How does an eradicated disease come back to life? - Let’s break it down!

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As mentioned in a previous article, an important factor in population wide immunization is the concept of herd immunity. Herd immunity is, by Merriam-Webster, defined as “a reduction in the probability of infection that is held to apply to susceptible members of a population in which significant proportion of the individuals are immune because the chance of coming in contact with an infected individual is less”.  Although the concept of herd immunity is widely known and referred to, the exact meaning  of it is interpreted in different ways [7]. Some refer to it as the complete resistance of a population to a disease when a threshold is reached, whereas others have described it as a reduction in frequency of infection in susceptible individuals. In general, one can consider herd immunity as a concept describing the ability of a population of mainly immune individuals to protect susceptible ones, by limiting the ability of a pathogen to spread.

 

The theory has multiple aspects to it, defining the calculus behind disease transmissions. In 1906, Hamer introduced the theoretical basis of herd immunity in the context of measles dynamics [7]. He defined the ability to infect for each case of measles as a function of the number of susceptible individuals in a population.

 

The law of mass action is [7]:

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where C is the number of cases of measles, t is the first period and t + 1 the following period. S is the number of susceptible individuals in the population, and r is the transmission parameter describing the proportion of all contacts between infected and susceptible individuals that lead to new cases. One that will be different for different diseases.

 

Thus,            is the number of new cases of measles per case of measles, defined as

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The transmission parameter can therefore be defined as a ratio between the total number of cases at a given time and the product of the number of susceptible individuals and cases in the previous period. The period is the serial interval, aka the time between successive cases in a chain of transmission.

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This led to the threshold theorem: Incidence increases when                                     , and thus

 

only when              . Thus, the epidemic threshold          is defined by     , and thus if the number

 

of susceptible individuals is kept below this value, incidents will drop.

The herd immunity threshold, aka the proportion of a population that needs to be immune for herd immunity to take place can thus be described as [7]:

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Although this is a very simple model, not taking into consideration many factors that may affect the ability to infect, it does show the kind of model that is used to describe transmission models for diseases. I urge you to read more about the different models in the review article [7].