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Abstract The interplay of exchange rates among nations significantly influences both international and domestic trade, underscoring the pivotal role of the foreign exchange market (Forex) in a country's economic landscape. Forex fluctuations have a significant impact on the everyday lives of both government agencies and the public population, directly influencing a country's prosperity or misfortune. This work proposes an advanced fuzzy time series model that incorporates domain universe sub-partitioning, parameter adjustment optimization methodologies, and interval forecasting methods. We utilized this model to examine annual exchange rate patterns between the Pakistani rupee (PKR) and the US dollar (US$), comparing its forecast accuracy to that of other models. Our proposed methodology outperformed existing methodologies in terms of forecasting precision, providing stakeholders with valuable insights for making informed, data-driven business decisions that benefit both individual firms and the country overall.

Abstract Toxoplasma infection in humans is considered due to direct contact with infected cats. Toxoplasma infection (an endemic disease) has the potential to affect various organs and systems (brain, eyes, heart, lungs, liver, and lymph nodes). Bilinear incidence rate and constant population (birth rate is equal to death rate) are used in the literature to explain the dynamics of Toxoplasmosis disease transmission in humans and cats. The goal of this study is to consider the mathematical model of Toxoplasma disease with harmonic mean type incident rate and also consider that the population of humans and cats is not equal (birth rate and the death rate are not equal). In examining Toxoplasma transmission dynamics in humans and cats, harmonic mean incidence rates are better than bilinear incidence rates. The disease dynamics are first schematically illustrated, and then the law of mass action is applied to obtain nonlinear ordinary differential equations (ODEs). Analysis of the boundedness, positivity, and equilibrium points of the system has been analyzed. The reproduction number is calculated using the next-generation matrix technique. The stability of disease-free and endemic equilibrium are analyzed. Sensitivity analysis is also done for reproduction number. Numerical simulation shows that the infection is spread in the population when the contact rate $$\beta_{h}$$ β h and $$\beta_{c}$$ β c increases while the infection is reduced when the recovery rate $$\delta_{h}$$ δ h increases. This study investigates the impact of various optimal control strategies, such as vaccinations for the control of disease and the awareness of disease awareness, on the management of disease.