This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to construct fractional integro-differential equations for the ruin probabilities in collective renewal risk models, with inter-arrival time distributions from the aforementioned family. Gamma-time risk models and fractional Poisson risk models are two specific cases among them, whose ruin probabilities have explicit solutions when claim size distributions exhibit rational Laplace transforms.

In this paper, we provide three equivalent expressions for ruin probabilities in a Cramér–Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag–Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.

The crisis caused by COVID-19 revealed the global unpreparedness for handling the impact of a pandemic. In this paper, we present a first quarter chronicle of COVID-19 in Hubei China, Italy and Spain, specifically their infection speed, death and fatality rates. By fitting distributions to these rates, we look for the effectiveness of government measures during the pandemic through a number of statistical approaches.