The number of passengers on the Diamond Princess cruise ship that were infected with the Wuhan coronavirus (SARS-cov-2) is perplexing. Why wasn't everyone, or nearly everyone infected? Michael Levitt, a Nobel-prize-winning researcher from Stanford University surmised that immunity to the virus is more widespread than thought. This is a workable explanation to a point, but does not explain why people are immune. Perhaps previous non-SARS coronavirus exposures conferred some degree of immunity. Maybe some people are just naturally immune, and maybe a little of both. This post and a couple of those that follow suggest how the proportion of a population that has little or no previous exposure to a virus can end up with an infected population that is only a fraction of the total population. From this, we can formulate a model that fits the existing data from the Diamond Princess, San Marino, South Korea, Washington state, etc. to get a prediction of how extensive coronavirus infection will be, as well as how we can recognize when the epidemic is receding.
To begin, here is a simple thought experiment to help establish some ideas:
Assume that there are two men on an island, and one of them becomes infected with a disease from someone who temporarily visited the island, but is now gone. Assume also that the other person on the island is not immune to the disease, and that the infected person can transmit the infection for only a limited time, for the sake of simplicity, say one day. Now assume that the uninfected and infected person encounter each other regularly, say, four times a day, and that during each one of these encounters the risk that the uninfected person will contract the infection is p=1/6. If the risk of infection from a particular encounter is independent of previous encounters, which is reasonable, then the risk that the uninfected person becomes infected during the time the infected person is capable of spreading the infection is
Pi=(1-(1-p)^4); = (1-(5/6)^4); = (1-0.482);
=0.528
Thus, under the schematic circumstances described above, the uninfected person has an approximately 53% chance, or slightly more likely than not, of becoming infected. If he alters his interactions with the infected person and spaces out his interactions so that the two now encounter each other only three times during the period the infected person can transmit the virus, the risk that the uninfected person will catch the infection is
Pi=(1-(1-p)^3); =(1-(5/6)^3); = (1-0.5787)
=0.4212.
Now the probability that the uninfected person becomes infected is only 42%; more likely than not that he will NOT that he will not become infected. Note that the number of exposures was decreased by 25%. If instead, we decrease the risk of exposure by 25% so that p is now pn=1/6*(3/4)=1/8, and we still have the original four interactions, the risk of infection is now
Pi=(1-(1-pn)^4); =(1-(7/8)^4); =(1-.586)
=0.413.
Decreasing the risk per exposure reduces the overall risk of infection by a little more than a proportional reduction in the number of exposures.
The key concepts illustrated by this thought experiment are that:
1.) The amount of time that an infected person can pass along the infection is limited;
2.) The risk of infection depends on the nature of the interactions that produce exposures;
3.) Modifying exposure risk decreases the likelihood of disease transmission, even when the risk of transmission is not completely eliminated.
4.) An exposed person may not become infected even if not immune.
These concepts should be easy to understand. Not all exposures carry the same risk of infection, for example, not everyone who shakes hands with an infected person will become infected, nor will everyone who passes an infected person on the street. It is understood as well, that apart form the artificial constraints imposed by having only two people on an island, an uninfected person is likely to have multiple encounters with many infected people, but the underlying principle is the same: each such encounter is associated with a risk of infection that is greater than zero but less than one. In the next post, the concept will be generalized to illustrate how the final number of people infected in a population will tend to a value other than 100%.
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