The main objective of this work is to develop the fuzzy fractional mathematical model that will be used to examine the dynamics of monkeypox viral transmission. The proposed dynamical model consists of human and rodents individuals and this monkeypox infection model is mathematically formulated by fuzzy fractional differential equation defined in Caputo’s sense. We provide results that demonstrate the existence and uniqueness of the considered model’s solution. We observe that our results are accurate, and that our method is applicable to the fuzzy system of fractional ordinary differential equations (ODEs). Furthermore, this monkeypox virus model has been identified as a generalization of SEIQR and SEI models. The results show that keeping diseased rodents apart from the human population reduces the spread of disease. Finally, we present brief discussions and numerical simulations to illustrate our findings.

The major goal of this study is to create an optimal technique for managing COVID‐19 spread by transforming the SEIQR model into a dynamic (multistage) programming problem with continuous and discrete time‐varying transmission rates as optimizing variables. We have developed an optimal control problem for a discrete‐time, deterministic susceptible class (S), exposed class (E), infected class (I), quarantined class (Q), and recovered class (R) epidemic with a finite time horizon. The problem involves finding the minimum objective function of a controlled process subject to the constraints of limited resources. For our model, we present a new technique based on dynamic programming problem solutions that can be used to minimize infection rate and maximize recovery rate. We developed suitable conditions for obtaining monotonic solutions and proposed a dynamic programming model to obtain optimal transmission rate sequences. We explored the positivity and unique solvability nature of these implicit and explicit time‐discrete models. According to our findings, isolating the affected humans can limit the danger of COVID‐19 spreading in the future.

This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The $\nu$th-fractional sum of a function f has two forms; closed form and summation form. Most authors in the previous literature focused on the summation form rather than developing the closed-form solutions, which is to say that they were more concerned with the summation form. However, finding a solution in a closed form requires less time than in a summation form. This inspires us to develop a new approach, which helps us to find the closed form related to nth-sum for a class of delta integrable functions, that is, functions with both discrete integration and nth-sum. By equating these two forms of delta integrable functions, we arrive at several identities (known as discrete fundamental theorems). Also, by introducing ∞-order delta integrable functions, the discrete integration related to the $\nu$th-fractional sum of f can be obtained by applying Newton’s formula. In addition, this concept is extended to h-delta integration and its sum. Our findings are validated via numerical examples. This method will be used to accelerate computer-processing speeds in comparison to summation forms. Finally, our findings are analyzed with outcomes provided of diagrams for geometric, polynomial and falling factorial functions.

This study investigates the dynamics and control of an infectious disease using an age-structured Susceptible-Vaccinated-Infected (SVI) model, incorporating imperfect vaccination and therapeutic treatment. We identify a backward bifurcation phenomenon, driven by treatment rates, which allows disease persistence even when the basic reproduction number R0 < 1, complicating eradication e orts. To address this, we formulate an optimal control problem with time-dependent vaccination and treatment rates to minimize infected individuals and control costs. Using bifurcation theory and integrated semigroup methods, we establish the model's well-posedness, equilibria stability, and bistability conditions. First-order optimality conditions are derived to characterize optimal controls. Numerical simulations,solved via the Forward-Backward Sweep method, demonstrate that combined vaccination and treatment signi cantly reduces disease prevalence, with early intervention being critical. These ndings underscore the importance of strategic resourceallocation in managing complex epidemic dynamics.

In the face of the COVID-19 pandemic, understanding the dynamics of disease transmission is crucial for effective public health interventions. This study explores the concept of symmetry within compartmental models, employing compartmental analysis and numerical simulations to investigate the intricate interactions between compartments and their implications for disease spread. Our findings reveal the conditions under which the disease-free equilibrium is globally asymptotically stable while the endemic equilibrium exhibits local stability. Additionally, we investigate the phenomenon of backward bifurcation, shedding light on the critical role of quarantine measures in controlling outbreaks. By integrating the concept of symmetry into our model, we enhance our understanding of transmission dynamics and provide a robust framework for evaluating intervention strategies. The insights gained from this research are vital for policymakers and health authorities aiming to mitigate the impact of infectious diseases in the future.