But why does the epidemic follow exponential growth? Typically, while modelling a real process, mathematicians make some reasonable assumptions. In this case, some of the assumptions were:

- A virus spreads without detection at the beginning (without any control measures in place);
- People are free to make contact with each other;
- Every day, on average, every infected person can infect k healthy persons before being quarantined;
- The population of a city/place is large enough (1 million, for example. In fact, in January 2020, there were 10 million people in Wuhan.)

With these assumptions, one possible model can be explained as follows

* The rate of new cases = k × existing cases yesterday*

Mathematically, it becomes:where P(t) denotes the number of existing cases at time t.

Solving this by standard methods used for differential equations:where the constant C represents the initial condition, that is, the number of infected cases (the first day in the model, i.e., 291 in Table 1, not the first day of the disease).

The expression for P(t) shows that the number of infected people increases exponentially. According to Graph 2, our C was 291 and k was estimated as 0.3067, which means every patient would infect “1.3589 healthy persons”. That’s a lot! It means the infected population (as well as the beds required in hospitals) will be doubled within 3 days. Such predictions gave a reason for the Chinese government to quarantine Wuhan City, in which cases clustered, just on the 4^{th} day (Jan. 23^{rd}) after announcing the epidemic. Three days later, the construction of Thunder God Hospital started.

#### Modelling after the lockdown and stay-at-home

Did we really need to quarantine Wuhan? Probably most people would say “yes” in answer to that question because things were the worst there and people in other cities wanted to keep their cities safe. The following graph shows most of the cases in China had occurred in Hubei Province. (Wuhan is the capital city of Hubei Province.)