The efficacy of a vaccination program is quantified by the fraction of the non-immunized population that gets sick in an epidemic. This fraction can also be thought as the probability that a particular individual will get sick in an epidemic.
A number of factors determine the probability of infection in an epidemic:
How long is the sick person contagious? What is the probability of infection given the contact with a sick person? What is the average rate of inter-personal contacts? How far does a person travel during the sickness? The answers to these questions depend on the type of virus, and the properties of the population such as its density and the patterns of movement.
The situation seems too complex for predictive modeling. Could a simplified model offer meaningful insight? Yes, if we pick a narrow aspect of the problem to look at. How about this? You probably heard the doomsday scenarios of a deadly virus spread around the world aboard airplanes. Is a this kind of talk just fear-mongering or a realistic prediction?
Let us construct a model to study whether the doomsday scenario is plausible. Let’s start with a 2D square lattice, or a board, whose sites (spaces) can be empty of occupied by “people” — let’s call them “entities.” The entities could be in three states: immune, vulnerable, and sick. The sick entities can infect the vulnerable but not the immune ones. We need to decide what to do with the sick entities. For example, some fraction of them can “die”–be removed from the board. The simplest thing is to just let them become immune after the disease has run its course. This is what is done in our model.
The entities can move around the board. The movement models the short range everyday movement of the population: commute, shopping, going to and from school, etc. I will use a turn based (like the Conway’s game of life) set of movement rules that are often used in simulating fluid-vapor interfaces. The result is a collection of dense clusters of varying sizes that float in a sparsely inhabited sea. There is little exchange of entities between the clusters. Since the infection is acquired on contact, global epidemics are impeded by the limited inter-cluster movement. One could think of these semi-isolated clusters as communities, cities, or even continents depending on your perspective.
Below is the movie of the model simulation in which the sick entities (red) infect the vulnerable entities (blue) and after a while become immune (green). A fraction of the population is already immune at the onset of the epidemic. Observe how the disease propagates quickly across the clusters and makes infrequent jumps between the clusters. In this particular simulation, 37% of the vulnerable population got sick before the epidemic fizzled out.
You probably noticed that the immunization rate in the above example is rather low, 30% to be exact. Since most entities are vulnerable, the epidemic has no trouble spreading. When the immunization rate is more than doubled to 70%, most epidemics fizzle out early. As you can see in the PDF (probability density function) plot below of the total epidemic size (defined as the fraction of the vulnerable population that go sick), all epidemics involve fewer than 4% of the populace. There is simply not enough population movement for the disease to spread.
Time to include airplanes and examine the plausibility of the doomsday scenario!
In addition to the short range movement, let’s allow at each turn a certain small fraction of the population to move anywhere on the board. The second graph below is the PDF of the epidemic size for the same parameters as the one above, but with the additional 5% of the population executing large scale movement each turn. Notice the radical change in the scale of the x axis. When a small fraction of the population travels long distances each turn, most epidemics grow to encompass the majority of the population. The bimodal nature of the epidemic size distribution suggests that there is a threshold size. If the epidemic hits a cluster that happens to be larger than the threshold, the disease can escape and infect almost all other clusters.
Let us now quantitatively examine the effect of the large scale movements on the probability of significant epidemics. In the graph below I will plot the probability of occurrence of an epidemic that involves > 10% of the vulnerable populace as a function of the immunization rate for two different magnitudes of the large scale movement. Significant epidemics become rare as the immunization rate increases. However, perhaps not surprisingly, greater immunization rate is required to avoid epidemics for a larger magnitude of large scale population movement.
Predicting how epidemics spread in the real world is a tricky business. However, the general conclusion of the simple model, I think, will stand. While 100% immunization rate is not strictly required to stem epidemics, as the extent of long distance travel increases, we will need a higher immunization rate. It would be unwise to be lax about immunization requirements to discover one day that not enough of the population is immunized.
The real issue, I think, is that the small fraction of people who refuse to be immunized are shielded from infection by those who took the risk of immunization (albeit a small risk). But that is a can of worms, I don’t really want to open…