This investigation focuses to develop an e-SEIRS (susceptible, exposed, infectious, recovered) epidemic computer network model to study the transmission of malicious code in a computer network and derive the approximate threshold condition (basic reproduction number) to examine the equilibrium and stability of the model. The authors have simulated the results for various parameters used in the model and Runge-Kutta Fehlberg fourth-fifth order method is employed to solve system of equations developed. They have studied the stability of crime level to equilibrium and found the critical value of threshold value determining whether or not the infectious free equilibrium is globally asymptotically stable and endemic equilibrium is locally asymptotically stable. The simulation results using MATLAB agree with the real life situations.

This paper deals with the investigation on spread of COVID-19 and its stability analysis (both local and global stability) in Odisha, India. Being the second most populous country in the world, It is urgent need to investigate the spread and control of disease in India .However, due to diversity of vast population, uncertainty of infection, varying rate of recovery, state wise different COVID-19 induced death rate and non uniform quarantine policy of the states, it is strenuous to predict the spread and control of disease accurately in the country. So, it is crucial to study the aspects of disease in each state for the better prediction. We have considered the state Odisha (India) having population nearly equal to the population of Spain because the entry of huge migrants to the state suddenly enhanced the number of COVID-19 patients from below two hundred to more than eight hundred within one week even after forty days of lockdown period. We have developed SIAQR epidemic model fabricated with influx of out-migrants diagnosed at compartment (A) , then sent to the compartment (I) for treatment those have confirmed the disease and the remaining healthy individuals are sent to quarantine compartment (Q) for a period of twenty one days under surveillance and observation. The set of ordinary (nonlinear) differential equations are formulated and they are solved using Runge -Kutta fourth order method. The simulation of numerical data is performed using computer software MATLAB. As there is no specific treatment, vaccine or medicine available for the disease till the date, so the only intervention procedure called quarantine process is devised in this model to check the stability behaviour of the disease. The numerical and analytical results of the study show that the disease free equilibrium is locally stable when basic reproduction number is less than unity and unstable when it is more than unity. Further the study shows that it persists to endemic equilibrium for global stability when basic reproduction number greater than unity. As per the current trends ,this study shows that the prevalence of COVID -19 would remain nearly 250 to 300 days in Odisha as for as the infected migrants would have been entering to the state. This mathematical modelling embedded with important risk factor like migration could be adopted for each state that will be helpful for better prediction of the entire country and world.

This paper pertains to investigate the heat transfer characteristics of two dimensional two phase flow in a porous medium over a stretching sheet with internal heat generation. The novelty of the present study is to consider the permeability parameter, space and temperature dependent internal heat generation along with various parameters like Froud number ,heat source/sink parameter, Grashof number, Prandtl number, Eckert number, Volume fraction, fluid interaction parameter etc. The method of solution involves similarity transformation which reduces the partial differential equations into non linear ordinary differential equations. These non linear ordinary differential equations have been solved by applying Runge-Kutta 4-th order method with help of shooting technique. The temperature profiles for different values of flow parameters are presented in figures. It is observed from all the figures that the boundary conditions are satisfied asymptotically in all the cases which supporting the accuracy of the numerical results. All the figures shows that increasing values of any parameter increase the thermal boundary layer except the prandtl number and permeability parameter. AMS classification 76T10, 76T15

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