A Model for Coronavirus Pandemic

A simple model is developed for spread of a pandemic disease. The model is based on simple, uninhibited population growth except that the rate of infection is assumed to be proportional to the existing infected population. The model is in agreement with the CDC data on COVID-19 for the United States in 2020.

December 2020 has infected nearly 79 million people worldwide and killed over 1.73 million [3]. This has generated a great number of investigations trying to understand the dynamics of the spread of the disease and to predict its future evolution [4][5][6][7][8].
The purpose of this article is not to suggest any preventive measures for COVID-19, but to propose a different yet simple model for the dynamics of the spread of the disease. The goal is to test an ab initio assumption that results in the spread of not just COVID-19 but any pandemic.

The Model
Let us first consider the problem of normal uninhibited population growth. At any time t, the rate of population growth is proportional to the existing population, Where the growth constant k is a positive number, which is the probability per unit time of a member of the population to double. This equation integrates to 0 kt N N e = (2) where 0 N is the initial population. We may also include the death rate in this problem by using In the population growth model, the parameter k is constant. In the spread of a pandemic disease, on the other hand, the growth parameter is not a constant. In fact, in a given population, the probability with which a contagious disease spreads increases with the fraction of the infected population. If we assume that this where k is a constant, n is the infected population, and N is the total population, then we have Where A is related to the integration constant. Using the initial condition and therefore, where 0 n is the initial infected population. Note that the functional form of Eq. (10) is quite different from that of simple exponential population growth Eq. (2). In fact, the graph of Eq. (10), shown in Figure 1, is generally known as the logistic growth curve [9].

Comparison with COVID-19 Data
To check the validity of our model, we have compared it to the COVID-19 data provided by CDC (Centers for Diseases Control and Prevention) for the United States in the period of January 22, 2020 to January 25, 2021 [10], shown in Table 1. We have used the data in 5-day steps. Thus, in Table 1, Day 0 corresponds to January 22, 2020 and Day 370 corresponds to January 25, 2021.
In equation (10), for the total population N we used the current population of the United States, 331 million. In this equation, k is an adjustable parameter that has to be fitted to the data. In addition, the initial data is highly uncertain as very few people were tested for the virus in the early stages of the pandemic. This is further evidenced by the fact that even the more recent CDC data are continuously revised up as more data come in. Therefore, we allowed the initial number of the infected population to be an adjustable parameter as well.  Figure 2 shows that the model presented here agrees with the coronavirus data of 2020 in the United States. The model is quite simple and only assumes the probability of infection is proportional to the fraction of the infected population. The model involves only two adjustable parameters which are obtained by fitting the data. This is in contrast to other models which involve Consequently, we performed a nonlinear least-squares analysis and fitted Eq. (10) with N = 3.31 × 10 8 to the CDC data via the parameters 0 n and k. For the best fit, we obtained the following values for these parameters to three significant figures, 0 n = 414000 ± 22600 and k = 0.0114 ± 0.0002 day −1 (11) Figure 2 shows the graph of Eq. (10) with these parameters as well as the CDC data for the COVID-19 pan-  several adjustable parameters. For example, the model developed by Kaxiras and Neofotistos for the COVID-19 pandemic involves 5 adjustable parameters.

Discussion and Conclusion
As can be seen from Figure 2, the pandemic data shows small variations in the slope at days number 50, 150, and 250. These are attributed to small changes in the parameters of the pandemic dynamics as a result of behavioral changes of the people such as wearing masks, social distancing, etc. However, the overall spread of the disease closely follows the logistic growth curve (10). Figure 2 and the agreement of Eq. 10 with the CDC data, we conclude that the assumption of probability of disease spread being proportional to fraction of the infected population is quite reasonable.