This research presents a new multi-stage innovation diffusion model that incorporates the delay effect due to the complexity and variability of actual diffusion processes as an evaluation time delay parameter, efficiently using the mathematical framework to provide insight into the progression of awareness of the product, decision-making stage, and finally the adoption of the product. In contrast to traditional diffusion models, this model considers the behavioral delay in the transition from awareness to adoption, therefore, adding realism and applicability to the diffusion process. Initially, a set of differential equations that incorporate the three stages with a single time delay parameter is formulated. The rigorous structure and well-defined nature of the model system is ensured by the positivity and boundedness of the solution to the model system. A unique equilibrium point is established, and the local stability conditions of the equilibrium point are acquired on analyzing the characteristic equation. Existence of Hopf bifurcation has been proven, and the direction and stability of bifurcating periodic solutions have also been obtained using the normal form and center manifold theorem. The global stability conditions for the equilibrium point are obtained on constructing a suitable Lyapunov function. Further, the research contributes by formulating an optimal control problem to maximize the number of adopters while minimizing promotional costs, linking theoretical results to marketing strategy. Numerical simulations, including parameter estimation with real-world data, are carried out to further analyze the model's behavior, support the theoretical results, and estimate the rate of adopters. The findings indicate that accounting for the adoption delay plays a crucial role in shaping the system stability and predicting long-term adoption dynamics.

Business firms and companies are constantly adopting the concept of market segmentation, which plays an important role in the promotion of a product. Additionally, advertising is another component that strengthens the company’s communication with customers. To have effective marketing strategies, the implementation of independent advertising strategies for each segmented market is important. This paper deals with an optimal control problem that aims to obtain a dynamic advertising policy for new customers as well as minimize the decay rate for existing customers in a segment-specific market. We will an derive explicit optimal dynamic advertising efforts policy using Pontryagin’s maximum principle. The analysis gives a deep insight into how the advertising effort should be planned by the decision-makers, designing strategies that maximize long-term profitability while effectively controlling advertising costs. The effectiveness of the suggested strategy is supported by numerical examples, along with parameter estimation to estimate the value of certain parameters.

Accurately forecasting how innovations are adopted is crucial for launching new products and long-term performance. While traditional diffusion models like the Bass model have been widely used to study adoption trends, they often assume deterministic behavior and overlook the randomness inherent in real-world markets. Stochastic differential equation (SDE)-based models offer a more flexible framework by incorporating random fluctuations, typically through additive noise. However, these models rarely account for multiplicative noise, where the intensity of uncertainty increases with the number of adopters. To address this gap, we propose two SDE-based innovation diffusion models that explicitly include multiplicative noise and consider constant and logistic time-dependent adoption rates. These models are evaluated using real-world sales data for technological products, and their performance is compared using established metrics. By incorporating multiplicative uncertainty, the proposed models offer a more realistic representation of adoption dynamics, making them valuable tools for understanding diffusion in uncertain and rapidly evolving markets.

Understanding how new products diffuse through the market is critical for effective planning and strategic decision-making. While traditional innovation diffusion models, such as the Bass model, have been widely used, they often assume a constant adoption rate and overlook real-world complexities such as delayed adoption, market heterogeneity, and repeat purchases. To address these limitations, this study introduces a dynamic product adoption framework that incorporates a time-varying Adoption Increment Factor (AIF). It encapsulates the speed at which the market responds to the introduction of a new innovation or product, reflecting the market’s receptivity to change and the success of adoption initiatives. Three model variants are developed based on different functional forms of AIF. These models are validated using real-world sales data from Acer personal computers and Samsung smartphones. Parameter estimation is performed using the Least Squares Estimation method, and the models are evaluated using MSE, MAE, and AIC as goodness-of-fit criteria. The results show that the sigmoid-based AIF model consistently provides the best fit, accurately reflecting both moderate and rapid adoption scenarios. This approach offers a more realistic representation of adoption behavior and holds significant value for both researchers and practitioners seeking data-driven insights into product diffusion dynamics.

Blockchain technology holds significant potential to revolutionize electronic medical records, owing to its core features of decentralization, transparency, and immutability. However, the suitability of different types of blockchain varies, each presenting unique benefits and limitations within the healthcare settings. Hence, selecting the most appropriate blockchain platform remains a complex decision, influenced by various conflicting criteria. This study presents a software engineering-driven decision-support framework to evaluate blockchain platforms for healthcare applications, using the Intuitionistic Fuzzy TOPSIS (IFS-TOPSIS) method. Through a literature review and expert consultations, we identified eight criteria for objective assessment. Among the three blockchain platforms evaluated, the results indicate that permissioned blockchain technology is the most suitable for the healthcare sector, primarily due to its strengths in regulatory compliance, data privacy, and system integration. The study’s findings would help practitioners identify and choose the best blockchain platform, thus contributing to a transition in the healthcare industry called “Smart Healthcare 5.0”.

Innovation diffusion remains a central theme in marketing and business research, offering insights into why some products achieve widespread adoption while others fail. Traditional models, such as the Bass formulation, capture the general S-shaped adoption curve by distinguishing between innovators influenced by external communication and imitators driven by social interaction. However, these models often assume that information and promotion eventually reach all potential consumers. In reality, adoption is constrained by market coverage. This is the extent to which advertising and distribution strategies effectively reach the population. This paper examines the role of market coverage in diffusion and reviews mathematical extensions that integrate this factor. By exploring exponential, Weibull, Erlang, and logistic forms of coverage and models linking coverage to advertising expenditure, we show how adoption trajectories shift under different assumptions. The discussion further highlights implications for sustainable innovations, such as electric vehicles and renewable energy, where broad coverage is essential to achieve environmental and social benefits. The study underscores coverage as both a strategic variable and a driver of sustainable business outcomes.

The available testing resources are usually restricted during the software testing process. It's usual to presume that these limitations are deterministic when discussing optimal control models. This isn't true in practice. For instance, a budget for the use of testing resources could be the first step in the testing process. However, it is possible that fault detection is accelerated during the fault testing process to meet unexpectedly high fault counts, which calls for additional funding for testing resources. These enhanced numbers are obviously unknown in nature. In this case, fuzzy set theory is a plausible model in sense of degree of uncertainty and hence the testing resource expenditure constraints i.e. total budget becomes imprecise in nature. By integrating fuzzy logic into the budgetary framework, this approach offers flexibility in decision-making, enabling more realistic and adaptable strategies. The aimof this research is to look into an optimal way to allocate testing resources in order to minimize software costs during the testing and operation phases while taking independent and dependent faults into account. The model is formulated as an optimal control problem for where the budget constraint is expressed as necessity and /or possibility type. The proposed problem is solved for an optimal testing effort policy employing Pontryagin's Maximum Principle. A numerical example is presented to support the theoretical optimal control model. The values of the optimal testing effort expenditure function are displayed in both tabular and graphic forms.

Software reliability plays a vital role in the today’s world as dependency on software system increases day by day. To determine reliability, various software reliability growth models have been proposed within the context of probability theory. Software failures include uncertainty, which is unable to represent completely by using probability theory. Further, software companies release their product early and then release the patch after sometime to remove the remaining number of the faults in the software. In this paper, we have developed a generalised testing cost model based on uncertainty theory by considering fault detection process and correction process as two-step process and investigate the software release, patch release time and belief reliability to minimise the testing cost. Numerical illustration is given to support the proposed model and validated on a real data set.

With an increase in market competition, the association between marketing and inventory management has become more important. The commercial activities are more rapid through social media, and advertising has played a crucial role in reaching the product to the consumers before it hits the market. It has thus become normal in an oligopolistic marketing system to increase sales through advertising effort and gain more profit from potential market. It is challenging, nevertheless, to calculate demand and costs related to advertising efforts. As a result, the purpose of this study is to identify the best advertising approach and its potential impact on demand in order to optimize the firm's overall profit. In this paper, we develop an inventory model for deteriorating items to obtain an optimal advertising and inventory strategy, where the consumer demand rate depends on advertising effort and inventory of the items displayed in the store. We have formulated two optimal control problems with the assumption that the replenishment cycle is longer than the fresh product time or not. It is assumed that products do not decay within the fresh product time interval, and inventory decreases due to consumer demand. Next, items will deteriorate and inventory level decreases because of the combined effects of customer demand and deterioration. The analytical solution for the optimal dynamic advertising effort strategies obtained by applying Pontryagin’s maximum principle to maximize overall profit over the planning period. The efficiency of the proposed model is demonstrated by numerical examples. A parameter sensitivity analysis is also performed, providing suggestions for enhancing the firm's profitability when dealing with deteriorating products.

The rapid utilization of computer-based automated systems for human tasks has caused a significant shift in society. Today's society places a high value on software stability. To create highly reliable software systems, software testing is required. Software reliability growth models estimate the number of software faults during the testing stage in order to assess software reliability. The fault removal process is usually assumed to be deterministic, but as software systems grow larger and more faults are discovered during testing phase. The number of faults that are discovered and removed during each debugging process decreases until it is insignificant compared to the fault content at the beginning of the testing phase. It is very likely that the method used in this scenario to find software faults is a stochastic process with a continuous state space. Dynamic indeterministic fluctuations such as testing efficiency, testing method, testing effort expenditure, and testing strategy. invariably have an impact on testing progress and incorporates the change point concept in fault detection rate. In this paper, we provide a software reliability growth model (SRGM) based on a stochastic differential equation (SDE). It also incorporates the change-point idea, which states that the rate of detection per remaining problem or fault may alter as a testing method shifts. The applicability and accuracy of the suggested model are demonstrated with software failure data sets. The model's validity was assessed using predictive validity and mean squared error. Finally, the suggested models were compared to existing continuous-state space SRGM using stochastic differential equations to determine their goodness of fit. It has been demonstrated that an SDE-based model with a change point performs significantly better than the existing model.

In this world of software technology, our dependency on software’s is increasing continuously. As a result, software industries are working hard to develop highly reliable software and to meet the expectation of customers. Generally, software companies release software early in market to take gain market share, but rigorous software testing is required for early release software to ensure reliability of software and meet the customer’s expectations. This requires a huge amount of resources, and it increases financial burden on the company, consequently, decreases the overall profit of company. Further, late release due to prolong testing of a software may improves reliability but results into a loss of market opportunity cost or may not be fulfil the customer’s aspirations. As a result, to stay competitive, companies release software early and release patches later to fix the bugs, improve the functionality of software, and to update the software. Software industries are improving the performance or usability of software by releasing patches which may increase the consumption of testing effort and consequently increase in cost. On the other hand, software firms also provide warranty on their products. To address the above said issues, we have developed a testing effort-based software reliability growth model, which incorporates warranty policy and estimates the optimal software release and patch time with the objective to minimise the total testing cost. Further, we have used Genetic Algorithm (GA) to estimate optimum software release and patch time. A numerical illustration has been presented on a real time data set to validate the proposed model.

The process of forming goodwill is intricate, and various factors such as advertising and product quality influence the formation of goodwill of a firm. In this paper, we formulate an optimal control model to derive the optimal control policies of advertisement, price and quality level for a product. The model is framed with the assumption that the firm employs one marketing strategy, namely, advertising effort as well as one operational strategy, namely, quality improvement through product innovation. The stock of goodwill increases in proportion to advertising effort and quality effort because of the positive effects of both advertising and product quality on goodwill. Using maximum-principle, we derive the optimal control effort policies, and the effectiveness of the proposed model is validated through numerical examples. The control theoretic analysis provides a deep insight of how the decision makers should plan the advertising and quality during the planning period.

This paper is a novel attempt to analyze different aspects of a two-strain epidemic model. The paper introduces a novel approach to analyzing a two-strain epidemic model, emphasizing the efficacy of combining non-pharmaceutical interventions and vaccination, particularly against new variants. Additionally, it presents a unique spatio-temporal model to assess spatial distribution of infections, offering fresh insights into how spatial factors influence disease transmission and control. The analysis involves stability analysis(local and global) and optimal control. Time-dependent social/non-pharmaceutical interventions coupled with vaccination of the susceptible class in the presence of both strains are analyzed using Pontryagin’s Maximum Principle. The numerical section shows the behavior of the infected class with and without control, the control intensity trend for the scenario when only non-pharmaceutical interventions (NPI) are practiced for the original strain, and when both NPI and vaccination are incorporated with the emergence of the new strain. The evaluation of the Incremental average ratio (IAR) and Incremental cost-effectiveness ratio (ICER) determines that NPI and vaccination as a combination is better and ideal in terms of costs incurred and effectiveness as a whole in averting infection in the presence of a new variant. Finally, we have also proposed a spatio-temporal pattern for the new strain model to analyze patterns using the finite element method by PDE Toolbox to show the effect of different initial conditions and geometry on the density of the infected population.

Market segmentation is one of the key marketing activities to target the potential market for a product, which allows the firm to have a better understanding of their customers. This paper considers an optimal control problem to determine the dynamic price and advertising policies of a new product introduction in a segment-specific market incorporating advertising-based goodwill. Under differentiated advertising and single-channel advertising, advertising efforts increase the stock of goodwill in each segment. Single-channel advertising starts in all segments with a fixed segment spectrum, while the differentiated advertising process deals with each segment independently. The explicit optimal dynamic advertising effort and price strategies are obtained by applying Pontryagin’s maximum principle, and local stability of equilibria have also been examined. The effectiveness of the proposed method is validated through numerical examples, and a local sensitivity analysis is performed to find the sensitive parameters that can affect the optimal values of price and advertising effort rates.

A new product's introduction to the market is greatly influenced by effective and enough advertising. In this paper, we have considered a problem in which the firm partitions the market into various segments to reduce costs associated with advertising and targets to maximize the total profit. By integrating single channel and differentiated advertising to segmented market, we extend Nerlove-goodwill Arrow's dynamic model in which advertising variables are control variables. It is assumed that the whole available budget is imprecise and fuzzy in nature in order to create a realistic model. Using necessity and possibility constraints, the optimal control model with fuzzy parameters is transformed into crisp form, and the Pontryagin Maximum principle is then used to solve the problem. Numerical examples are provided to support the theoretical analysis.

In a world of raging epidemic spreads, HIV has a major impact on the health of the global public. Immune system cells play a major role in defending the body from the impact of the virus. HIV's major effect is on CD4+ cells (or T cells) and CD8+ cells (or Z cells). The activation of CD8 cells has a major impact as it helps in fighting against this virus. Therefore, this paper focuses on the development of a novel mathematical model incorporating the immune cells(T and Z cells) and studies the dynamics of the model. The study of the dynamics of HIV-immune cells and the CD8 cell response can provide a rational selection of strategies for treatment and cure based on CD8 cells. A critical threshold (basic reproduction number) ρ0 is obtained along with the existence of disease-free equilibrium, and endemic equilibrium without and with the immune response for the model. Further, a Lyapunov function is constructed using the graph-theoretic approach to establish the global dynamics for endemic equilibrium(with and without immune response) point and matrix theoretic method for disease-free equilibrium. Local sensitivity analysis for ρ0 and E2 has also been carried out to recognize the sensitive parameters which may help in controlling the disease. The numerical discussion is carried out to validate our theoretical results. Sensitivity analysis has also been carried out for basic reproduction number and endemic equilibrium point with the immune response to recognize the sensitive parameters which may help in controlling the disease. Finally, the impact of the immune response of activated cells directly helping in suppressing the viral replication is shown by uncertainty analysis using PRCC. We were motivated to develop the novel model of HIV with immune response and study the dynamics of the model in the presence of CD8 cells as activated CD8-cells based immune response treatments can help control the disease at a mild stage itself due to its cytotoxic potential.

In a world of raging epidemic spreads, HIV has a major impact on the health of the global public. Immune system cells play a major role in defending the body from the impact of the virus. HIV's major effect is on CD4+ cells (or T cells) and CD8+ cells (or Z cells). The activation of CD8 cells has a major impact as it helps in fighting against this virus. Therefore, this paper focuses on the development of a novel mathematical model incorporating the immune cells(T and Z cells) and studies the dynamics of the model. The study of the dynamics of HIV-immune cells and the CD8 cell response can provide a rational selection of strategies for treatment and cure based on CD8 cells. A critical threshold (basic reproduction number) rho(0) is obtained along with the existence of disease-free equilibrium, and endemic equilibrium without and with the immune response for the model. Further, a Lyapunov function is constructed using the graph-theoretic approach to establish the global dynamics for endemic equilibrium(with and without immune response) point and matrix theoretic method for disease-free equilibrium. Local sensitivity analysis for rho(0) and E-2 has also been carried out to recognize the sensitive parameters which may help in controlling the disease. The numerical discussion is carried out to validate our theoretical results. Sensitivity analysis has also been carried out for basic reproduction number and endemic equilibrium point with the immune response to recognize the sensitive parameters which may help in controlling the disease. Finally, the impact of the immune response of activated cells directly helping in suppressing the viral replication is shown by uncertainty analysis using PRCC. We were motivated to develop the novel model of HIV with immune response and study the dynamics of the model in the presence of CD8 cells as activated CD8-cells based immune response treatments can help control the disease at a mild stage itself due to its cytotoxic potential.

This paper addresses the problem of determining the optimal promotional policy for a diffusion model in a segment-specific market under the assumption that the additional demand of the new product also improves brand image in the form of goodwill of the firm. The model is framed with the assumption that the firm uses the mass and differentiated promotion effort for each segment. The differentiated promotional efforts target each market segment independently and the mass promotional effort reaches different segments with a fixed spectrum. We derive the optimal promotional effort policy for each segment using maximum-principle and also analyze the stability of the dynamical system by constructing a Lyapunov function through the graph theoretic approach. The analysis gives a deep insight into how the promotional effort should be planned by the decision makers keeping in mind the financial constrains without hindering the promotional effort at the end of the planning period.