A simple kinetic modelling of coronavirus infection in a system

In this paper, provide an analysis of data on the characteristics of patients with confirmed COVID-19 infection throughout Iran and other countries such as China, S. Korea,USA, Italy, Spain, Germany, France, UK, Austria, and Argentina, and presented a kinetic modelling for prediction of infection, death, recovery and patients populations.The predict results of infection, death and recovery rate constants helps to interpret of public sentiment on disseminating related health information, and comparison different governments behavior to overcoming the outbreak of the coronavirus (COVID-19). Calculation results are in agreement with the census data of the coronavirus infection in many regions of the world.

Since December 2019, novel pneumonia cases were reportedin China. 1 Thesecases have been identified as infections with a novel strain of coronavirus, called COVID-19 by the World Health Organization (WHO). [2][3][4][5][6] In January 2020, the coronavirus spread over China and reached to other countries.Daily situation contain the infection and death cases were reported by WHO from January 21.
Commonsymptoms include fever, coughand shortness of breath. Other symptoms may include fatigue,muscle pain, diarrhea, sore throat, loss of smell and abdominal pain. While the majority of cases result in mild symptoms, some progress to viral pneumonia and multi-organ failure. 7 In 19 February 2020, first cases of coronavirusinfectionsin Iran werereported fromQom province. 8 The number of infection cases to COVID-19 have spreadexponentially since March, 2020 in most countries in Europe and USA. As of 16 April 2020, more than 2 millioncases have been reported in more than 200 countries and territories, resulting in more than 95,800 deaths and more than 542,000 people have recovered. 9 For many researchers and specialists it is crucial to understand the time profile of infection population. Therefore, analysis and modelling can inform predictions about potential future growth of infection.
Lin and et al. utilize Susceptible-Exposed-Infectious-Recovered (SEIR) modelling to forecast COVID-19 outbreak within and outside of China based on daily observations. 10 Kucharski and co-workers reported a mathematical modelling study for dynamics of transmission and control of COVID-19. 11 Leungprovide an impact assessment of the transmissibility and severity of COVID-19 during the first wave in mainland Chinese locations outside Hubei. 12 AlsoBayham used data from the monthly releases of the US Current Population Survey to characterize the 3 family structure and probable within-household child-care options of US health-care workersby a modelling analysis. 13 Koo and co-workerswere adapted an influenza epidemic simulation model to estimate the likelihood of human-to-humantransmission of severe acute respiratory syndrome coronavirus 2 in a simulated Singaporean population. 14 In this work, kinetic modelling of coronavirus infection has been provided based on an autocatalytic behavior. The proposed model was used for data on cases of coronavirus disease  in China as a sample. The number of infection, death, recovery and patients (timeprofile) were predicted for China cases by provided model. With regarding to very good adaptationof calculated and census data, model were used for Iran and other countries to estimate how transmission had varied over time during in January, 2020, and April, 2020.

Modelling
The scheme 1 shows an autocatalysis reaction, which is just the appearance of one of the products of the reaction as a reactant in the same (HV) or a coupled reaction. The following kinetic model (Scheme 1) adapted to coronavirus infection which can depict the dynamic of infection, recovery and death processes in one and two steps (one-wave and two-waves). In this model H, HV, R and D are number of health, infected,recovered and death cases, respectively. (1) The number of recovery cases can be obtained by solving Eq.(5). By rearrangement of Eq. (7) and substituting in Eq.(5),Eq.(11) is obtained: As can be seen from Eq.(11), the solving of differential equation seems to be difficult. Thus, by considering theassumption given in Eq. (12) With assumption similar to Eq. (12), [D] can be calculated by Eq.(15).
In order to assess the performance of the obtained kinetic model, the number of infected coronavirus calculated from Eq.(10) were compared whit the infection census data for China and is indicated in Fig. 1a. respectively and compared whit the corresponding census data in Fig.1b, Fig.1c, and Fig.1d, respectively.The calculated kinetic parameters regression using kinetic model is summarized in Table 1.
An additional set of data containing infected,recovery,deathand patients cases for other countries such as S. Korea, Iran, USA, Italy, China, Spain, Germany, France, UK, Austriaand Argentina were used to compare with predictionsof the kinetic model.Kinetic estimated parameters of 10 different countries are given in Table 1. Themodel was fitted with all the available census data(before17 April, 2020)and kinetic parameters areshown in Table 1.  Fig. 2a, Fig.2b,Fig.2c, Fig.2d  A typically, comparison of predicted and infection census data for Iran and USA are also shown in Fig. 3(a-d) and Fig.4(a-d). The excellent match between the census and predicted results for different countriesindicates that the proposed kinetic model in Scheme 1 represents the coronavirus infection satisfactorily.The results of prediction and calculated kinetic parameters are summarized in Table 1. shown in Fig. 2(a-e), Fig. 3(a-d) and Fig.4(a-d) S9c and 9c').The main assumption of this kinetic model is the individual behaviors and governmental action, effect on kinetic parameters and is not affected on model validity.

Discussion
As summarized in Table 1 However, the highest value of first order recovery rate constantis also belongs to S. Korea.
The highest and lowest values of first order death rate constant belong to France and USA, respectively. However, the lowest value of first order recovery rate constant is also belongs to USA.
In conclusion,kinetic modelling of coronavirus infectionbased on autocatalytic chemical reaction idea can be applied for simulation of infection casesthat originated in a system to estimate how transmission had varied over time during and not depended to region of infection. The merit of our model is that we predicted some essential parameters, including maximum number of infection, first and second order infection rate constants, recovery, death rate constants, 12 recovery and death fractions and plateau time of infection. Meanwhile, proposed model is relatively simple and our estimates for different countries are in good agreements with census data at during selected times.However, individual behaviors and governmental action, effect on deviation of calculation and census data, but these values compensate by kinetic parameters and is not affected on validity of model.