APR 25, 2020

SIRU ODE Model

Venky P Krishnan, Koushik Ramachandran and Sreekar Vadlamani

Date May 5 May 6 May 7 May 8 May 9
Predicted total cases 49793 54147 59133 64879 71545

Predicted total number of COVID-19 cases in India for the next 5 days

We are working on a mathematical model to predict the number of COVID-19 infections in India. This effort is a work in progress, and we seek your participation in a short online survey which will help us obtain a quantifiable "social distancing parameter" for different districts/states in India.

Why are we interested in this online survey?

Currently, our choice of the social distancing parameter is determined by the best fit to the official reported data for a certain amount of time to predict future infection numbers. In the short survey that we have prepared, you will encounter four questions (all multiple-choice), after entering your state/district/pin code information. This survey is a joint work of the contributors in collaboration with Subhashini Sadasivam.

If we receive a large number (several thousands we hope!) of accurate responses to these questions from various parts of the country, we hope to derive a so-called "social distancing parameter" that can better inform the model and make it more realistic. A diverse and well-represented survey is crucial to the success of this effort. To this end, we seek your help in drawing the attention of your relatives and friends from the length and breadth of the country to this survey. All it takes is a WhatsApp message these days.

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The Web site is created and operated by Shrikant G

The details of the model we are working on are as below. The source code will be available on GitHub soon.

1. Mathematical Model

We use the SIRU ODE model introduced in [2, 3], to predict the total number of reported COVID-19 cases in India. This model from [2, 3] is as follows:

\(S'(t)=-\tau(t) S(t)(I(t)+U(t))\)

\(I'(t)= \tau(t) S(t) (I(t)+U(t)) -\nu I(t)\)       (1)

\(R'(t)=\nu_1 I(t)-\eta R(t)\)

\(U'(t)=\nu_2 I(t)-\eta U(t)\),

where

  • \(S(t)\) - number of susceptible individuals at time \(t\),
  • \(I(t)\) - number of asymptomatic infectious individuals at time \(t\),
  • \(R(t)\) - number of reported symptomatic infectious individuals at time \(t\),
  • \(U(t)\) - number of unreported symptomatic infectious individuals at time \(t\),
  • \(1∕\nu\) - average time during which individuals are infectious but asymptomatic; \(nu\) is taken to be 1/7 here,
  • \(f\) - fraction of asymptomatic infectious that report to the health authorities, thereby becoming symptomatic infectious; taken to be 0.4 here,
  • \(\nu_1 = f\nu\) - rate of asymptomatic infectious individuals becoming reported symptomatic infectious individuals,
  • \(\nu_2 = (1 - f)\nu\) - rate of asymptomatic infectious individuals staying as unreported symptomatic individuals,
  • \(1/\eta\) - average time for symptomatic infectious individuals to develop symptoms; \(\eta\) is taken to be 1/7,
  • \(\tau(t)\) - this function takes into account social distancing; described in more detail below.

The data for the simulations is taken from COVID-19 Dashboard by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University. The csv file given at this link below has the total number of reported cases for several countries of the world including India https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases.

Even though the first reported case of coronavirus in India was on Jan 30, 2020, the first three cases were quarantined. Due to this reason, in our simulations, we take the data of total reported cases starting from March 01, 2020. Similar to what is done in [2, 3], we use the data during the time-period, March 03, 2020 to March 25, 2020 to fit an exponential curve. The end date is chosen as March 25 due to the fact that the lockdown announced by the government started from this date.


Figure 1. Exponential curve fitting of the total reported data from March 3 - 25, 2020

The exponential curve fitting the data during the time period March 4-25, 2020 is shown in Figure 1. The transmission function in the system of ODEs is of the form \(\tau(t)\) = \(\tau_0g(t)\), where \(g(t)\) takes into account efforts like social distancing and lockdown by the government to mitigate the spread of the disease. The value \(\tau_0\) is computed as in [1, 4] which involves the basic reproduction number (which in our case is computed to be approximately 3) and the initial susceptible population of India (approximately 1.3 billion). Using this method, we arrive at the value of \(\tau_0 = 1.92 \times 10^{-10}\). The remaining constants such as the initial starting time of the disease and the initial condition \(I_0\) and \(U_0\) corresponding to \(I\) and \(U\) are computed based on the exponential fitting curve as in [2]. These are found to be approximately 12.5 and 3.9 respectively. We have chosen the initial value of \(R\) to be \(R_0 = 0\).

Scenario 1: Indefinite lockdown

As in [2], we choose the function \(g(t)\) to be an exponentially decreasing function of the form given in Figure 2. This constitutes an indefinite lockdown scenario. Under this condition, we arrive at Figure 3 for the total number of reported cases.


Figure 2. Social distancing function where the lockdown starts on March 25, 2020 and is indefinite


Figure 3. Plots of the simulated and reported cases when the lockdown that started on March 25, 2020 is indefinite

Scenario 2: Lockdown ends May 3, 2020

The notion of an indefinite lockdown is unrealistic. As it stands, the lockdown is slated to end on May 3, 2020. The transmission function we choose in this case would be as in Figure 4. Under this scenario, we arrive at Figure 5 for the total number of reported cases.


Figure 4. Social distancing function where the lockdown starts on March 25, 2020 and ends on May 3, 2020


Figure 5. Plots of the simulated and reported cases when the lockdown that started on March 25, 2020 ends on May 3, 2020

2. Estimating the transmission function

Estimating \(\tau(t)\) is critical to modelling the spread of any infectious disease. In this study, the transmission function is chosen to be of the form \(e^{-\mu x}\) and this value of \(\mu\) indicates how strong or weak the social distancing measures of a country are. In our simulations, we have chosen this value of \(\mu\) to be 0.035 so as to get the best fit for our predicted data with the reported data, and use this in turn to predict future data.

For instance, based on Model 1, we predict that the total number of reported cases for the next 5 days to be as in Table 1.

Date May 5 May 6 May 7 May 8 May 9
Predicted total cases 49793 54147 59133 64879 71545

Table 1. Predicted total number of COVID-19 cases in India for the next 5 days

However, an important question that still remains to be understood is the following. What does a particular value of \(\mu\) for a country represent in terms of its social distancing measures? In other words, we would like to quantify the social distancing efforts currently going on in our country. We aim to gain a better understanding of this by means of a crowd-sourcing effort. More precisely, we seek the reader's help in filling the online survey found at the following link. The realistic data that you provide would help us in gaining an insight into quantifying the social distancing efforts currently underway. This survey should only take a couple of minutes and we thank you for your time. Also we seek your help in drawing the attention of your relatives and friends from different parts of the country to this survey. A diverse and well-represented survey reflecting the length and breadth of our country is crucial to the success of this effort.

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3. Acknowledgments

We would like to thank Shyam Ghoshal, Debabrata Karmakar and Debayan Maity for the stimulating discussions about this work. They introduced and helped us understand the References [2, 3], as well as shared Python codes. We extend our sincere thanks.

References

  • [1] O. Diekmann, J.A.P. Heesterbeek, and J.A.J. Metz. On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J. Math. Biol., 28:365-382, 1990.
  • [2] Zhihua Liu, Pierre Magal, Ousmane Seydi, and Glenn Webb. A COVID-19 epidemic model with latency period, 2020. Infectious Disease Modelling (to appear).
  • [3] Zhihua Liu, Pierre Magal, Ousmane Seydi, and Glenn Webb. Predicting the cumulative number of cases for the covid-19 epidemic in China from early data, 2020. arXiv, 2002.12298.
  • [4] P. Van der Driessche and J. Watmough. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29-48, 2002.

Disclaimer

The information provided here is the result of an academic exercise. The projections are only indicative, and the contributors do not claim that their projections are accurate. The opinions and views expressed here are not endorsed in any shape or form by the organization in which the contributors work.