL and Neo4j graph methods that do not consider hierarchical parallel computing.

Nonstandard finite difference schemes for some epidemic optimal control problems

We construct and analyse nonstandard finite difference (NSFD) schemes for two epidemic optimal control problems. Firstly, we consider the well-known MSEIR system that can be used to model childhood diseases such as the measles, with the vaccination as a control intervention. The second optimal control problem is related to the 2014–2016 West Africa Ebola Virus Disease (EVD) outbreak, that came with the unprecedented challenge of the disease spreading simultaneously in three different countries, namely Guinea, Liberia and Sierra Leone, where it was difficult to control the considerable migrations and travels of people inbound and outbound. We develop an extended SEIRD metapopulation model modified by the addition of compartments of quarantined and isolated individuals. The control parameters are the exit screening of travelers and the vaccination of the susceptible individuals. For the two optimal control problems, we provide the results on: (i) the (global) stability of the disease-free and/or endemic equilibria of the state variable systems; (ii) the positivity and boundedness of solutions of the state variables systems; (iii) the existence, uniqueness and characterization of the optimal control solutions that minimizes the cost functional. On the other hand: (iv) we design Euler-based nonstandard finite difference versions of the Forward-Backward Sweep Method (NSFD-FBSM) that are dynamically consistent with the state variable systems; (v) we provide numerical simulations that support the theory and show the superiority of the nonstandard approach over the classical FBSM. The numerical simulations suggest that significantly increasing the coverage of the vaccine with its implementation for adults as well is essential if the recurrence of measles outbreaks is to be stopped in South Africa. They also show that the optimal control vaccination for the 2014-2016 EVD is more efficient than the exit screening intervention.

Sensitivity assessment of optimal control strategies and cost-effectiveness analysis of a novel Candida Auris environmental transmission model in intensive care facilities

Candida Auris is an emerging fungal pathogen flagged by CDC as a serious global health threat among nosocomial infections in the recent times. As an evolving pathogen that often goes misidentified or unidentified under standard laboratory tests, it has the ability to cause fatal infections among the target population involving patients with serious medical conditions admitted to intensive care facilities, due to its capacity to resist anti-fungal treatment and the ability to persist in the hospital environment for long periods. The subject of this paper is to develop a deterministic model to study the transmission nature of Candida Auris wherein measures like apt admission screening methods with weekly screening follow-ups, transmission prevention, proper treatment protocols and environmental disinfection procedures are introduced as constant mitigating controls into the model initially which are later redefined as variable control functions during the optimal control analysis. The theory of optimal control implemented into the model helps us to understand the sensitivity of each control strategy upon the behaviour of each state variable. Further, cost-effectiveness analysis is rigorously conducted using incremental cost-effectiveness ratio (ICER) to identify and rank the control strategies involved based on their economic efficiency. Numerical simulation for the optimal control analysis is performed in MATLAB using the Forward–Backward Sweep Method and the findings are illustrated graphically.

Optimal control and bifurcation analysis of a delayed fractional-order epidemic model for the COVID-19 pandemic

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.

Optimal Rotation Age in Fast Growing Plantations: A Dynamical Optimization Problem.

Forest plantations are economically and environmentally relevant, as they play a key role in timber production and carbon capture. It is expected that the future climate change scenario affects forest growth and modify the rotation age for timber production. However, mathematical models on the effect of climate change on the rotation age for timber production remain still limited. We aim to determine the optimal rotation age that maximizes the net economic benefit of timber volume in a negative scenario from the climatic point of view. For this purpose, a bioeconomic optimal control problem was formulated from a system of Ordinary Differential Equations (ODEs) governed by the state variables live biomass volume, intrinsic growth rate, and area affected by fire. Then, four control variables were associated to the system, representing forest management activities, which are felling, thinning, reforestation, and fire prevention. The existence of optimal control solutions was demonstrated, and the solutions of the optimal control problem were also characterized using Pontryagin's Maximum Principle. The solutions of the model were approximated numerically by the Forward-Backward Sweep method. To validate the model, two scenarios were considered: a realistic scenario that represents current forestry activities for the exotic species Pinus radiata D. Don, and a pessimistic scenario, which considers environmental conditions conducive to a higher occurrence of forest fires. The optimal solution that maximizes the net benefit of timber volume consists of a strategy that considers all four control variables simultaneously. For felling and thinning, regardless of the scenario considered, the optimal strategy is to spend on both activities depending on the amount of biomass in the field. Similarly, for reforestation, the optimal strategy is to spend as the forest is harvested. In the case of fire prevention, in the realistic scenario, the optimal strategy consists of reducing the expenses in fire prevention because the incidence of fires is lower, whereas in the pessimistic scenario, the opposite is true. It is concluded that the optimal rotation age that maximizes the net economic benefit of timber volume in P. radiata plantations is 24 and 19 years for the realistic and pessimistic scenarios, respectively. This corroborates that the presence of fires influences the determination of the optimal rotation age, and as a consequence, the net economic benefit.

Bi-Basis Function Second Derivative Block Method for the Solution of Optimal Control Problems

This paper determines the state trajectory of a dynamic system over a period to optimize a given performance index. A one-step second derivative hybrid method based on the combination of Lucas and first Boubaker polynomials as basis functions is formulated for the solution of optimal control problems via the forward-backward sweep method. The technique is used to generate a set of hybrid schemes at selected grid and off-grid points with implementation in block form. The stability and convergence of the new method is discussed and the accuracy of the method is tested on some numerical examples. The results obtained correspond to a large extent with the results obtained by the exact solution. Bi-bases function, Block method, First Boubaker polynomials, Lucas polynomial, Optimal control problem

Impact of infectious density-induced additional screening and treatment saturation on COVID-19: Modeling and cost-effective optimal control

This study introduces a novel SI2HR model, where “I2” denotes two infectious classes representing asymptomatic and symptomatic infections, aiming to investigate and analyze the cost-effective optimal control measures for managing COVID-19. The model incorporates a novel concept of infectious density-induced additional screening (IDIAS) and accounts for treatment saturation. Furthermore, the model considers the possibility of reinfection and the loss of immunity in individuals who have previously recovered. To validate and calibrate the proposed model, real data from November–December 2022 in Hong Kong are utilized. The estimated parameters obtained from this calibration process are valuable for prediction purposes and facilitate further numerical simulations. An analysis of the model reveals that delays in screening, treatment, and quarantine contribute to an increase in the basic reproduction number R0, indicating a tendency towards endemicity. In particular, from the elasticity of R0, we deduce that normalized sensitivity indices of baseline screening rate (θ), quarantine rates (γ, αs), and treatment rate (α) are negative, which shows that delaying any of these may cause huge surge in R0, ultimately increases the disease burden. Further, by the contour plots, we note the two-parameter behavior of the infectives (both symptomatic and asymptomatic). Expanding upon the model analysis, an optimal control problem (OCP) is formulated, incorporating three control measures: precautionary interventions, boosted IDIAS, and boosted treatment. The Pontryagin's maximum principle and the forward-backward sweep method are employed to solve the OCP. The numerical simulations highlight that enhanced screening and treatment, coupled with preventive interventions, can effectively contribute to sustainable disease control. However, the cost-effectiveness analysis (CEA) conducted in this study suggests that boosting IDIAS alone is the most economically efficient and cost-effective approach compared to other strategies. The CEA results provide valuable insights into identifying specific strategies based on their cost-efficacy ranking, which can be implemented to maximize impact while minimizing costs. Overall, this research offers significant insights for policymakers and healthcare professionals, providing a framework to optimize control efforts for COVID-19 or similar epidemics in the future.

The effect of curative and preventive optimal control measures on a fractional order plant disease model

In this paper, we present a novel mathematical model of fractional order for epidemic dynamics in plants, this fractional order model (FOM) in the sense of Caputo derivatives governing fractional differential equations (FDEs), introducing modified parameters to coincide both sides dimensions for FOM that means enhancing its accuracy in representing the real-life scenarios to control the spread of the epidemic. Our contributions include a comprehensive qualitative analysis of the proposed FOM, such as proving the existence and uniqueness of the projected solution by transforming our problem into a fixed point problem and applying a Banach contraction principle and Schauder’s fixed-point theorem. The positivity and boundedness of this solution are also demonstrated. We precisely evaluate all equilibrium points (EPs), examine their local and global stability by using Routh–Hurwitz conditions and fractional LaSalle’s invariance principle (LIP), respectively and shed light on the dynamic behavior of the FOM. Furthermore, we discuss the sensitivity analysis for parameters of the control reproduction number (CRN), with insightful plots to illustrate the model’s response to parameter changes (replanting and roguing parameters). From this foundation, we formulate a fractional optimal control problem (FOCP) by incorporating preventive u1 and curative u2 controls to effectively eliminate the propagation of the epidemic in plants. The fractional necessary optimality conditions (FNOCs) are derived attribution to Pontryagin’s maximum principle (PMP). We employ the forward–backward sweep method (FBSM) based on fractional Euler method (FEM) in order to facilitate control implementation and show the different suggested strategies. By enhancing this method, optimal control efforts are projected onto the FOM and yield three various strategies to impact the epidemic’s trajectory. For each strategy, we explain how the presence (i.e. with control cases) and absence (without control cases) of the proposed controls affected the susceptible, protected and infected plants. Some attractive figures with various values of the weight factor and fractional order α for the CRN are also presented, allowing a visual assessment of their impact on the dynamics of the epidemic in plants. In addition, we calculate the objective function for controlled and uncontrolled cases to provide a quantitative measure of its effectiveness in containing the outbreak in the plants. The results gained from our analyses and simulations provide valuable guidance for the management and elimination of epidemics in plants by offering various scenarios to implement the proposed controls.

Distribution State Estimation and Its Impact of Load Modeling

To maintain system security and other essential parameters like reliability and quality, continuous monitoring of the system is very important. Considering the distribution network, state estimation (SE) methods can be adopted. The purpose of the method is to identify and estimate unknown variables based on the online measurements of test data. The primary objectives considered in this paper are: To choose the exact SE method and the artificial neural network (back propagation algorithm), which can be used for determination in the islanding mode of distribution network states, the composite load model is considered for the estimation of states and further enhancement. By adopting the system, state variables in terms of error are measured in the 12-bus distribution network with precise measurements and compared with practical values. The SE proposed includes results with the load flow backward-forward sweep method to satisfy the system state variables. Numerical results indicate that the model performs better for error measurement data with states and in the case of state forecasting.

OPTIMAL CONTROL ON FRACTIONAL ENZYME KINETICS WITH INHIBITORS

In enzyme kinetic chemical reaction, inhibitors are working as a regulators of the metabolism of the systems and the biochemical activities as well. The main focus of this study is to explore the fractional dynamics of enzyme kinetic biochemical reactions with competitive and uncompetitive inhibitors. Further, the analysis continuous with the optimal controls on the system which accelerate biochemical reactions and maximize product generation. Eventually, the numerical analysis have been done by the Adams predictor–corrector method and the forward–backward sweep method for the dynamics and optimal control study, respectively.

Optimal control and cost-effectiveness analysis for the human melioidosis model

In this work, we formulated and investigated an optimal control problem of the melioidosis epidemic to explain the effectiveness of time-dependent control functions in controlling the spread of the epidemic. The basic reproduction number (R0c) with control measures is obtained, using the next-generation matrix approach and the impact of the controls on R0c is illustrated numerically. The optimal control problem is analyzed using Pontryagin's maximum principle to derive the optimality system. The optimality system is simulated using the forward-backward sweep method based on the fourth-order Runge-Kutta method in the MATLAB program to illustrate the impact of all the possible combinations of the control interventions on the transmission dynamics of the disease. The numerical results indicate that among strategies considered, strategy C is shown to be the most effective in reducing the number of infectious classes compared to both strategy A and strategy B. Furthermore, we carried out a cost-effectiveness analysis to determine the most cost-effective strategy and the result indicated that the strategy B (treatment control strategy) should be recommended to mitigate the spread and impact of the disease regarding the costs of the strategies.

A Comparative Study of Optimal PV Allocation in a Distribution Network Using Evolutionary Algorithms

The growing distributed energy resource (DER) penetration into distribution networks, such as through residential and commercial photovoltaics (PV), has emerged through a transition from passive to active networks, which takes the complexity of planning and operations to the next level. Optimal PV allocation (sizing and location) is challenging because it involves mixed-integer non-linear programming with three-phase non-linear unbalanced power flow equations. Meta-heuristic algorithms have proven their effectiveness in many complex engineering problems. Thus, in this study, we propose to achieve optimal PV allocation by using several basic evolutionary algorithms (EAs), particle swarm optimization (PSO), artificial bee colony (ABC), differential evolution (DE), and their variants, all of which are applied for a study of their performance levels. Two modified unbalanced IEEE test feeders (13 and 37 bus) are developed to evaluate these performance levels, with two objectives: one is to maximize PV penetration, and the other is to minimize the voltage deviation from 1.0 p.u. To handle the computational burden of the sequential power flow and unbalanced network, we adopt an efficient iterative load flow algorithm instead of the commonly used and yet highly simplified forward–backward sweep method. A comparative study of these basic EAs shows their general success in finding a near-optimal solution, except in the case of the DE, which is known for solving continuous optimization problems efficiently. From experiments run 30 times, it is observed that PSO-related algorithms are more efficient and robust in the maximum PV penetration case, while ABC-related algorithms are more efficient and robust in the minimum voltage deviation case.

Game-theoretic modeling and analysis of cyberbullying spreading on OSNs

The rapid development of online social networks (OSNs) has enabled seamless cyberbullying activities, which not only cause psychological harm to victims but also bring disharmony to society. From the perspective of OSN platform administrators, strategic decision making towards when/how to launch an anti-cyberbullying campaign is the key to mitigating its negative impact. Hence, this paper develops a dynamic anti-cyberbullying investment solution to mitigate the adverse impact of strategic cyberbullying phenomena cost-effectively. Specifically, this paper first quantifies the payoffs of the attacker/bullying instigator and the defender/platform administrator based on a newly proposed propagation model. Then, considering the budget-constrained factor for both sides, a differential game-theoretical model is proposed to simulate each party's optimal trade-off between cost and benefit. Next, by utilizing Pontryagin Maximum/Minimum Principle, a set of necessary conditions for a Nash equilibrium strategy profile have been proposed. Based on the forward-backward sweep method, an iterative algorithm (SCR optimization algorithm) is tailored to solve the conditions numerically. Through experiments conducted under Facebook, Youtube and Twitter sub-networks, the algorithm's performance has been evaluated by inspecting its convergence speed and payoff compared with randomized profiles. Consequently, the resulting strategies are verified to exhibit superior performance in dealing with cyberbullying spreading.

A deterministic transmission model for analytics-driven optimization of COVID-19 post-pandemic vaccination and quarantine strategies.

This study developed a deterministic transmission model for the coronavirus disease of 2019 (COVID-19), considering various factors such as vaccination, awareness, quarantine, and treatment resource limitations for infected individuals in quarantine facilities. The proposed model comprised five compartments: susceptible, vaccinated, quarantined, infected, and recovery. It also considered awareness and limited resources by using a saturated function. Dynamic analyses, including equilibrium points, control reproduction numbers, and bifurcation analyses, were conducted in this research, employing analytics to derive insights. Our results indicated the possibility of an endemic equilibrium even if the reproduction number for control was less than one. Using incidence data from West Java, Indonesia, we estimated our model parameter values to calibrate them with the real situation in the field. Elasticity analysis highlighted the crucial role of contact restrictions in reducing the spread of COVID-19, especially when combined with community awareness. This emphasized the analytics-driven nature of our approach. We transformed our model into an optimal control framework due to budget constraints. Leveraging Pontriagin's maximum principle, we meticulously formulated and solved our optimal control problem using the forward-backward sweep method. Our experiments underscored the pivotal role of vaccination in infection containment. Vaccination effectively reduces the risk of infection among vaccinated individuals, leading to a lower overall infection rate. However, combining vaccination and quarantine measures yields even more promising results than vaccination alone. A second crucial finding emphasized the need for early intervention during outbreaks rather than delayed responses. Early interventions significantly reduce the number of preventable infections, underscoring their importance.

Adjoint-based optimal control of contractile elastic bodies. Application to limbless locomotion on frictional substrates

In nature, limbless locomotion is adopted by a wide range of organisms at various length scales. Interestingly, undulatory, crawling and inching/looping gait constitutes a fundamental class of limbless locomotion and is often observed in many species such as caterpillars, earthworms, leeches, larvae, and C. elegans.In this work, we developed a computationally efficient 3D Finite Element (FE) unified framework for the locomotion of limbless organisms on frictional substrates. Muscle activity is simulated with a multiplicative decomposition of the deformation gradient, which allows mimicking a broad range of locomotion patterns in 3D contractile elastic solids.We propose a two-field FE formulation based on positions and velocities. Governing partial differential equations are transformed into equivalent time-continuous differential–algebraic equations (DAEs). Next, the optimal locomotion strategies are studied in the framework of optimal control theory. We resort to adjoint-based methods and deduce the first-order optimality conditions, that yield a system of DAEs with two-point end conditions. Hidden symplectic structure is discussed, and Symplectic Euler time integration of optimality conditions are employed. The resulting discrete first-order optimality conditions form a non-linear programming problem that is solved efficiently with the Forward Backward Sweep Method. Finally, some numerical examples are provided to demonstrate the comprehensiveness of the proposed computational framework and investigate the energy-efficient optimal strategy out of distinct locomotion patterns adopted by limbless organisms.

PSO‐based optimal placement of electric vehicle charging stations in a distribution network in smart grid environment incorporating backward forward sweep method

AbstractThe transition from conventional fossil‐fuel vehicles to electric vehicles (EVs) is critical for mitigating environmental pollution. The placement of electric vehicle charging stations (EVCS) significantly impacts the utility operator and electrical network. Inappropriately placed EVCS lead to challenges such as increased load, unbalanced generation, power losses, and reduced voltage stability. Incorporating distributed generation (DG) helps mitigate these issues by maximizing EV usage. This study focuses on optimizing EVCS and DG placement in radial distribution networks. The methodology employs a backward and forward sweep method for load flow analysis and utilizes the particle swarm optimization (PSO) algorithm to determine optimal EVCS and DG locations and sizes. This approach, validated on the IEEE‐33 bus system, outperforms existing methods. Results indicate a 2.5 times greater power loss reduction compared to simulated annealing (SA), 1.6 times better than artificial bee colony, and parity with genetic algorithm (GA). Overall, the PSO algorithm demonstrates superior optimization effectiveness and computational efficiency, showcasing 1–2.5 times better performance than other methodologies. Employing this approach yields significantly improved results, making it a promising technique for optimizing EVCS and DG placement in distribution networks.

NUMERICAL SOLUTION TO OPTIMAL CONTROL PROBLEMS USING COLLOCATION METHOD VIA PONTRYAGIN’S PRINCIPLE

In this study, Lucas polynomial approximate solution is considered to develop a collocation technique for solving optimal control problems with implementation in block using forward backward sweep method. The collocation block method developed is stable and convergent. The method is implemented using MATLAB code, and the examples show that forward-backward sweep methods with the collocation method is an efficient technique for solving optimal control problems as compared with some existing methods.

Optimal control and cost-effectiveness analysis for leptospirosis epidemic

This paper aims to apply an optimal control theory for the autonomous model of the leptospirosis epidemic to examine the effect of four time-dependent control measures on the model dynamics with cost-effectiveness. Pontryagin's Maximum Principle was used to derive the optimality system associated with the optimal control problem. Numerical simulations of the optimality system were performed for different control strategies and the results were presented graphically with and without controls. The optimality system was simulated using the Forward–Backward Sweep method in the Matlab programme. The numerical results revealed that the combination of all optimal control measures is the most effective strategy for minimizing the spread and impact of disease in the community. Furthermore, a cost-effectiveness analysis was performed to determine the most cost-effective strategy using the incremental cost-effectiveness ratio approach and we observed that the rodenticide control-only strategy is most effective to combat the spread of disease when available resources are limited.

Cost-effective optimal control analysis of a COVID-19 transmission model incorporating community awareness and waning immunity

Abstract This article presents a cost-effective optimal control analysis of interventions applied to a S2EI2RS type deterministic compartmental model of COVID-19, considering community awareness and immunity loss. We introduce two time-dependent controls, namely, home quarantine and treatment, to the model for defining an optimal control problem (OCP). In addition to some basic qualitative properties, we obtain the reproductive threshold R 0 {R}_{0} by using the next-generation method and see the impact of controls on it. We also investigate the effect of community awareness and waning immunity, when no controls are applied. The existence and characterization of optimal controls is proved to establish the optimality system, and the OCP is solved using the forward–backward sweep method. The results are simulated using MATLAB. Our comparative cost-effective analysis indicates that implementing both control strategies simultaneously, along with community awareness, is the most optimal and sustainable way to flatten COVID-19 curves in a short period of time than that of implementing single controls. This article offers valuable insights that can assist policymakers and public health experts in designing targeted and effective control measures for COVID-19 and future epidemics in the post-COVID era. Therefore, this piece of work could be a valuable contribution to the existing literature.