With every passing day, one comes to know that cases of the corona virus disease are increasing. This is an alarming situation in many countries of the globe. So far, the virus has attacked as many as 188 countries of the world and 5 549 131 (27 May 2020) human population is affected with 348 224 deaths. In this regard, public and private health authorities are looking for manpower with modeling skills and possible vaccine. In this research paper, keeping in view the fast transmission dynamics of the virus, we have proposed a new mathematical model of eight mutually distinct compartments with the help of memory-possessing operator of Caputo type. The fractional order parameter of the model has been optimized so that smallest error can be attained while comparing simulations and the real data set which is considered for the country Pakistan. Using Banach fixed point analysis, it has been shown that the model has a unique solution whereas its basic reproduction number  0 is approximated to be 6.5894. Disease-free steady state is shown to be locally asymptotically stable for  0 < 0, otherwise unstable. Nelder-Mead optimization algorithm under MATLAB Toolbox with daily real cases of the virus in Pakistan is employed to obtain best fitted values of the parameters for the model's validation. Numerical simulations of the model have come into good agreement with the practical observations wherein social distancing, wearing masks, and staying home have proved to be the most effective measures in order to prevent the virus from further spread.

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Abstract. The main aim of this paper is to obtain a general theorem on existence of continuous branch of solutions of equations which depend on a parameter by using the monotone minorant method in conjunction with the theory of fixed point index. As an application, we apply this theorem to prove the existence of a positive eigen-pair of multivalued homogeneous increasing operators. The simplicity and uniqueness of the eigen-pair are also investigated in this paper.

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