15.6: Analysis of the I model

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With \(v\) as the prevalence of vaccination in a population, what will the I model reveal about vaccination? In an uninfected population the prevalence of infection will be zero \((p\,=\,0)\), so the number of new infections produced per infected individual per time unit will be \(\beta\,(1\,−\,v)\).

On average, a fraction \(\alpha\) will die per time unit, so the average duration of infection will be 1/\(\alpha\), assuming complete randomness. If 1/10 die per year, for example, the average duration of infection will be 10 years.

This makes \(R_0\,(v)\,=\,\beta\,(1\,−\,v)\,\times\,(1/\alpha)\,=\,(\beta/\alpha)\,(1\,−\,v)\). And the disease will decline to extinction if \(R_0\,(v)\,\lt\,1\) —that is, if \(R_0\,(v)\,=\,(\beta/\alpha)\,(1\,−\,v)\,\lt\,1\), which you can work out algebraically in a few steps to \(v\,\gt\,1\,−\,\alpha/\beta\).

Look what this means. A disease that infects 4 individuals per year in a totally uninfected population \(\beta\,=\,4), and which remains infectious for one year \(1/\alpha\,=\,1), will decline to extinction if \(v\,\gt\,1\,−\,1/4\,=\,3/4\). If only slightly more than 3/4 of the population is vaccinated, that disease will eventually vanish. Remarkably, a disease can be eradicated even if the whole population cannot be vaccinated! Largely because of this, society can develop programs striving toward the conquest of disease.

What does the \(I\) model reveal about the evolution of infectious disease? The pathogen has many more generations and can therefore evolve biologically more rapidly than the host, and \(\alpha\) and \(\beta\) can evolve to benefit the pathogen.

Because \(\beta\) enters the equation with a plus sign and \(\alpha\) enters with a minus sign, the disease will spread more rapidly— \((1/p)\,(dp/dt)\) will be larger—if \(\beta\) increases and \(alpha\) decreases.

This means that, genetics permitting, a successful disease operating according to this or any similar model will tend to become more infectious (higher \(\beta\)) and less virulent (lower \(alpha\)) over time. At the limit, nearly everybody will be infected but the effect on anybody will be minimal. Polio in humans before the 20^th century, and SIV in monkeys, are examples.

In the ultimate limit, a disease could evolve to negative virulence—that is, to be a mutualism with the host. Rhizobial bacterial in legumes may be an example.

As usual, there are refinements to this idea, in part because infectivity and virulence are not independent. Diseases that evolve to be more infectious may have to use more of their victims’ metabolic resources, and consequently may become more virulent in the process. Again, such refinements can be addressed in more specific models.