1. Introduction
The mathematical modeling of infectious diseases is a way in which to study the spread of diseases and their behavior to predict the future trajectories of an epidemic, as well as to help guide public health planning and disease control. The models developed use stochastic processes to estimate the number of infected cases that could occur in the coming weeks or months. This methodology helps researchers simulate real-world possibilities in a virtual environment. On the other hand, diffusion stochastic processes (SDPs) are good mathematical models that describe probabilistic phenomena in several domains, like environment, biology, economics, medicines and others. Some of the SDPs studied in this sense are found in the works of Bertalanffy [
1], Gamma [
2], Weibull [
3] and Lundqvist-Korf [
4], who used SDPs to study growth patterns because of their exponential behavior. In addition, Capocelli and Ricciardi [
5] were the first to consider a diffusion process that was associated with the Gompertz curve, and this has been applied to several fields of study, such as population growth or neural activity modeling [
6]. The logistic diffusion process has been applied to a diverse range of scientific areas; specifically, Capocelli and Ricciardi [
7] derived a new diffusion process from a re-parameterization of the logistic model. Giovanis and Skaidas [
8] proposed a stochastic logistic model that was analytically solved by using the theoretical framework of reducible stochastic differential equations (SDEs), and these were then applied to study the consumption of electricity in Greece and the United States.
The new virus of SARS-CoV-2, named COVID-19 (Coronavirus 2019) by the World Health Organization (WHO), is believed to have originated from an animal source in the city of Wuhan, Hubei region, China, in December 2019. Since its first appearance, the disease has spread worldwide. On 11 March 2020, the WHO officially classified the epidemic as a pandemic. As of 22 March 2022, the virus (including its variants) has infected 492 million people, with 6.5 million deaths. Several research studies have been developed in the context of modeling the pandemic. For example, see Ref. [
9] for the stochastic COVID-19 Levy jump model with an isolation strategy; Ref. [
10] introduced a new model for the spread of COVID-19 and the improvement of safety; Ref. [
11] constructed a new random infectious disease system under the environmental noise of the infection rate, as well as studied the probability density function (PDF) of the stochastic system; and Ref. [
12] applied a mathematical model to describe the behavior of the number of cases with respect to time in Italy.
The Rayleigh distribution was developed by the physicist Lord Rayleigh [
13], and it is widely used in physics-related fields to model processes such as sound and light radiation, wind speed, and wave height. It is also used in communications theory to describe the hourly median and instantaneous peak power of received radio signals (see, [
14]). In addition, it plays a crucial role in the field of land mobile radio as it can accurately describe wind speed. Thus, the Rayleigh distribution has a wide use in many fields, which makes its study from dynamic and stochastic point of views interesting (see for example, [
15]).
The problem of statistical inference in SDPs has attracted a great deal of interest in recent years, and this applies both when the process is considered continuously or discretely. In general, the estimation of parameters in stochastic models is not direct apart from in simple cases, and one possible method is based on the approximation of the maximum likelihood (ML) function. In this context, various methods have been developed to address this problem. The basic case for this approach can be found in Bibby and Sorensen [
16], as well as in Ait-Sahalia [
17]. The method of estimating the ML of the parameters when using likelihood equations can be difficult to implement, which is the reason why we propose to use the simulated annealing (SA) method for estimating the parameters in an SDP (see, for instance, [
3,
4]).
In this work, based on previous research in this field, we introduce a stochastic Rayleigh diffusion process (SRDP) whose mean is proportional to the Rayleigh density function (RDF). The SRDP is used to model the evolution of the active cases of COVID-19, and this is the first attempt to use the stochastic Rayleigh diffusion process to model a pandemic. The SRDP was introduced by [
18] to model female and male life expectancy at birth in Spain. A brief version of this process was introduced by [
19], and we specifically call this method the Ornstein–Uhlenbeck radial process. This process has also been studied and applied by [
20] to model the production of thermal energy in the Maghreb. The Rayleigh process has also been used to model the exchange rate dynamics when using the Swiss Franc against the Euro under a floating-rate regime [
21], as well as for the statistical inference and computational aspects of the stochastic Rayleigh diffusion model [
22].
In the present paper, we introduce a new SRDP that is different from the one studied in [
22]. This SRDP is based on RDF, and the paper is organized in the following way: In
Section 2, we describe how the explicit form of this SRDP model is given by Itô’s lemma. Moreover, we give all the main characteristics of the proposed process, such as the transition PDF (TPDF), mean function (MF) and the conditional mean function (CMF). In
Section 3, we adopt the ML to find the estimators of the parameters of our model. The parameter estimators are found by solving the ML equations. That said, in this case, we cannot find the solution directly, so we use the numerical method of simulated annealing (SA). In
Section 4, we simulate the SRDP through using its explicit form. We also observe the behavior of the SRDP trajectories for the different values of parameters of the diffusion coefficient. We then use the simulated data to obtain the estimators of our given parameters by applying the adopted numerical method. Furthermore, we predict some of the realizations by using the estimated MF (EMF) and conditional CMF (ECMF).
Section 5, offers an application of this process through an exploration of the development of the active COVID-19 cases in Morocco from January to February 2022. In the last section, we detail the conclusion of our findings.