In light of the new animal cases of H5N1 found in Hong Kong, where it was first discovered that the virus jumped barriers from birds to humans in 1997, let’s talk about how modelers describe epidemics of infectious disease, shall we?
First of all, there are a couple of reasons why mathematical models may be useful to epidemiologists.
1. For predictions. Being able to predict the future has always been one of the perpetual dreams of mankind. Thus, we have weather forecasts, economical forecasts, palm readings, fortune telling, astrology, etc. In medical science, some epidemiologists specialize in creating mathematical models to predict occurrences of new cases.
2. For understanding. Mathematical models are often a simplified version of the actual scenario. By removing factors that are less important, these models help investigators to understand the actual scenario, which are often rather complex to comprehend.
3. To anwer questions. Many mathematical models are being used to answer questions that cannot be answered in other ways. Often because the scenario that is modeled has not happened (like a disease outbreak), and we do not want to make it happen just so that we can do measurements, so investigators make use of mathematical models and change various factors on them in order to find out the effects of, say certain interventions, environmental factors, etc.
To begin, let’s start with some basic terms, shall we?
Basic reproductive rate, Ro.
What the heck is that?! Well, by definition, Ro is the average number of individuals being directly infected by an infectious case in a totally susceptible population (i.e., everyone in the population is susceptible to be infected).
Say, a chicken (whom we call it a primary case) is infected by chicken flu (the above definition is applicable to humans as well as to poultry). But the poultry workers do not know that it is sick and put it into a big cage with a whole bunch of other chickens, that are well, to get ready for sell. This primary case chicken has no face mask, cannot cover its mouth when it sneezes, and it is just basically being very, very sick. So it cannot help but sneezes at these other chickens that are living in the cage with him. Sneeezz-chu. Excuse me, he says. Despite apologizing, these sneezes nevertheless get other chickens next to him, say, two of them, infected. In this case, then Ro = 2.
Once an outbreak is underway, the reproductive rate is denoted by R, the net reproductive rate, which is the average number of individuals being directly infected by an infectious case in a partially susceptible population.
From this above reasoning, it can be deduced that the necessary condition for an epidemic is that R is greater than 1.
R > 1 => there will be an epidemic
R = 1 => the disease will become an endemic
R < 1 => the disease will eventually disappear
Now, what does Ro (or R) depends upon? According to literature, the basic formula for Ro is:
Ro = c * p * d, where
c = number of potential infectious contacts (per unit time)
p = transmission probability per contact
d = duration of infectiousness of an infected case
In this scenario, the c would be the number of chickens in the cage.
The p is different for difference diseases, and I do not have the number for chicken flu among chickens. However, take HIV infectious (among humans) as an example, the p is somewhere between 0.001 and 0.1 for sexual intercourse; but the p is virtually 1.0 for blood transfusion (Giesecke, 1995).
The d is a biological constant for any disease. For human flu, antibiotics can often shorten the d.
Implication
To lower Ro, we need to lower the c, p, or d.
For instance, if we totally isolate this speaking, yet very sick chicken, then c would become zero. If there is no other flu-infected chickens in the cage, nor are we putting more flu-infected chicken into the cage, then the disease will eventually die out as far as the chicken population within this cage is concerned. However, things will get a little more complicated if we have problem identifying which one is the sick chicken. (Opps!)
Vaccinating chickens essentially lower the proportion of susceptible chickens within the population.
Or else, having the sick chicken wearing face mask should reduce p.
Otherwise, d will be shortened by providing antiviral drugs to this sick chicken (if available).
Fun, huh? Obviously, I am only scratching the surface of the subject, but that’s one of the applications of mathematics in medical science, and that’s why we have the kind of public health measures that we have.
Reference:
Giesecke, Johan. Modern Infectious Disease Epidemiology. Oxford University Press, New York. 1995.