Ensconced in the apparent safety of the 4 walls the mind looks out into the completely silent streets with hardly a soul or even a passing ratha — a mere 120 days have made the world look and sound different. With that realization, we place here the 3rd in this series of notes recording these times. This note will cover some very basic stuff on visualizing disease progression in a population which we used to explain things to laypeople (bālabodhana).
We hear laypeople express surprise over things like: (1) “Just a few weeks ago there were just 10s of cases of the Wuhan disease now it has just exploded.” (2) “What is the point in staying at home? How can it reduce the disease?” (3) “There are all these predictions of millions of people dying and now it has come down to tens of thousand. There must be some over-reaction due to X or Y.” As long as the person is willing to have a patient discussion we found that we could explain the basics relating to these using some illustrations which don’t use any sophisticated mathematics and just a simple program which we wrote for the purpose. One can easily reproduce similar illustrations with ones own assumptions in whichever is the language of ones choice. When we try to clarify point (3) we tell people that what we are showing them is not like any of the complex models that they hear of in the news. We explain to them, extrapolating from our simple program, that the forecasting models have many, many ways in which they can be parameterized. We do not know the correct values of several of these parameters and as result of that there can be tremendous diversity of outcomes due to the combinatorial interaction of the alternative values of parameters. Hence, we simply cannot be certain which forecast of the model is going to actually play out. However, the simple program does explain some of the essential outcomes that a layperson needs to understand.
Our model for how disease can progress in a population is a very simple one: (1) It assumes a population starting a certain fixed size , which undergoes decrement from deaths due to infection but beyond that there are no other factors changing its size. This simple assumption is good enough for a fast-moving epidemic like the one we are saddled with. In what we illustrate below . (2) We initiate the disease with a single infected individual introduced into the population and let the disease run for a certain maximum number of time units. One can take them to be days if one wants; in our example. (3) If a person in the population catches the disease he remains infectious for 7 days, i.e., can transmit it to someone who is in the vicinity. Within that period the infected individuals can die with a certain probability; in our below illustration. (4) After that period if the person has not died he is considered as recovered and is immune to reinfection. This again is a reasonable assumption for a fast-moving disease. Thus, our process features 4 states for individuals in the population: Uninfected; Infected; Recovered; Dead. These are respectively shown in different colors: white; brown; green; black. (5) Not every person who has gotten the infection spreads it uniformly to others in the vicinity. Here, we assume a power-law distribution of the capacity of an infected individual to infect others in the vicinity in an encounter. One example goes thus; an infected person in an encounter infects:
0 persons 0.747475 of the times
1 persons 0.204900 of the times
2 persons 0.025200 of the times
3 persons 0.008725 of the times
4 persons 0.004550 of the times and so on till,
124 persons 0.000025 of the times. They all add up to 1.
This assumption captures the fact that most encounters do not actually result in an infection but some rare people are “super-spreaders” so they infect a large number of people in an encounter.
In its default state, people in our population randomly move at each time unit with little restriction on movement. Infection is transmitted with the power-law-defined frequency to neighbors when they encounter an infected individual. This goes on till the set maximum of time units is attained. But we can set a restriction parameter, which decides how freely the people in the population can randomly move to encounter others. The greater this restriction parameter the less they can move about to bump into others.
For each run, the upper panel shows the actual numbers of the 4 states in the population at each time unit of the simulation. The lower panels illustrates the state of the population at each of the 25 time units as 1 dot per person, colored based on the individual’s state.
Figure 1 shows a run where there is essentially no restriction on movement. It begins innocuously with most of the population in the uninfected state and within 25 time units the entire population goes through infection and ends up as recovered or dead. This helps bring home to a layperson how the disease progresses: the initial stages always look innocuous and how an explosion happens by exponentiation and can overwhelm any treatment system. An analogy can also be made to a fire: as long as there is fuel it continues to burn but as the fuel decreases there is a down turn in the burn rate and finally the fire ends.
Figure 2 shows a run where right from early on free-movement of the population is somewhat restricted. Here, we see some mitigation of the deaths and infections. Still majority of the population gets infected.
Figure 3 shows a situation where much stronger restriction is imposed on the free movement of people. Here, a considerable fraction of the population is uninfected, deaths are considerably reduced and the infection can be brought to zero in the end.
The latter two scenarios illustrate to the layperson the value of early mitigation measures in the form of restriction of social interactions or free movement in controlling such epidemics. Of course, this is a very simple illustration and in no way should be taken as a forecasting model. However, as noted above forecasting models suffer from great parametric uncertainty and a combinatorial increase in search space when the multiplicity of alternatives are taken into account. Thus, a simple model like this, which provides some basic qualitative understanding of what seems puzzling to laypeople is good enough for them to get some intuitive feel for the situation.