A new nonlinear mathematical model for forecasting the waves of the COVID-19 epidemic

Social distance and Hospitals' capacity are two very important factors in the COVID-19 pandemic, which usually change over time. Nevertheless, studies on forecasting this pandemic have not approximately considered these changes in their models. In this study, we accordingly provided a new nonlinear diffusion model by considering these changes for forecasting the COVID-19 outbreak in Iran. This new model generally indicated that it can well follow and predict the trends of the new cases, new recoveries and daily deaths relevant to Iran’s COVID-19 epidemic. In this model, some outcomes also indicated that this model can detect some-things about the management style of the COVID-19 epidemic in Iran. Therefore, since this model could follow three waves of the COVID-19 outbreak in Iran, it can be used in mitigation strategies for the better management of the COVID-19 epidemic in Iran and even other world countries.


Introduction
After identifying the COVID-19 in late December 2019 in Wuhan, Hubei Province, China and then the Middle Eastern and European countries [1], various mathematical models have been provided for COVID-19 disease transmission. These models are very important, because their outcomes can be considered in mitigation strategies for the better management of the COVID-19 pandemic in the future, if their parameters and outcomes can be compatible with reality. 1 In recent months, many research works have been reported on forecasting the COVID-19 diffusion in Iran and other world countries. For example, Ndaïrou and his colleagues [2] provided a compartment model for the COVID-19 transmission in Wuhan, China. These researchers generally stated that their model is suitable for the COVID-19 outbreak, which occurred in this city. Sarkar and his colleagues [3,4] also proposed a classical SEIR (susceptible exposed infectious recovered)-type epidemiological model that predicts the dynamics of COVID-19 in 17 provinces of India and overall India. In a similar study, Samui and his colleagues [5] also employed a four-compartment SEIR model for the COVID-19 transmission in India with epidemic data up to April 30, 2020. Reports provided by Mandal, Chatterjee, Alaraj and their colleagues [6][7][8] about the corona-virus disease 2019 transmission, which was based on the SEIR models, are also other similar studies about the outbreak of this disease in India. Fortunately, these reports generally include significant results for predicting this disease in India, and it is remarkable that the reported results are also suitable for predicting the COVID-19 disease in this country, because the COVID-19 outbreak in this country still contains one wave.
According to the SEIR models, Kuniya [9] also applied a compartmental model to the daily reported cases of COVID-19 in Japan from 15 January to 29 February and outcomes have been only suitable for forecasting the first wave of the disease in Japan. Similarly, the stochastic SEIR model to predict the spread of COVID-19 in Kuwait, which is reported by Almeshal and his colleagues [10], could not completely predict the procedure of the outbreak in this country. About forecasting the COVID-19 outbreak in Iran, there are also similar studies that provided mathematical models based on the daily reported cases of COVID-19 in the early outbreak of this disease after 20 February in Iran [11][12][13][14]. Nevertheless, although, the results of these models may be generally suitable for predicting the first wave of the COVID-19 outbreak in this country, these models cannot completely predict the procedure of the COVID-19 outbreak in Iran, which consists of three waves. The mathematical model provided by Ghanbari [15], which nearly followed the second wave of the COVID-19 outbreak in Iran, also seems that is only developed for forecasting the first and second waves, because this researcher has approximately provided no information about this issue that its model can predict later waves. In other words, although these researchers tried to provide the mathematical models for forecasting the COVID-19 outbreak in Iran and different world countries, they have forgotten that the parameters of their models such as social distance and Hospitals' capacity are dependent on the behavior of the humans, and these parameters actually change over time. Accordingly, in this study, we provided a new diffusion model for forecasting the COVID-19 outbreak in Iran, in which the changes of these parameters, i.e. social distance and Hospitals' capacity, are the engine of the proposed model.
The remainder of this paper is also organized as follows: Section 2 present the details of the new diffusion model proposed for forecasting the COVID-19 outbreak in different world countries. Section 3 provided the results of the proposed diffusion model for forecasting the COVID-19 outbreak in Iran. Finally, section 3 also presents a brief conclusion about the results obtained in this research.

The proposed diffusion model
In engineering, the spread of a pandemic similar to the COVID-19 is a diffusion problem [16], in which the concentration changes of compartments are dependent on force entered the compartments and the output rate of the compartments. Accordingly, as shown in Fig 1, we considered three compartments (new cases, new recoveries and daily deaths) in our diffusion model. Therefore, we wrote the diffusion equations for each of the three compartments as [16]: where F C , F R and F D are forces entered the compartments of the new cases, new recoveries and daily deaths, respectively. K C , K R and K D are coefficients that set the output rate of these compartments. These forces and coefficients are generally are influenced by two fundamental factors, i.e. social distance and Hospitals' capacity. Since, these factors have different effects on three compartments of the new cases, new recoveries and daily deaths, we defined six variables (s C , s R , s D , c C , c R and c D ) to model the effects of these two factors on the mentioned compartments. Also, since, these factors usually change between their best and worst states (0 to 100%), we can simply model the change of these six variables by using a first-order differential equation as follows: As explained, s i and c i are also variables that represent the status of the social distance and the Hospitals' capacity for each of the compartments (new cases, new recoveries and daily deaths). The multiplication of these two variables (Equation 3) accordingly provides an overall status of the social distance and the Hospitals' capacity, which the force entered the compartments and the output rate of the compartments are directly dependent on it.
From Equations 1 and 3, therefore we can deduce that Actually, we modeled the coefficients of K C , K R and K D as 1-ϕ i (t), because the value of these coefficients is reduced, when force entered compartments is increased. In other words, force entered a compartment increases the internal components of the compartment. Therefore, since, the mentioned coefficients are calculated from the ratio of the internal components of the compartment to the exited components of the compartment, the coefficients of K C , K R and K D should be decreases, if force entered the compartment is increased. In the next section, we set parameters the proposed model (a i , b i , c i , d i , e i , f i , α, β and θ) for forecasting the COVID-19 outbreak in Iran.
New cases New recoveries Daily deaths Currently Fig.1 . The proposed diffusion model

Forecasting the COVID-19 outbreak in Iran by using the proposed diffusion model
As stated earlier, the social distance and the Hospitals' capacity change over time, because they are dependent on the behavior of the humans. Therefore, the parameters of a i , b i , c i , d i , e i and f i in the system of equations 2 change over time. Table I provides the values of these parameters within 335 days, which we determined based on the knowledge of first-order differential equations for forecasting the procedure of the COVID-19 outbreak in Iran from 20 February 2019 to 26 December 2020. In addition, since the parameters of α, β and θ are dependent on the maximum value of the new cases, new recoveries and daily deaths in the third wave of the COVID-19 outbreak in Iran, we rewrote the system of equations 4 as follows: where A is the maximum value of the new cases in the third wave of the COVID-19 outbreak in Iran (A = 14000) .   Fig 2a and 2b show the outcomes of the proposed diffusion model (system of equations 5) for predicting the new cases, new recoveries and daily deaths within 365 from 20 February 2019 for Iran. As shown in this figure, this differential model could well model every three waves, which occurred in the real data of the COVID-19 outbreak in Iran. In the remaining two months of the solar year 2000 (i.e. January and February), this model has also forecasted the beginning of a new small wave for the COVID-19 outbreak in Iran. It is also interesting that the changes of the variables s i and c i , which is represented in Fig 2c, detected some-things about the management style of the COVID-19 epidemic in Iran. In other words, as shown in this figure, the variables s i (s C , s R and s D ), which represent the status of the social distance, are reduced at the beginning of the first wave. This reduction is the reason for the first wave in the curves of the new cases, new recoveries and daily deaths. In the following, the variables c i (c C , c R and c D ), which represent the status of the Hospitals' capacity, are incremented, which its outcome is the completion of the first wave. After this phase, while variables c i are incremental, the variables s i experiences a further reduction. The result of this reduction is the beginning of the second wave. After the completion of the second wave, which its reason is the increase in the variables s i , the variables s i are reduced again in addition to decreasing the variables c i due to the fatigue of hospital staff. Its result is also the third wave. To limit this big wave, the variables c i is exponentially increased, which its reason is a mitigation strategy applied by Iranian authorities. Therefore, as mentioned, the variables s i and c i , can detect somethings about the management style of the COVID-19 outbreak.
Where I(t) is the currently Infected. Δt is also equal to one, because I(t) is calculated in each day. Fig 3 shows the currently infected, which is predicted by the proposed model. As shown in this figure, this model could well follow the curve of the currently infected.
In the next section, we provided more information about the results obtained in this research.

Discussion and conclusion
In most world countries, the trend of the COVID-19 epidemic generally depends on social interactions and Hospitals' capacity, although other behavior of the people and government interventions can affect this trend. In other words, social distance and Hospitals' capacity are two fundamental factors in the spread of epidemics similar to the COVID-19 epidemic in a community. Accordingly, these two factors not only affect the infection rate, but also they can directly affect the rate of new recoveries and daily deaths.
In the diffusion model, which we proposed according to this viewpoint in this paper, these effects are clearly visible in variables s i and c i (Fig 2c). Actually, these variables, which are candidates from social distance and Hospitals' capacity and generally adjust the input and output rate of the model's compartments, are the motion engine of the proposed model.
In is remarkable that according to the estimated trend for these variables by the proposed model (Fig  2c), the social distance in the two stages has reduced for forecasting the trend of COVID-19 epidemic in Iran, which their outcome are also the first and third waves in this epidemic. This reduction has been generally felt by the Iranian people and interventions have been also taken to fix them.
As explained earlier, for the second wave, especially in the trend of daily deaths, variables s i is along with a further reduction, so that variables c i could only fix this reduction with a delayed increase, which had been applied by the Iranian authorities. In other words, increasing the Hospitals' capacity could fix this problem (low social distance).
Therefore, these outcomes along with the model's outcomes (Fig 2a and 2b) for forecasting the new cases, new recoveries and daily deaths confirm that the model proposed in this study can be employed for predicting the COVID-19 outbreak in Iran in the next months, because this model has predicted that a new wave will likely occur in the coming months. Of course, as summarized in Table I, the parameters of a i , b i , c i , d i , e i and f i , especially the parameters of c i and f i , which are candidates for social distance and Hospitals' capacity, should be approximately set according to the condition of these factors (social distance and Hospitals' capacity). Therefore, a more comprehensive study on these parameters may improve the outcomes of the proposed model. in addition, there are other factors that affect the spread of the COVID-19 pandemic such as vaccination. Therefore, these factors can be investigated in future studies by adding a set of new compartments to the present model.

Conflict of Interest
The authors declare that they have no conflict of interest.