Mathematical modelling, analysis and numerical simulation of social media addiction and depression.

We formulate a mathematical model of social media addiction and depression (SMAD) in this study. Key aspects, such as social media addiction and depression disease-free equilibrium point (SMADDFEP), social media addiction and depression endemic equilibrium point (SMADEEP), and basic reproduction number (R0), have been analyzed qualitatively. The results indicate that if R0 < 1, the SMADDFEP is locally asymptotically stable. The global asymptotic stability of the SMADDFEP has been established using the Castillo-Chavez theorem. On the other hand, if R0 > 1, the unique endemic equilibrium point (SMADEEP) is locally asymptotically stable by Lyapunov theorem, and the model exhibits a forward bifurcation at R0 = 1 according to the Center Manifold theorem. To examine the model's sensitivity, we calculated the normalized forward sensitivity index and conducted a Partial Rank Correlation Coefficient (PRCC) analysis to describe the influence of parameters on the SMAD. The numerical results obtained using the Fourth-order Runge-Kutta (RK-4) scheme show that increasing the number of addicted individuals leads to an increase in the number of depressed individuals.

Fractional Modeling of Cancer with Mixed Therapies.

BACKGROUND: Cancer is the biggest cause of mortality globally, with approximately 10 million fatalities expected by 2020, or about one in every six deaths. Breast, lung, colon, rectum, and prostate cancers are the most prevalent types of cancer. METHODS: In this work, fractional modeling is presented which describes the dynamics of cancer treatment with mixed therapies (immunotherapy and chemotherapy). Mathematical models of cancer treatment are important to understand the dynamical behavior of the disease. Fractional models are studied considering immunotherapy and chemotherapy to control cancer growth at the level of cell populations. The models consist of the system of fractional differential equations (FDEs). Fractional term is defined by Caputo fractional derivative. The models are solved numerically by using Adams-Bashforth-Moulton method. RESULTS: For all fractional models the reasonable range of fractional order is between ß = 0.6 and ß = 0.9. The equilibrium points and stability analysis are presented. Moreover, positivity and boundedness of the solution are proved. Furthermore, a graphical representation of cancerous cells, immunotherapy and chemotherapy is presented to understand the behaviour of cancer treatment. CONCLUSIONS: At the end, a curve fitting procedure is presented which may help medical practitioners to treat cancer patients.

A novel fractional model for the projection of households using wealth index quintiles.

Forecasting household assets provides a better opportunity to plan their socioeconomic activities for the future. Fractional mathematical models offer to model the asset-holding data into a piece of scientific evidence in addition to forecasting their future value. This research focuses on the development of a new fractional mathematical model based on the wealth index quintile (WIQ) data. To accomplish the objective, we used the system of coupled fractional differential equations by defining the fractional term with the Caputo derivative and verified it with the stability tests considering the steady-state solution. A numerical solution of the model was obtained using the Adams-Bashforth-Moulton method. To validate the model, we used real-time data obtained from the household series of surveys in Punjab, Pakistan. Different case studies that elucidate the effect of quintiles on the population are also presented. The accuracy of results between real-world and simulated data was compared using absolute and relative errors. The synchronization between the simulated results and real-time data verifies the formulation of the fractional WIQ model. This fractional model can be utilized to predict the approximation of the asset-holding of the households. Due to its relative nature, the model also provides the opportunity for the researchers to use the WIQs of their respective regions to forecast the households' socioeconomic conditions.

Artificial neural network-based heuristic to solve COVID-19 model including government strategies and individual responses.

The current work aims to design a computational framework based on artificial neural networks (ANNs) and the optimization procedures of global and local search approach to solve the nonlinear dynamics of the spread of COVID-19, i.e., the SEIR-NDC model. The combination of the Genetic algorithm (GA) and active-set approach (ASA), i.e., GA-ASA, works as a global-local search scheme to solve the SEIR-NDC model. An error-based fitness function is optimized through the hybrid combination of the GA-ASA by using the differential SEIR-NDC model and its initial conditions. The numerical performances of the SEIR-NDC nonlinear model are presented through the procedures of ANNs along with GA-ASA by taking ten neurons. The correctness of the designed scheme is observed by comparing the obtained results based on the SEIR-NDC model and the reference Adams method. The absolute error performances are performed in suitable ranges for each dynamic of the SEIR-NDC model. The statistical analysis is provided to authenticate the reliability of the proposed scheme. Moreover, performance indices graphs and convergence measures are provided to authenticate the exactness and constancy of the proposed stochastic scheme.

Numerical Investigations through ANNs for Solving COVID-19 Model.

The current investigations of the COVID-19 spreading model are presented through the artificial neuron networks (ANNs) with training of the Levenberg-Marquardt backpropagation (LMB), i.e., ANNs-LMB. The ANNs-LMB scheme is used in different variations of the sample data for training, validation, and testing with 80%, 10%, and 10%, respectively. The approximate numerical solutions of the COVID-19 spreading model have been calculated using the ANNs-LMB and compared viably using the reference dataset based on the Runge-Kutta scheme. The obtained performance of the solution dynamics of the COVID-19 spreading model are presented based on the ANNs-LMB to minimize the values of fitness on mean square error (M.S.E), along with error histograms, regression, and correlation analysis.

A novel mathematical model for COVID-19 with remedial strategies.

Coronavirus (COVID-19) outbreak from Wuhan, Hubei province in China and spread out all over the World. In this work, a new mathematical model is proposed. The model consists the system of ODEs. The developed model describes the transmission pathways by employing non constant transmission rates with respect to the conditions of environment and epidemiology. There are many mathematical models purposed by many scientists. In this model, " α E " and " α I ", transmission coefficients of the exposed cases to susceptible and infectious cases to susceptible respectively, are included. " δ " as a governmental action and restriction against the spread of coronavirus is also introduced. The RK method of order four (RK4) is employed to solve the model equations. The results are presented for four countries i.e., Pakistan, Italy, Japan, and Spain etc. The parametric study is also performed to validate the proposed model.

A modified artificial neural network based prediction technique for tropospheric radio refractivity.

Radio refractivity plays a significant role in the development and design of radio systems for attaining the best level of performance. Refractivity in the troposphere is one of the features affecting electromagnetic waves, and hence the communication system interrupts. In this work, a modified artificial neural network (ANN) based model is applied to predict the refractivity. The suggested ANN model comprises three modules: the data preparation module, the feature selection module, and the forecast module. The first module applies pre-processing to make the data compatible for the feature selection module. The second module discards irrelevant and redundant data from the input set. The third module uses ANN for prediction. The ANN model applies a sigmoid activation function and a multi-variate auto regressive model to update the weights during the training process. In this work, the refractivity is predicted and estimated based on ten years (2002-2011) of meteorological data, such as the temperature, pressure, and humidity, obtained from the Pakistan Meteorological Department (PMD), Islamabad. The refractivity is estimated using the method suggested by the International Telecommunication Union (ITU). The refractivity is predicted for the year 2012 using the database of the previous ten years, with the help of ANN. The ANN model is implemented in MATLAB. Next, the estimated and predicted refractivity levels are validated against each other. The predicted and actual values (PMD data) of the atmospheric parameters agree with each other well, and demonstrate the accuracy of the proposed ANN method. It was further found that all parameters have a strong relationship with refractivity, in particular the temperature and humidity. The refractivity values are higher during the rainy season owing to a strong association with the relative humidity. Therefore, it is important to properly cater the signal communication system during hot and humid weather. Based on the results, the proposed ANN method can be used to develop a refractivity database, which is highly important in a radio communication system.

Analytical solutions and moment analysis of chromatographic models for rectangular pulse injections.

This work focuses on the analysis of two standard liquid chromatographic models, namely the lumped kinetic model and the equilibrium dispersive model. Analytical solutions, obtained by means of Laplace transformation, are derived for rectangular single solute concentration pulses of finite length and breakthrough curves injected under linear conditions. In order to analyze the solute transport behavior by means of the two models, the temporal moments up to fourth order are calculated from the Laplace-transformed solutions. The limiting cases of continuous injection and negligible mass transfer limitations are evaluated. For validation, the analytical solutions are compared with the numerical solutions of models using the discontinuous Galerkin finite element method. Results of different case studies are discussed for linear and nonlinear adsorption isotherms. The discontinuous Galerkin method is employed to obtain moments for both linear and nonlinear models numerically. Analytically and numerically determined concentration profiles and moments were found to be in good agreement.

A discontinuous Galerkin method to solve chromatographic models.

This article proposes a discontinuous Galerkin method for solving model equations describing isothermal non-reactive and reactive chromatography. The models contain a system of convection-diffusion-reaction partial differential equations with dominated convective terms. The suggested method has capability to capture sharp discontinuities and narrow peaks of the elution profiles. The accuracy of the method can be improved by introducing additional nodes in the same solution element and, hence, avoids the expansion of mesh stencils normally encountered in the high order finite volume schemes. Thus, the method can be uniformly applied up to boundary cells without loosing accuracy. The method is robust and well suited for large-scale time-dependent simulations of chromatographic processes where accuracy is highly demanding. Several test problems of isothermal non-reactive and reactive chromatographic processes are presented. The results of the current method are validated against flux-limiting finite volume schemes. The numerical results verify the efficiency and accuracy of the investigated method. The proposed scheme gives more resolved solutions than the high resolution finite volume schemes.