Global dynamics and computational modeling for analyzing and controlling Hepatitis B: A novel epidemic approach.

Hepatitis B virus (HBV) infection is a global public health issue. We offer a comprehensive analysis of the dynamics of HBV, which can be successfully controlled with vaccine and treatment. Hepatitis B virus (HBV) causes a significantly more severe and protracted disease compared to hepatitis A. While it initially presents as an acute disease, in approximately 5 to 10% of cases, it can develop into a chronic disease that causes permanent damage to the liver. The hepatitis B virus can remain active outside the body for at least seven days. If the virus penetrates an individual's body without immunization, it may still result in infection. Upon exposure to HBV, the symptoms often last for a duration ranging from 10 days to 6 months. In this study, we developed a new model for Hepatitis B Virus (HBV) that includes asymptomatic carriers, vaccination, and treatment classes to gain a comprehensive knowledge of HBV dynamics. The basic reproduction number [Formula: see text] is calculated to identify future recurrence. The local and global stabilities of the proposed model are evaluated for values of [Formula: see text] that are both below and above 1. The Lyapunov function is employed to ensure the global stability of the HBV model. Further, the existence and uniqueness of the proposed model are demonstrated. To look at the solution of the proposed model graphically, we used a useful numerical strategy, such as the non-standard finite difference method, to obtain more thorough numerical findings for the parameters that have a significant impact on disease elimination. In addition, the study of treatment class in the population, we may assess the effectiveness of alternative medicines to treat infected populations can be determined. Numerical simulations and graphical representations are employed to illustrate the implications of our theoretical conclusions.

A fractional modeling approach for the transmission dynamics of measles with double-dose vaccination.

Measles, a member of the Paramyxoviridae family and the Morbillivirus genus, is an infectious disease caused by the measles virus that is extremely contagious and can be prevented through vaccination. When a person with the measles coughs or sneezes, the virus is disseminated by respiratory droplets. Normally, the appearance of measles symptoms takes 10-14 d following viral exposure. Conjunctivitis, a high temperature, a cough, a runny nose, and a distinctive rash are some of the symptoms. Despite the measles vaccination being available, it is still widespread worldwide. To eradicate measles, the Reproduction Number (i.e. R0<1) must remain less than unity. This study examines a SEIVR compartmental model in the caputo sense using a double dose of vaccine to simulate the measles outbreak. The reproduction number R0 and model properties are both thoroughly examined. Both the local and global stabilities of the proposed model are determined for R0 less and greater than 1. To achieve the model's global stability, the Lyapunov function is used while the existence and uniqueness of the proposed model are demonstrated In addition to the calculated and fitted biological parameters, the forward sensitivity indices for R0 are also obtained. Simulations of the proposed fractional order (FO) caputo model are performed in order to analyse their graphical representations and the significance of FO derivatives to illustrate how our theoretical findings have an impact. The graphical results show that the measles outbreak is reduced by increasing vaccine dosage rates.

Cattaneo-Christov heat flow model at mixed impulse stagnation point past a Riga plate: Levenberg-Marquardt backpropagation method.

Applications of artificial intelligence (AI) via soft computing procedures have attracted the attention of researchers due to their effective modeling, simulation procedures, and detailed analysis. In this article, the designing of intelligence computing through a neural network that is backpropagated with the Levenberg-Marquardt method (NN-BLMM) to study the Cattaneo-Christov heat flow model at the mixed impulse stagnation point (CCHFM-MISP) past a Riga plate is investigated. The original model CCHFM-MISP in terms of PDEs is converted into non-linear ODEs through suitable similarity variables. A data set is generated for all scenarios of CCHFM-MISP through Lobatto IIIA numerical solver by varying Hartman number, velocity ratio parameter, inverse Darcy number, mixed impulse variable, non-dimensional constraint, Eckert number, heat generation variable, Prandtl number, thermal relaxation variable. To find the physical impacts of parameters of interest associated with the presented fluidic system CCHFM-MISP, the approximate solution of NN-BLMM is carried out by performing training (80 %), testing (10 %), and validation (10 %), and then the results are equated with the reference data to ensure the perfection of the proposed model. Through MSE, state transition, error histogram, and regression analysis, the outcomes of NN-BLMM are presented and analyzed. The graphical illustration and numerical outcomes confirm the authentication and effectiveness of the solver. Moreover, mean square errors for validation, training and testing data points along with performance measures lie around 10-10 and the solution plots generated through deterministic (Lobatto IIIA) approach and stochastic numerical solver are matching up to 10-6, which surely validate the solver NN-BLMM. The outcomes of M and B on velocity present the similar impacts. The velocity of material particles decreases under Da while, it increases through velocity ratio and magnetic parameters.

Numerical treatment of thermal nanofluid flow for energy enhancement over a porous stretching sheet impact of slip and buoyancy force.

The advancement of nanofluid innovation is a crucial area of research for physicists, mathematicians, manufacturers, and materials scientists. In engineering and industries, the fluid velocity caused by stretching sheets and nanofluids has a lot of applications such as refrigerators, chips, heat exchangers, hybrid mechanical motors, food development, and so on. The originality of the current study is the analysis of the thermal nanofluid in the existence of a porous matrix, and buoyancy force over the stretched sheet, so in limiting cases, the existing work is equated with the available effort, and excellent correspondence is originated. The governing equations in terms of PDEs are changed to the convection differential by utilizing the appropriate transformation and then solved by the ND-solved method along with bvph2. The thermal boundary layer thickness upsurges as the radiation and temperature factors are improved. It is observed that with the growing amount of volume fraction factor the velocity profile declines. When the velocity slip factors and permeability are enhanced the velocity profile augments. It is examined as the values of permeability factor, Biot number, and velocity slip factor are increased the inner temperature of the fluid improves. For the increasing values of θ_r, ϕ, and Nr, the temperature is increasing. In the future, the present model can be extended by using the hybrid nanofluid for the activation of thermal conductivity and heat enhancement analysis.

Impact of variable thermal conductivity on flow of trihybrid nanofluid over a stretching surface.

The present article describes the impact of variable thermal conductivity on the flow of ternary hybrid nanofluid with cylindrical shape nanoparticles over a stretching surface. Three nanoparticles combine in base fluid polymer. The assumption made will be used to model an equations. Modeled equations are in the form of a system of partial differential equations are difficult to solve can be converted to system of an ordinary differential equations, through resemblance substitutions, and will be solved numerically. Numerical scheme of Runge-Kutta order four is coupled with the shooting method to solve the resulting equations. The graphs in the study illustrate how physical quantities, such as magnetic field, injection/suction, nanoparticles volume fraction, and variable thermal conductivity, affected the velocity, skin friction, temperature, and local Nusselt number. The velocity profiles deflate as the volume fraction rises. While the temperature rises with an increase in the volume fraction of nanoparticles for both injection and suction, the velocity profiles also decline as the injection and suction parameter increases. Furthermore, as the magnetic field increases, the temperature profile rises while the velocity profile falls. The temperature curves increase as thermal conductivity increases. Finally, as the magnetic field is strengthened, the Nusselt number and skin friction decrease. The combination of mathematical modeling, numerical solution techniques, and the analysis of physical quantities contributes to the advancement of knowledge in this ternary hybrid nanofluid.

Mathematical modeling of magnetized nanofluid flow over extending and contracting pipe with heat source and chemical reaction: Error estimation and stability analysis.

The aim of the current study is to examine the magnetohydrodynamic (MHD) flow over the permeable pipe containing the nanoparticles with a heat transport mechanism. The leading equations of flow are obtained in terms of partial differential equations (PDEs). The suitable transformation is applied to alter PDEs to ordinary differential equations (ODEs) and solved numerically via the RK4 method. The novelty of the current work is to investigate the MHD nanofluid overextending and shrinking pipe enclosing the chemical reaction with a heat reservoir. The outcomes of different factors are depicted through graphs and tables on the flow phenomena. It is observed that for both situations of wall extension or contraction with injection, the temperature is the increasing function of the thermophoresis and Brownian motion factors. It is explored that the heat is upraised when the heat generation is augmented but decays for heat absorption. The point to be noted here is that the Hartmann number and Prandtl number enhance the heat curves in the presence of a heat source. It is also observed that when the thermophoresis factor is increased the nanoparticles concentration is also enhanced via heat source/sink. The error estimation is computed for different order approximations. For the confirmation of the mathematical modeling, the current study is validated with the previous.

Supporting Public Health Research Capacity, Quality, and Productivity in a Diverse Region.

Public health research plays a critical role in strengthening health systems and improving their performance and impact. However, scholarly production in public health coming from the Eastern Mediterranean Region (EMR) remains well below the world average and lacks a tangible growth trend over time. During the seventh Eastern Mediterranean Public Health Network Regional Conference, a roundtable session brought together a panel of public health experts representing Global Health Development/Eastern Mediterranean Public Health Network affiliates, universities or academia, and research institutions from the region, where they shared insights on the current situation of public health research; challenges and barriers to research facing the different countries in the EMR and the region in general; and how research agendas, productivity, and quality can be supported through strengthening research capacity in the region. Although the region is diverse in terms of health system capacity and socioeconomic development, several common challenges were identified, including a lack of strategic prioritization to guide health research, insufficient funding, ineffective transfer of knowledge to policy and practice, limited availability of research facilities, and limited national and international research collaboration. Occupied countries and countries in a state of conflict, such as Palestine, face additional barriers, such as personal and social security, lack of control of borders and natural resources, travel and movement restrictions, and confidentiality challenges because of the continuing war conditions and occupation. However, there have been success stories in the EMR regarding research publications and their positive and effective impact on policy and decision-makers. To improve research resilience and public health care in the region, a collaborative approach involving institutions, policymakers, and relevant stakeholders is critical.

Two-dimensional nanofluid flow impinging on a porous stretching sheet with nonlinear thermal radiation and slip effect at the boundary enclosing energy perspective.

In the current analysis, we examine the heat transmission analysis of nanofluid (NF) movement impinging on a porous extending sheet. The influence of nonlinear thermal radiation (TR), buoyancy force, and slip at the boundary are also examined. The leading partial differential equations (PDEs) are altered to convectional differential equation (ODEs) by suitable transformation. The ODEs are then transformed to first order by introducing the innovative variables and elucidated numerically using bvph2. The Skin Friction (SF) and Nusselt number (NN) are elaborated in detail for Al2O3, Cu, and TiO2 nanoparticles. For validation of the code, ND-solve approach is also applied. The novelty of the current effort is inspect NF flow with heat transfer over extending sheet enclosing thermal and slip effect at the boundary numerically. The thickness of boundary layer increases as the temperature and radiation factors are increased. It is perceived that the fluid velocity decays with the growing values of volume fraction parameter. When permeability and velocity slip parameters are improved the velocity outline enhances. It is investigated that the temperature inside the fluid enhances as the values of velocity slip factor, permeability factor and Biot number are augmented. For the growing values of temperature ratio, volume friction, and thermophoresis factor the temperature is enhances. It is detected that the slip factor causes the friction factor to decrease. Furthermore, the existent study is associated with the preceding.

A memory effect model to predict COVID-19: analysis and simulation.

On 19 September 2020, the Centers for Disease Control and Prevention (CDC) recommended that asymptomatic individuals, those who have close contact with infected person, be tested. Also, American society for biological clinical comments on testing of asymptomatic individuals. So, we proposed a new mathematical model for evaluating the population-level impact of contact rates (social-distancing) and the rate at which asymptomatic people are hospitalized (isolated) following testing due to close contact with documented infected people. The model is a deterministic system of nonlinear differential equations that is fitted and parameterized by least square curve fitting using COVID-19 pandemic data of Pakistan from 1 October 2020 to 30 April 2021. The fractional derivative is used to understand the biological process with crossover behavior and memory effect. The reproduction number and conditions for asymptotic stability are derived diligently. The most common non-integer Caputo derivative is used for deeper analysis and transmission dynamics of COVID-19 infection. The fractional-order Adams-Bashforth method is used for the solution of the model. In light of the dynamics of the COVID-19 outbreak in Pakistan, non-pharmaceutical interventions (NPIs) in terms of social distancing and isolation are being investigated. The reduction in the baseline value of contact rates and enhancement in hospitalization rate of symptomatic can lead the elimination of the pandemic.

Epidemiologic and clinical characteristics of diabetic foot ulcer among patients with diabetes in Afghanistan: An IDF supported initiative.

AIM: The aim is to illustrate epidemiological and clinical characteristics of diabetic patients with foot ulcer (DFU) in Kabul diabetic medical center (KDMC), Afghanistan. METHOD: It is a descriptive study explaining the characteristics of diabetic patients with DFU admitted to KDMC, between 1/9/2019 to 31/8/2020 which is a center for management of diabetic patients including DFU. The university of Texas diabetic classification for DFU was used. RESULTS: Totally 3159 patients admitted to KMDC of whom 47.4% were females and 96.7% type 2 diabetes. The proportion of DFU was 9.2%. The patients' mean age was 55.4 ± 10.6 years and 78% were coming from Kabul. Prevalence of smoking and snuff use were 8.6% and 5.6% respectively. Majority of females 93% were housewives. The duration of diabetes was 5-19 years. Almost two-third were under glycaemia and HbA1c control and 9.2% had history of amputation. The common symptoms were burning, aching, numbness and tingling. The most common cause of DFU was both neuropathy and arteriopathy. After treatment 16% were referred for orthopedic procedures. CONCLUSIONS: DFU affects almost one-tenth of diabetics while a significant number of patients attend at late stage requiring orthopedic treatment. Monitoring of diabetic patients to prevent DFU is important is recommended.

Numerical Study of the Electromagnetohydrodynamic Bioconvection Flow of Micropolar Nanofluid through a Stretching Sheet with Thermal Radiation and Stratification.

The current work explores the bioconvection micropolar nanofluid through a stretching surface subjected to thermal radiation, stratification, and heat and mass transmission. Bioconvection contains the gyrotactic (random movement of microorganism in the direction of gravity with weak horizontal verticity) unicellular microorganism in aqueous environments. Heat and mass transfer assists the bioconvection to occur. The aim of this research is to evaluate the heat transfer rate of nanofluid in the presence of a unicellular microorganism. Self-similar variables are induced to reduce the governing equations into a non-linear differential system which is further solved via the bvp4c algorithm to tackle the fluid problem. Using visual representations, the effects of a number of dimensional less factors arising from the dimensional less differential system are determined. For a range of limiting conditions, the obtained results of this model correspond precisely to those in the literature. This study's findings are highly regarded in the evaluation of the impact of key design factors on heat transfer and, therefore, in the optimization of industrial processes. Skin friction, local Nusselt number, Sherwood number, and density of microorganism concentrations are also studied for various parameters. Buoyancy ratio factor supports skin friction and density of microorganism profile to increase. Local Nusselt number drops due to the thermal radiation factor. Brownian motion speeds up the Sherwood number.

Electromagnetic Trihybrid Ellis Nanofluid Flow Influenced with a Magnetic Dipole and Chemical Reaction Across a Vertical Surface.

The purpose of this study is to evaluate the augmentation of thermal energy transfer in trihybrid Ellis nanofluid flow in the occurrence of magnetic dipole passes over a vertical surface. The ternary hybrid nanofluid is prepared by the dispersion of ternary nanoparticles (Al2O3, SiO2, and TiO2) in the Carreau Yasuda fluid. The velocity and heat transportation has been examined in the existence of the Darcy Forchhemier influence and heat source/sink. The phenomena of fluid flow have been mathematically designed for energy and fluid velocity in the form of a nonlinear partial differential equation (PDE)-based system. The system of PDEs is further refined to the set of ordinary differential equations via suitable similarity substitutions. The acquired dimensionless equations are numerically solved with the help of the HAM. It has been noticed that the energy contour is enhanced versus the variation of viscous dissipation and heat generation. A significant contribution of a magnetic dipole is observed to elevate the production of the thermal energy field, and an opposite trend is noticed versus the flow profile. The accumulation of Al2O3, SiO2, and TiO2 nanomaterials in the base fluid "engine oil" improves the velocity and energy profiles.

Neuro-Computing for Hall Current and MHD Effects on the Flow of Micro-Polar Nano-Fluid Between Two Parallel Rotating Plates.

The present works focus on the effects of electric and magnetic fields on the flow of micro-polar nano-fluid between two parallel plates with rotation under the impact of Hall current (EMMN-PPRH) has considered by using Artificial Neural Networks with the scheme of Levenberg-Marquardt backpropagation (ANN-SLMB). The nonlinear PDEs are transformed into nonlinear ODEs by employing similarity variables. By varying different parameters such as coupling parameter, electric parameter, rotation parameter, viscosity parameter, Prandtl number and the Brownian motion parameter, a dataset for recommended ANN-SLMB is produced for numerous scenarios through utilizing homotopy analysis method (HAM). The ANN-SLMB training, testing and validation technique have been used to analyze the approximate solution of individual cases, and the recommended model has matched for confirmation. After that, regression analysis, MSE, and histogram investigations were utilized to validate the proposed ANN-SLMB. The recommended technique is distinguished nearest of the suggested and reference findings, with an accuracy level ranging from 10-09 to 10-11.

Electromagnetohydrodynamic bioconvective flow of binary fluid containing nanoparticles and gyrotactic microorganisms through a stratified stretching sheet.

Bioconvection has recently been the subject of dispute in a number of biotechnological fields that depend on fluids and their physical properties. When mixed nanofluids are subjected to heat and mass transmission, the process of bioconvection occurs. This attempt conveys the theoretical analysis of two-dimensional electrically conducting and magnetically susceptible binary fluid containing nanoparticles and gyrotactic microorganisms past a stratified stretching surface. Furthermore binary chemical reaction, thermal radiation, and activation energy are taken into assumptions. The analytical solution based on HAM has been performed. The convergence of HAM is presented with the help of figures. The present study is compared with previously published results and has established an excessive agreement which validate the present study. It is perceived that the presence and absence of an electric field influences the variations in fluid velocities due to presence of magnetic field. The micropolar constant heightens the velocity and microrotation of the fluid flow. The buoyancy parameter and bioconvection Rayleigh number diminish the velocity function while these parameters show dual impact on microrotation function. The skin friction and couple stress escalates with the increasing buoyancy ratio parameter and bioconvection Rayleigh number.

The intelligent networks for double-diffusion and MHD analysis of thin film flow over a stretched surface.

This study presents a novel application of soft-computing through intelligent, neural networks backpropagated by Levenberg-Marquardt scheme (NNs-BLMS) to solve the mathematical model of unsteady thin film flow of magnetized Maxwell fluid with thermo-diffusion effects and chemical reaction (TFFMFTDECR) over a horizontal rotating disk. The expression for thermophoretic velocity is accounted. Energy expression is deliberated with the addition of non-uniform heat source. The PDEs of mathematical model of TFFMFTDECR are transformed to ODEs by the application of similarity transformations. A dataset is generated through Adams method for the proposed NNs-BLMS in case of various scenarios of TFFMFTDECR model by variation of rotation parameter, magnetic parameter, space dependent heat sink/source parameter, temperature dependent heat sink/source parameter and chemical reaction controlling parameter. The designed computational solver NNs-BLMS is implemented by performing training, testing and validation for the solution of TFFMFTDECR system for different variants. Variation of various physical parameters are designed via plots and explain in details. It is depicted that thin film thickness increases for higher values of disk rotation parameter, while it diminishes for higher magnetic parameter. Furthermore, higher values of Dufour number and the corresponding diminishing values of Soret number causes enhancement in fluid temperature profile. Further the effectiveness of NNs-BLMS is validated by comparing the results of the proposed solver and the standard solution of TFFMFTDECR model through error analyses, histogram representations and regression analyses.

Magnetized and non-magnetized Casson fluid flow with gyrotactic microorganisms over a stratified stretching cylinder.

This study presents the magnetized and non-magnetized Casson fluid flow with gyrotactic microorganisms over a stratified stretching cylinder. The mathematical modeling is presented in the form of partial differential equations and then transformed into ordinary differential equations (ODEs) utilizing suitable similarity transformations. The analytical solution of the transformed ODEs is presented with the help of homotopy analysis method (HAM). The convergence analysis of HAM is also presented by mean of figure. The present analysis consists of five phases. In the first four phases, we have compared our work with previously published investigations while phase five is consists of our new results. The influences of dimensionless factors like a magnetic parameter, thermal radiation, curvature parameter, Prandtl number, Brownian motion parameter, Schmidt number, heat generation, chemical reaction parameter, thermophoresis parameter, Eckert number, and concentration difference parameter on physical quantities of interests and flow profiles are shown through tables and figures. It has been established that with the increasing Casson parameter (i.e. [Formula: see text]), the streamlines become denser which results the increasing behavior in the fluid velocity while on the other hand, the fluid velocity reduces for the existence of Casson parameter (i.e. [Formula: see text]). Also, the streamlines of stagnation point Casson fluid flow are highly wider for the case of magnetized fluid as equated to non-magnetized fluid. The higher values of bioconvection Lewis number, Peclet number, and microorganisms' concentration difference parameter reduces the motile density function of microorganisms while an opposite behavior is depicted against density number.

A stochastic numerical analysis based on hybrid NAR-RBFs networks nonlinear SITR model for novel COVID-19 dynamics.

BACKGROUND: Mathematical modeling of vector-borne diseases and forecasting of epidemics outbreak are global challenges and big point of concern worldwide. The outbreaks depend on different social and demographic factors based on human mobility structured with the help of mathematical models for vector-borne disease transmission. In Dec 2019, an infectious disease is known as "coronavirus" (officially declared as COVID-19 by WHO) emerged in Wuhan (Capital city of Hubei, China) and spread quickly to all over the china with over 50,000 cases including more than 1000 death within a short period of one month. Multimodal modeling of robust dynamics system is a complex, challenging and fast growing area of the research. OBJECTIVES: The main objective of this proposed hybrid computing technique are as follows: The innovative design of the NAR-RBFs neural network paradigm is designed to construct the SITR epidemic differential equation (DE) model to ascertain the different features of the spread of COVID-19. The new set of transformations is introduced for nonlinear input to achieve with a higher level of accuracy, stability, and convergence analysis. METHODS: Multimodal modeling of robust dynamics system is a complex, challenging and fast growing area of the research. In this research bimodal spread of COVID-19 is investigated with hybrid model based on nonlinear autoregressive with radial base function (NAR-RBFs) neural network for SITR model. Chaotic and stochastic data of the pandemic. A new class of transformation is presented for the system of ordinary differential equation (ODE) for fast convergence and improvement of desired accuracy level. The proposed transformations convert local optimum values to global values before implementation of bimodal paradigm. RESULTS: This suggested NAR-RBFs model is investigated for the bi-module nature of SITR model with additional feature of fragility in modeling of stochastic variation ability for different cases and scenarios with constraints variation. Best agreement of the proposed bimodal paradigm with outstanding numerical solver is confirmed based on statistical results calculated from MSE, RMSE and MAPE with accuracy level based on mean square error up to 1E-25, which further validates the stability and consistence of bimodal proposed model. CONCLUSIONS: This computational technique is shown extraordinary results in terms of accuracy and convergence. The outcomes of this study will be useful in forecasting the progression of COVID-19, the influence of several deciding parameters overspread of COVID-19 and can help for planning, monitoring as well as preventing the spread of COVID-19.

A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load.

COVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained R 0 ≈ 1.50 . Finally, an efficient numerical scheme of Adams-Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.

Dynamics of fractional order COVID-19 model with a case study of Saudi Arabia.

The novel coronavirus disease or COVID-19 is still posing an alarming situation around the globe. The whole world is facing the second wave of this novel pandemic. Recently, the researchers are focused to study the complex dynamics and possible control of this global infection. Mathematical modeling is a useful tool and gains much interest in this regard. In this paper, a fractional-order transmission model is considered to study its dynamical behavior using the real cases reported in Saudia Arabia. The classical Caputo type derivative of fractional order is used in order to formulate the model. The transmission of the infection through the environment is taken into consideration. The documented data since March 02, 2020 up to July 31, 2020 are considered for estimation of parameters of system. We have the estimated basic reproduction number ( R 0 ) for the data is 1.2937 . The Banach fixed point analysis has been used for the existence and uniqueness of the solution. The stability analysis at infection free equilibrium and at the endemic state are presented in details via a nonlinear Lyapunov function in conjunction with LaSalle Invariance Principle. An efficient numerical scheme of Adams-Molten type is implemented for the iterative solution of the model, which plays an important role in determining the impact of control measures and also sensitive parameters that can reduce the infection in the general public and thereby reduce the spread of pandemic as shown graphically. We present some graphical results for the model and the effect of the important sensitive parameters for possible infection minimization in the population.

Modeling and analysis of the dynamics of novel coronavirus (COVID-19) with Caputo fractional derivative.

The new emerged infectious disease that is known the coronavirus disease (COVID-19), which is a high contagious viral infection that started in December 2019 in China city Wuhan and spread very fast to the rest of the world. This infection caused million of infected cases globally and still pose an alarming situation for human lives. Pakistan in Asian countries is considered the third country with higher number of cases of coronavirus with more than 200,000. Recently, many mathematical models have been considered to better understand the coronavirus infection. Most of these models are based on classical integer-order derivative which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using the non-integer Caputo derivative. In the absence of vaccine or therapy, the role of non-pharmaceutical interventions (NPIs) is examined on the dynamics of theCOVID-19 outbreak in Pakistan. First, we construct the model in integer sense and then apply the fractional operator to have a generalized model. The generalized model is then used to present the detailed theoretical results. We investigate the stability of the model for the case of fractional model using a nonlinear fractional Lyapunov function of Goh-Voltera type. Furthermore, we estimate the values of parameters with the help of least square curve fitting tool for the COVID-19 data recorded in Pakistan since March 1 till June 30, 2020 and show that our considered model give an accurate prediction to the real COVID-19 statistical cases. Finally, numerical simulations are presented using estimated parameters for various values of the fractional order of the Caputo derivative. From the simulation results it is found that the fractional order provides more insights about the disease dynamics.