Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model’s dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if ${\mathcal{R}}_v$ is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter ${\beta }_2$ that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when $\psi =1$ for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free (${\mathcal{R}}_0\le 1$) and endemic cases (${\mathcal{R}}_0>1$). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number ${\mathcal{R}}_0\approx 1.2591$. We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail.

We construct a model to investigate HIV/AIDS dynamics in real cases and study its mathematical analysis. The study examines the qualitative outcomes and confirms the local and global asymptotic stability of both the endemic equilibrium and the disease-free equilibrium. The model’s criteria for exhibiting both local and global asymptotically stable behavior are examined. We compute the endemic equilibria and obtain the existence of a unique positive endemic equilibrium. The data is fitted to the model using the idea of nonlinear least-squares fitting. Accurate parameter values are achieved by fitting the data to the model using a 95% confidence interval. The basic reproduction number is computed using parameters that have been fitted or estimated. Sensitivity analysis is performed to discover the influential parameters that impact the reproduction number and the eradication of the disease. The results show that implementing preventive measures can reduce HIV/AIDS cases.

The parametric analyses for the Darcy-Forchheimer CNTs hybrid nanofluid (HNF) flow over a permeable spinning disc have been elaborated in this article. The effect of thermal slip and viscous dissipation on the surface of a spinning disc has been evaluated. The HNF is synthesized in the presence of an inorganic compound namely ferric oxide and carbon nanotubes. The nanofluid flow phenomena have been modeled in the form of the system of PDEs along with the Maxwell equation. Using a similarity strategy, the nonlinear PDEs system is simplified to ODEs. The ordinary set of ODEs are then computed using the parametric continuation approach (PCM). The outcomes are evaluated by comparing to boundary value solver (bvp4c) package, for reliability and validity purposes. Multiple flow parameters have been displayed and addressed in relation to energy, mass, and velocity profiles. The superparamagnetic property of magnetic nanoparticles and week van der Waals force interactions between C–C atoms make the current study more plausible. It has been perceived that the use of CNTs and magnetic nanoparticles in the water significantly enhances its thermophysical properties and heat transition properties. The energy profile accelerates with the effect of Eckert number, Forchheimer term, and volume friction parameters.

We consider a new mathematical model for the COVID-19 disease with Omicron variant mutation. We formulate in details the modeling of the problem with omicron variant in classical differential equations. We use the definition of the Atangana–Baleanu derivative and obtain the extended fractional version of the omicron model. We study mathematical results for the fractional model and show the local asymptotical stability of the model for infection-free case if ${\mathcal{R}}_0<1$. We show the global asymptotically stable of the model for the disease free case when ${\mathcal{R}}_0\le 1$. We show the existence and uniqueness of solution of the fractional model. We further extend the fractional order model into piecewise differential equation system and give a numerical algorithm for their numerical simulation. We consider the real cases of COVID-19 in South Africa of the third wave March 2021–Sep 2021 and estimate the model parameters and get ${\mathcal{R}}_0\approx 1.4004$. The real parameters values are used to show the graphical results for the fractional and piecewise model.

Purpose: This paper studies a simple SVIR (susceptible, vaccinated, infected, recovered) type of model to investigate the coronavirus’s dynamics in Saudi Arabia with the recent cases of the coronavirus. Our purpose is to investigate coronavirus cases in Saudi Arabia and to predict the early eliminations as well as future case predictions. The impact of vaccinations on COVID-19 is also analyzed. Methods: We consider the recently introduced fractional derivative known as the generalized Hattaf fractional derivative to extend our COVID-19 model. To obtain the fitted and estimated values of the parameters, we consider the nonlinear least square fitting method. We present the numerical scheme using the newly introduced fractional operator for the graphical solution of the generalized fractional differential equation in the sense of the Hattaf fractional derivative. Mathematical as well as numerical aspects of the model are investigated. Results: The local stability of the model at disease-free equilibrium is shown. Further, we consider real cases from Saudi Arabia since 1 May–4 August 2022, to parameterize the model and obtain the basic reproduction number ${\mathcal{R}}_0^v\approx 2.92$. Further, we find the equilibrium point of the endemic state and observe the possibility of the backward bifurcation for the model and present their results. We present the global stability of the model at the endemic case, which we found to be globally asymptotically stable when ${\mathcal{R}}_0^v>1$. Conclusion: The simulation results using the recently introduced scheme are obtained and discussed in detail. We present graphical results with different fractional orders and found that when the order is decreased, the number of cases decreases. The sensitive parameters indicate that future infected cases decrease faster if face masks, social distancing, vaccination, etc., are effective.