Sonari College

The baking system serves as the backbone of a country’s economy and plays a crucial role in its development. The banking system supports economic growth by mobilizing savings, granting loans, handling payments, supporting government policies, and maintaining financial stability. A distressed bank is one facing financial hardship due to high Non-Performing Assets, losses, mismanagement, or liquidity problems. Such banks often require regulatory assistance from a healthy, operating bank (an undistressed bank) to regain stability. Due to the interconnectedness of banks, the failure of one bank may trigger financial instability in others, a phenomenon known as systemic risk, and may lead to a banking crisis. So, it is important to have effective policies that can mitigate the spread of banking crises and thereby stabilize the country’s overall economy. This paper aims to develop an optimal strategy to eliminate systemic risk contagion in the banking sector. For this purpose, we propose a mathematical model consisting of four ordinary differential equations with two controls: measures taken by undistressed banks to provide liquidity and to guide risk management. In contrast, the other control represents financial support extended by the central bank, such as emergency credit lines, strengthening monitoring and supervision, and strict guidelines to reduce NPA. The model’s basic properties, such as non-negativity and boundedness of solutions, are established to demonstrate its financial feasibility. The basic reproduction number (ℛ_0) of the model is calculated using the next generation matrix method. The local and global stability results are obtained for both risk-free and risk equilibrium points. It is found that there is no chance of systemic risk contagion if ℛ_0<1 . Further, a delay model has been formulated and studied to investigate the impact of time delay in declaring an exposed bank as a distressed bank using numerical simulation. A semi-relative sensitivity analysis is performed to examine the most influential parameters affecting undistressed and distressed banks. Moreover, an optimal control problem is formulated to obtain a strategy that minimizes the number of contagious banks while incurring minimal associated costs. The existence of the optimal controls is shown, and the optimality conditions are obtained analytically. Finally, the proposed model is simulated to substantiate the analytical results, and the optimal control problem is solved numerically using a forward-backwards iterative method. This study provides an optimal control approach to mitigate the banking crisis by minimizing the number of distressed banks while incurring minimal investment.

Abstract Cancer is the second leading cause of death worldwide and occurs when cells uncontrollably divide, grow, and invade surrounding tissue. Several treatment strategies have been introduced to increase survival rates; however, each approach has potential side effects. Consequently, demand for alternative treatment approaches, such as immune system enhancement with vitamins, is growing. Here, we introduce a mathematical framework based on ordinary differential equations to examine the interactions among nutrients, immune cells, and cancer cells. Unlike many existing tumor–immune models, the proposed framework reveals that variations in nutrient levels can induce qualitative changes in system dynamics, including bistability and regime transitions between tumor-dominant and immune-controlled states. Long-term behavior, stability, and bifurcation techniques are employed. Numerical simulations are performed using the odeint solver in SciPy, which is based on the LSODA algorithm and is capable of handling both stiff and non-stiff systems. Our results suggest that optimizing nutrient levels can enhance the immune response, potentially leading to safer cancer treatment strategies.

In this study, we develop a delay differential equation model for terrorism dynamics by extending the framework of Kumar et al. (2023). The population is divided into susceptible individuals, active terrorists, and those who have deserted or resisted recruitment, with an additional component incorporating economic stress to capture the influence of socioeconomic conditions such as unemployment and inequality on radicalization. A time delay is introduced to represent the ideological incubation period associated with the transition to terrorist activity. We investigate the qualitative dynamics of the model, including the existence and stability of equilibrium points. The analysis shows that the terror-free equilibrium is locally asymptotically stable when the basic reproduction number R-0 < 1, while a terrorism-persistent equilibrium exists and remains stable when R-0 > 1. Notably, numerical simulations indicate that time delay has a limited influence on the system dynamics for the selected parameter set, with no significant qualitative changes or sustained oscillations observed in the long-term behavior. Furthermore, a physics-informed neural networks (PINNs) framework is employed to solve the model and estimate key parameters from world data. In particular, terrorism data from Pakistan over the period 2014-2020 are used to validate the model, and the PINN approach demonstrates high accuracy in capturing the observed dynamics. The results highlight the role of economic stress in influencing terrorism persistence within the modeling framework. However, these findings should be interpreted as qualitative insights rather than direct policy recommendations. Overall, the proposed framework provides a data-informed and theoretically grounded approach for analyzing terrorism dynamics under socio-economic influences.

Financial systems often exhibit complex, nonlinear behavior influenced by multiple interrelated economic factors. In this paper, a four-dimensional financial model is developed, incorporating the money supply as a new factor alongside interest rate, investment demand, and the price index. The model is initially formulated in integer order and later extended to fractional order to capture the memory effects characteristic of real-world financial processes. A comprehensive dynamic analysis is performed to investigate equilibrium points, stability conditions, and the impact of varying monetary injection efficiency. The effect of fractional order variation on system dynamics is also examined to understand its role in stability and complexity. To address instability, a feedback control strategy is proposed within the fractional framework to regulate the system’s behavior. Numerical simulations are used to validate the theoretical approach and demonstrate the potential utility of the control mechanism in maintaining financial system stability.

This study develops compartmental epidemic models based on systems of ordinary and delayed differential equations to investigate the transmission dynamics of COVID-19. A normalized model incorporating a discrete time delay associated with the latent period of infection is analyzed, and the basic reproduction number R_0 is derived to examine its influence on the stability of disease-free and endemic equilibria. It is shown that the disease-free equilibrium is locally asymptotically stable when R_0 < 1 , whereas the endemic equilibrium is stable for R_0> 1 , independently of the length of the delay. The model is subsequently extended to include vaccination, allowing the impact of different vaccination rates on disease transmission to be examined. Numerical simulations are presented to support the analytical results and to illustrate how time delay and vaccination jointly affect the system dynamics. The results highlight the critical role of vaccination in reducing the basic reproduction number below unity and achieving disease elimination. The framework provides a theoretical basis for designing time-sensitive vaccination policies in future pandemics.