Research on linear and nonlinear IS-LM models has been resonating under synonymous perspectives, confined to bifurcations and intangible relations to economic work systems. Trifle discussion exists on how choice of linear/nonlinear models affects policy making and almost no elaboration on framing an economic system within a linear and nonlinear structure to analyze their effect separately. Parameters surrounding IS-LM model like adjustment coefficients, depreciation of capital stock etc. have not been given due spotlight, given the audacity they possess to modulate system dynamics. In counteraction, we have investigated an augmented IS-LM model with two-time delays in capital accumulation equation. This model is subjected to linear and nonlinear arguments of investment, savings and liquidity function giving rise to $M_1$(linear) and $M_2$(nonlinear) models. They undergo hopf bifurcation for different values of delay parameters $\tau_{1}$ and $\tau_{2}$. Our study accentuates the following aspects(1) In a neophyte attempt, comparing the dynamics of a linear and nonlinear business cycle model in an environment as similar as possible, when $\tau_{1}$ and $\tau_{2}$ are the bifurcating parameters. (2) Parameter sensitivity analysis for both models. (3) Non linearity in savings function, which is a sparse event so far. Our findings reveal that (1) Non-linearity elevates system sensitivity and $M_2$ model attains stability easily in the long run for dual delays, while for single delay $M_1$ model has this feat. (2) $M_2$ model encapsulates recurring cyclic behavior while $M_1$ model is not capable of generating the same and demonstrates motifs of either stability or instability. (3) Parameter sensitivity analysis reveals that both the models are most vulnerable when (3a)Value of depreciation of capital stock is decreased. (3b) Money supply and propensities to investment are increased. (4) how aforementioned information can be utilized for crafting economic policies for linear/nonlinear economies, especially curated for their modus operandi. Numerical simulations follow.

Consistently, influenza has become a major cause of illness and mortality worldwide and it has posed a serious threat to global public health particularly among the immuno-compromised people all around the world. The development of medication to control influenza has become a major challenge now. This work proposes and analyzes a structured model based on two geographical areas, in order to study the spread of influenza. The overall underlying population is separated into two sub populations: urban and rural. This geographical distinction is required as the immunity levels are significantly higher in rural areas as compared to urban areas. Hence, this paper is a novel attempt to proposes a linear and non-linear mathematical model with adaptive immunity and compare the host immune response to disease. For both the models, disease-free equilibrium points are obtained which are locally as well as globally stable if the reproduction number is less than 1 (R₀₁ < 1 & R₀₂ < 1) and the endemic point is stable if the reproduction number is greater then 1 (R₀₁ > 1 & R₀₂ > 1). Next, we have incorporated two treatments in the model that constitute the effectiveness of antidots and vaccination in restraining viral creation and slow down the production of new infections and analyzed an optimal control problem. Further, we have also proposed a spatial model involving diffusion and obtained the local stability for both the models. By the use of local stability, we have derived the Turing instability condition. Finally, all the theoretical results are verified with numerical simulation using MATLAB.

The COVID-19 vaccine has been available in India since January 2021, although many individuals have refused to take the vaccine for various reasons. Vaccination plays a crucial role in disease control by preventing a substantial number of cases and associated disabilities. However, vaccine hesitancy poses a barrier that hinders these efforts. Our article presents a novel approach by proposing a mathematical model for COVID-19 that incorporates vaccine hesitancy, vaccine efficacy and behaviour compensation post-vaccination. The model is calibrated with COVID-19 incidence data for India from 13 February 2021 to 12 January 2022, using the Markov chain Monte Carlo method. The analysis examines the effects of hesitancy and social interventions through a series of practical simulations. The simulation results show that while COVID-19-infected individuals may have natural immunity, vaccination post-recovery is crucial to reduce cases by up to 64.1%. Social interventions, such as face masks and distancing, remain essential to prevent a rise in cases and ensure effective disease control. The model demonstrates that vaccination, combined with continued social interventions, is crucial for effectively reducing COVID-19 cases and preventing future outbreaks. Addressing vaccine hesitancy and maintaining preventive measures are key to successfully controlling the pandemic.