Mathematical Modelling of Coronavirus disease (COVID-19) Outbreak in India using Logistic Growth and SIR Models


 The mathematical modelling of the Coronavirus disease (COVID-19) outbreak in India is done by using the logistic growth model and the Susceptible-Infectious-Recovered (SIR) framework. Karnataka, Kerala and Maharashtra, three states of India, are selected based on the pattern of the disease spread and the prominence in being affected in India. The parameters of the models are estimated by utilizing real-time data. The models predict the ending of the pandemic in these states and estimate the number of people that would be affected under the prevailing conditions. The models classify the pandemic into five stages based on the nature of the infection growth rate. According to the estimates of the models it can be concluded that Kerala is in a stable situation whereas the pandemic is still growing in Karnataka and Maharashtra. The infection rate of Karnataka and Kerala are lesser than 5% and reveal a downward trend. On the other hand, the infection rate and the high predicted number of infectives in Maharashtra calls for more preventive measures to be imposed in Maharashtra to control the disease spread.


Introduction
The novel Corona virus disease-2019 (COVID-19) is a deadly infectious disease which was first reported in Wuhan, China by the end of the year 2019. Later, it spread across continents leaving only a few countries unaffected. The large spread and increasing number of deaths made the World Health Organization (WHO) to declare it as a pandemic on 11 March 2020 [1]. The governments of the affected countries have imposed various preventive measures including lockdown, seal down, social distancing, quarantining infected and susceptible individuals, promotion of self-hygiene measures and usage of masks. Recently, Fang et al. [2] reported that the governmental interventions proved to be effective in containing the infection rate.
In India, the pandemic was first reported on 30 January 2020 in Thrissur, a district of Kerala for three individuals who had come from Wuhan. In March, the number of infected individuals multiplied due to individuals who had travelled to India from various affected countries. Initially, the government imposed preventive measures including thermal screening of passengers arriving at the airport, shut down of educational institutions and suspending tourist visas. Further, on 24 March, the government imposed a nationwide lock down as a major preventive measure to contain the spread of the virus. This restriction on the human mobility was reported to be effective for various infectious diseases [3,4]. The Indian context of the disease spread and the impact of the public health interventions were mathematically modelled by Mandal et al. [5].
During these hard times of the pandemic, researchers from various disciplines conduct intensive research to analyze it and to curb its ill-effects [6][7][8]. Apart from virologic and experimental studies, mathematical and statistical explorations are also important for the decision-makers and policy makers to evaluate the current situation and to take the necessary actions for the future. These models also provide an estimate of the ending time and possible number of people that can be infected by this pandemic. The logistic growth model is an efficient technique for epidemic forecasting. Chowell et al. [9] used this model to estimate the transmission of the Ebola virus. They concluded that the predictions from the model were inconsistent in the initial period of the epidemic. The model was also adopted by Pell et al. [10] to estimate the final size and the peak time of the infection. The SIR model (compartmental disease model) is a mathematical method to describe the epidemic growth through a system of time dependent differential equations. The differential equations are based on the compartments into which the population is divided. The SIR model and various modified SIR models were widely used by researchers to model Ebola [11] and AIDS [12]. Recently, such models were used to model the coronavirus epidemic spreading. Khrapov and Loginova [13] utilized the modified SIR model for the analyzing the dynamics of the COVID-19 pandemic in China. In this direction, the modified SEIR model was used by Berger et al. [14] to include the effects of quarantining and infection testing. May 2020 are used for the modelling. The final size of the pandemic in the states is estimated and the date when the pandemic becomes stable is also predicted.

Mathematical models 2.1 Logistic growth model
The population growth can be modelled using the logistic growth model which is a sigmoid curve. Pierre-Francois Verhulst pioneered the usage of logistic growth models for biological systems.
where, -accumulated number of cases, -infection rate, -time and is the final epidemic size.
If the initial number of cases is given by | =0 = 0 , then the solution (1) is i.e., = 1+ − where, = ( For estimating the maximum number of affected people, the cases follow the Weibull function. Further, the highest growth rate occurs at the time = ln . At , the number of cases is 2 . The epidemy is modelled using five stages based on the time . This is based on the nature of the growth rate. The five stages are as follows:

Stage 1: Slow growth stage
The spread of any infectious disease is slow initially which is accounted in this stage. This is the exponential growth phase of the logistic growth model. The growth at the time < − 2 is considered in this stage.

Stage 2: Accelerating growth stage
Here the spread of the disease spreads largely in the population. At this stage the number of positive cases multiplies several folds. This accelerating phase is at the time − 2 < < .

Stage 3: Decelerating growth stage
At this stage the spread of the disease decelerates due to various preventive measures taken.
The spread of the disease is controlled and lesser than that of Stage 2. The decelerating growth is at the time < < + 2 .

Stage 4: Transition stage
This is the stage when the disease growth rate slows down. At this stage the epidemy is said to be under control. The preventive measures taken prove to be effective. The time duration of this stage is + 2 < < 2 .

Stage 5: Steady stage
This stage can be regarded as the culminating stage of the epidemy. There is no growth in the disease spread. This stage is at the time > 2 .

SIR model
The Susceptible-Infected-Recovered (SIR) model is a mathematical model also known as compartmental disease model to describe the disease spread in a population. Here, the considered population belongs to any one of the three compartments: Susceptible, Infected or Recovered as shown in Fig. 1.
Susceptible are the individuals without immunity to the disease, hence they can be infected.  [15], it comprises of following set of time related nonlinear ordinary differential equations to simulate the growth of a disease. The following suppositions are involved in the model: • The population is considered to be unvarying during the phase of modelling.
• All the infected beings have an equal chance to be recovered.
• Secondary waves of infections and any other unusual outbreak of the infection are not considered in these models.
• The real time data of the officially reported positive cases are used for the models.
Solving the equations one can get, To find the total number of susceptible individuals at the end of the disease spread, the limit to ∞ is taken to get: where, ∞ is the final number of recovered individuals. In order to estimate the model parameters and 0 with the available real time data of the number of positive cases (denoted by for each day ), the initial values are taken to be 0 = 1 and 0 = 0. Now, the parameters are found such that the error sum of square of the predicted and the actual number of cases is minimum.

Methodology
The number of positive novel corona virus (COVID-19) cases in Karnataka, Kerala and Maharashtra from 9 March 2020 to 9 May 2020 were recorded. The data source was based on the daily reports from the respective health departments of the states. These states were selected due to the significant difference in the disease spread patterns. This data was used to estimate the parameters for the logistic growth model and the SIR model. The models and the visualizations are done using the program developed by Batista [16,17] using MATLAB.   ( 2 ) suggest the accuracy of the fitted models.

(b) SIR Model
The estimated parameters for the SIR model of the three states are given in Table 2.
The contact rate quantifies the mean number of contacts per infected individual per day. It is seen that the contact rate is the highest for Maharashtra. The exclusion of the infected population by immunization, death, quarantining or isolation is quantified using the removal  Table 2. The basic reproduction number (ℛ 0 ) quantifies the expected number of secondary infections by one typical infection in a population that is completely susceptible.
Based on the data it is estimated that the ℛ 0 of Kerala is the highest among the three states.
The high predicted number of cases of Maharashtra indicates that the state must enforce more strict measures to curb the spread of the virus.

Concluding Remarks
The COVID-19 outbreak in India was analyzed by comparing the dynamics of the pandemic in Karnataka, Kerala and Maharashtra using the logistic growth and SIR models. It was observed that the states experience a totally different pattern of the disease transmission.