India imposed a nationwide lockdown from 25th March 2020 onwards to combat the spread of COVID-19 pandemic. To model the spread of a disease and to predict its future course, epidemiologists make use of compartmental models such as the model. In order to address some of the assumptions of the standard model, a new modified version of model is proposed in this paper that takes into account the percentage of infected individuals who are tested and quarantined. This approach helps overcome the assumption of homogenous mixing of population which is inherent to the conventional model. Using the available data of the number of COVID-19 positive cases reported in the state of Kerala, and in India till 26th April, 2020 and 12th May 2020, respectively, the parameter estimation problem is converted into an optimization problem with the help of a least squared cost function. The optimization problem is then solved using differential evolution optimizer. The impact of lockdown is quantified by comparing the rising trend in infections before and during the lockdown. Using the estimated set of parameters, the model predicts that in the state of Kerala, by using certain interventions the pandemic can be successfully controlled latest by the first week of July, whereas the value for India is still greater than 1, and hence lifting of lockdown from all regions of the country is not advisable.

This paper presents a new approach for describing the shape of 11-year sunspot cycles by considering the monthly averaged values. This paper also brings out a prediction model based on the analysis of 22 sunspot cycles from the year 1749 onward. It is found that the shape of the sunspot cycles with monthly averaged values can be described by a functional form of modified binary mixture of Laplace density functions, modified suitably by introducing two additional parameters in the standard functional form. The six parameters, namely two locations, two scales, and two area parameters, characterize this model. The nature of the estimated parameters for the sunspot cycles from 1749 onward has been analyzed and finally we arrived at a sufficient set of the parameters for the proposed model. It is seen that this model picks up the sunspot peaks more closely than any other model without losing the match at other places at the same time. The goodness of fit for the proposed model is also computed with the Hathaway -Wilson -Reichmann χ measure, which shows, on average, that the fitted model passes within 0.47 standard deviations of the actual averaged monthly sunspot numbers.

No abstract

A stochastic prediction model for the sunspot cycle is proposed. The prediction model is based on a modified binary mixture of Laplace distribution functions and a movingaverage model over the estimated model parameters. A six-parameter modified binary mixture of Laplace distribution functions is used for the modeling of the shape of a generic sunspot cycle. The model parameters are estimated for 23 sunspot cycles independently, and the primary prediction-model parameters are derived from these estimated model parameters using a moving-average stochastic model. A correction factor (hump factor) is introduced to make an initial prediction. The hump factor is computed for a given sunspot cycle as the ratio of the model estimated after the completion of a sunspot cycle (post-facto model) and the prediction of the moving-average model. The hump factors can be applied one at a time over the moving-average prediction model to get a final prediction of a sunspot cycle. The present model is used to predict the characteristics of Sunspot Cycle 24. The methodology is validated using the previous Sunspot Cycles 21, 22, and 23, which shows the adequacy and the applicability of the prediction model. The statistics of the variations of sunspot numbers at high solar activity are used to provide the lower and upper bound for the predictions using the present model.