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It is acknowledged that dengue infection has a significant economic impact due to healthcare costs and lost productivity. Research can provide insights into the economic burden of the disease, guiding policymakers in their allocation of resources for prevention and control interventions. In this work, we structured a novel mathematical model that describes the spread of dengue with the effects of carriers, an index of memory and vaccination. To show the effect of treatment on the dynamics of dengue, we have incorporated medication-related treatment into the system. The proposed dynamics are represented by using fractional derivatives to capture the role of memory in the control of the infection. We introduced the fundamental principles and notions of non-integer derivatives for the analysis of the model; moreover, the existence and uniqueness results for the solution of the system have been established with the help of mathematical skills. The theory of fixed points has been utilized for the analysis and examination of the system. We have established Ulam-Hyers stability for the recommended system of dengue infection. Regarding the numerical findings, a numerical method is presented to highlight the solution pathways for the system of dengue infection. Several simulations have been performed to visualize the contribution of the input parameters of the system to the prevention and control of the infection. The index of memory, vaccination, and treatment are suggested to be attractive parameters which can reduce the level of infection while the biting rate, asymptomatic carriers and transmission rate are critical as they can increase the risk of the infection in society. Our findings not only provide information for the effective management of the infection they also possess valuable insights that can improve public health. [ABSTRACT FROM AUTHOR]

In this study, we formulate a mathematical model in the framework of the Atangana-Baleanu fractional derivative in Caputo sense to study the transmission of tungiasis. In this formulation, interactions between the human host and the sand fleas are taken into consideration, including factors like infestation rate, incubation duration, and recovery rate. We calculate the basic reproduction parameter for the system, symbolized by R0 with the help of the next-generation matrix technique. A novel numerical scheme for encapsulating the non-local and memory-dependent aspects of the system is conceptualized via the Atangana-Baleanu fractional derivative. We prove the existence and uniqueness of the solution of the model of the infection and establish stability of the steady-states of the model. In addition to this, numerical simulations are carried out to evaluate the efficiency of interventions like campaigns for better sanitation and treatment, and to investigate the influence of various management techniques on the prevalence of tungiasis. The outcomes of the numerical simulations give us information about the possible efficacy of different control strategies in lowering the incidence of tungiasis. This research gives quantitative tools to enhance decision-making processes in public health treatments and advances our understanding of the dynamics of the tungiasis. [ABSTRACT FROM AUTHOR]