**The Lockdown**

India was under a lockdown to promote social distancing so that the spread of the Chinese Coronavirus or Covid-19 would slow down. On March 22 there was a self imposed curfew. The lockdown started on March 24 and was to go for three weeks until April 14. Like the rest of the family, the 17 year old Sarasvati (Sara) and her 16 year old brother Neal were also not allowed to go anywhere. After being locked up in the house for one week, Sara and Neal started to have this discussion. By this time, the virus had take over the world with over 800,000 cases but India remained lucky that it had a low rate of infection. Some blamed the low number in India to inadequate screening and others credited it to BCG vaccination during childhood or to the presence hsa-miR-27b gene in the Indian population. Here is the discussion.

**Exponential Spread Of The Disease**

\ Neal: Do you know when this dreadful Chinese virus will go away ?

Sara: Nobody knows but here is a graph showing that India has been very lucky.

Neal: This is an exponential curve in which the American numbers dominates and you can hardly see the Indian data.

Sara: Remember, you learned about logarithms. If the exponent value is a^{x }where a is the base, and x = log_{a} a^{x}.

I plotted log_{10} (number of cases) versus the days. It is the same data as the first graph.

Neal: Yes, now I can see data for all the three countries. It is interesting that the slopes of the three lines are different. The USA line is way steeper than the India line. Is it because the USA has many more corona patients than India ? The value of log_{10} (number of cases) is around 4 on March 19 and 5 around on March 26. Since this is log with base 10, the value of the number of cases increased 10 – fold in 7 days. For India, it took 14 days from March 15 to March 24 to get a 10-fold increase.

Sara: Yes. The steeper line means that log(number of cases) increases much faster.

Neal: Is it not that an exponential function is: Initial value x a^{t} ? If a =2, then the values of the exponent will become Initial value x 2^{t/doubling time}. From your data, I just calculated that in India it took an average of approximately 4.5 days for the confirmed case numbers to double compared to only 2.6 days in USA with Canada being in between at 3 days. It is because, India is still in the Phase 2 of the coronavirus growth where you can trace who infected whom. In contrast, USA is in the Phase 3. There are a lot of infected people in a population and you cannot tell from whom you got the virus. Thus the infections are communal and far more rapid. Canada is somewhere in between.

**Confirmed vs. Active Cases**

Sara: I guess, now you want to do some modeling. Before, we do the modeling let us try to understand the Indian data. The graph we have is for the cumulative number of confirmed cases. Some of the patients have already recovered and a fewer patients are also dead. So the active cases will be the total number minus those who recovered or died. Here is a graph comparing all these values from March 22 to March 31.

Neal: What else do we need to know for the modeling and to make some predictions ?

Sara: We need to know three things. First, how many new active cases will come up.

Neal: Mind you, this assumes that the lockdown is followed by all. Movement of large number of workers and gathering in a in Nizamuddin already led to aberrations. Still, I guess we have to go with some assumptions. The success of the lockdown plan seems to be fair one. What about the recovery rates ?

**Recovery As An Exponential Decay**

Sara: In China, it seems that new cases nearly stopped by the beginning of March. As a result of the recovery, the number of active cases started decreasing. Here is a graph for the number of remaining active cases on different dates. This is also an exponent but a decreasing one. One way to write it is:

Initial number x 2^{(-t/t1/2)} where t^{1/2 }is the half life of the number of active cases and t is the time.

Neal: The data seem to fit well into a half life of 7.7 days whose graph is in blue. I we will use this half life of 7.7 days in our modeling.

**Mathematical Modeling **

Sara: We will start with the Sunday of March 22 when Modiji called for self imposed curfew for all the Indian people.

Neal: We can first graph the available data for active cases from March 22 to March 31. Then what do we do ?

Sara: On any given day, we can calculate the active cases that disappeared due to recovery. Because the recovery half life is 7.7 days, the active cases remaining after one day will be: number on previous day x 2^{(-1/7.7)}.^{ }

In the past 9 days, an average of 130 new cases came up per day and two deaths per day occurred. We make the assumption that Modiji’s plan succeeds and the number of new cases per day drops to zero by the end of the lockdown. If it was a linear decrease, it would mean 130, 120,110… cases per day until April 14 when it will come down to zero on April 14. Let us also think that the death rate will remain at an average of 2 per day. So on any given day, the number of active cases will be:

(number on previous day x 2^{(-1/7.7))} – number of deaths in last 24 hours + new cases in last 24 hours.

Note that the number of active cases is consistently smaller than the number of confirmed cases. This calculation will give us the prediction for the future graphed here until April 30, 2020.

Neal: This graph means that the number of active cases will never reach zero but we can always quarantine the remaining small number of patients. They will recover or die in 14 days after that. By May 15, the country should be free of this Chinese virus. I am going to tell this result to dad and then to all my Whatapp friends.

**A Model Is Only As Good As The Assumptions You Make**

Sara: Neal, slow down because we could be wrong. We only had the data until March 31and for the rest we made assumptions. We don’t know that the number of new cases will follow this trend. All it will take will be a couple of large gatherings to screw things up. The data on the effect of mass exodus of laborers is yet to come, Also, the half life assumption is based on data from China. We don’t know how reliable it is. Considering how good the fit was, I would say that it was fudged because normally one would expect some scatter. *A model is only as good as the assumptions you make.*

Neal: That is true but I am glad we did this. I got to learn about exponents and logs. Besides, it was fun.

**Challenge**

**Emotions**

Rationality is fine but it cannot overcome emotions. The recent Nizamuddin case was an example. It was a large religious gathering at a time when social distancing was needed to slow the spread of the coronavirus. Large scale exodus of labor from Delhi led to crowds at a bus stop. Some gurudwaras such as the one in Majnu ka teela still had people gathered in it. Vaisakhi is coming on April 13. Vaisakhi being the birthday of Khalsa, expect processions and fairs. Expect more new cases at times than in the model. This could make the coronavirus hang around longer. Therefore, calculate what would happen every week rather than every day. Model what would happen in the next 12 weeks after March 31. For practice, assume the number of new cases per week to be 1500, 1200, 1800, 1200, 800, 600, 500, 400, 300, 200, 100, 0, and deaths to be 20 per week.

Solution: If the half life remains 7.7 days as in the story, at the end of any given week, the number of active cases will be

(number on previous week x 2^{(-7/7.7))} – number of deaths in last week + new confirmed cases in the last week.

The derived number of active cases each week is given in the Table below and the graph is at the end.

**Emotions and therapy**

Although a vaccine will take a long time to develop, there are several potential therapies which could shorten the half life of the virus. Examples are a combination of chloroquine with other drugs, human plasma therapy, BCG vaccine boosts, over-expression of the micro RNA hsa-miR-27b. Assume that one of these therapies will be employed and the half life will drop to 4 days instead of 7.7 days. Model the progress assuming the number of new cases and death to be the same as the Emotions model and a half life of 4 days.

Compare the model in the story with the Emotions model and the Emotions and therapy model.

Solution: (number on previous week x 2^{(-7/4))} – number of deaths in last week + new cases in last week.

Table of the derived number of active cases each week is below and the graph is at the end.

Number of deaths in last week + new confirmed cases in the last week.

The derived number of active cases each week is given in the Table below and the graph is at the end.

**Emotions and therapy**

Although a vaccine will take a long time to develop, there are several potential therapies which could shorten the half life of the virus. Examples are a combination of chloroquine with other drugs, human plasma therapy, BCG vaccine boosts, over-expression of the micro RNA hsa-miR-27b. Assume that one of these therapies will be employed and the half life will drop to 4 days instead of 7.7 days. Model the progress assuming the number of new cases and death to be the same as the Emotions model and a half life of 4 days.

Compare the model in the story with the Emotions model and the Emotions and therapy model.

Solution: (number on previous week x 2^{(-7/4))} – number of deaths in last week + new cases in last week. Table of the derived number of active cases each week is below and the graph is at the end.

Number of active cases in the three models |
||||

Week after March 31 | From the story | Emotions | Emotions +Therapy | |

0 | 1109 | 1109 | 1109 | |

1 | 1423 | 2081 | 1820 | |

2 | 904 | 2298 | 1731 | |

3 | 468 | 3014 | 2305 | |

4 | 235 | 2795 | 1875 | |

5 | 2278 | 1347 | ||

6 | 1803 | 991 | ||

7 | 1450 | 785 | ||

8 | 1162 | 623 | ||

9 | 909 | 475 | ||

10 | 674 | 331 | ||

11 | 449 | 188 | ||

12 | 229 | 46 |

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