Science Spotlight

Proof: math explains seasonal vaccination efficiency

The effects of limiting the duration of an epidemic. We modeled epidemics with an R0 (average number of people that a typical infected individual infects in a fully susceptible population) of 2.0 with serial intervals of 5, 10, or 15 days. The models are initialized with a population of 1000 individuals, one of whom is initially infected. (a) The number of infected individuals over time with when the serial interval is 5 days (blue circles), 10 days (red triangles), and 15 days (purple xs). The epidemics are stopped at day 90, indicated by the vertical dashed line, to represent the end of an epidemic season. Note that the y-axis is the number of infected individuals and not infection incidence. (b) The attack rate of an epidemic vs R0. The black line shows the effect of increasing R0 on the infection attack rate when outbreaks are not interrupted using the final size expression. The same relationship is plotted when the epidemics are stopped on day 90 and the serial interval is 5 days (blue circles), 10 days (red triangles), and 15 days (purple xs). Other initial conditions were the same as in panel (a). (c) The effect of vaccine coverage on the attack rate. Initial conditions were the same as in panel (a), and vaccinated individuals are infected at rate 65% less than non-vaccinated susceptibles. The final size expression was used to compute the attack rate of uninterrupted outbreaks. The critical vaccination threshold is indicated by the vertical dashed gray line. (d) The efficiency of mass vaccination vs coverage. We define efficiency to be the number of infections averted per vaccination administered, which we would like to maximize. The horizontal dotted line represents the simplifying assumption that mass vaccination only reduces the risk of infection among vaccinated individuals (by 65%) and not among non-vaccinated individuals.
Image provided by Dr. Dennis Chao

Mathematical models have been used to understand disease outbreaks. They can provide insight into the relationship between disease transmission and the fraction of the population necessary to vaccinate to prevent outbreaks. One key aspect of disease outbreaks is seasonality. Transmission of many pathogens is seasonal and can be driven either by cyclical environmental forces such as rainfall or behavioral and demographic forces such as the beginning of the school year. When a pathogen’s transmissibility is not constant over time, modeling this relationship becomes more complicated.

To tackle this issue, Drs. Dennis Chao and Dobromir Dimitrov used mathematical modeling to explore how seasonality could impact the effectiveness and efficiency of mass vaccination. The results of their study were recently published in Mathematical Biosciences and Engineering.

The investigators used a Susceptible–Infectious–Recovered (SIR) model. This epidemiological model computes the theoretical number of people infected with a contagious disease in a closed population over time. The model involves a system of ordinary differential equations (an equation involving derivatives or the rate of change of a variable) relating the number of susceptible people, S(t), the number of people infected, I(t), and the number of people who have recovered, R(t); t represents that these components are time-varying. The investigators added an additional compartment to represent vaccinated individuals. In their model, when a sufficient fraction of a population is vaccinated, an outbreak cannot grow in size, called a critical vaccination threshold. In addition, they assume that protection conferred by vaccines is leaky, meaning vaccinated individuals have a reduced rate of infection but may be partially susceptible.  The pathogen’s transmissibility and serial interval (the average time between the onset of symptoms in an individual and the onset in secondary cases) can be used to compute the speed and magnitude of an epidemic.

In the model where epidemics do not depend on the season, the proportion of the population infected and the disease attack rate (or incidence) does not depend on their speed of transmission.  In this scenario, the attack rate is a concave-shaped function of the vaccination coverage up to the critical vaccination threshold, i.e., the attack rate of the epidemic drops more and more rapidly with increasing coverage until the critical vaccination threshold is reached.

Conversely, in the model where epidemics are seasonal, the speed of transmission can affect the relationship between transmissibility and the attack rate. Limiting the time during which a pathogen can be transmitted also affects the relationship between vaccination coverage and the size of the epidemic. Their analysis demonstrates that mass vaccination can become either more or less efficient as vaccination coverage increases, depending on the speed of transmission relative to the length of the epidemic season. Using mathematical modeling, they show that mass vaccination is most efficient, in terms of infections prevented per vaccine administered, at high levels of coverage for pathogens that have relatively long epidemic seasons, like influenza (sigmoid function of vaccination coverage), and at low levels of coverage for pathogens with short epidemic seasons, like dengue (convex function of vaccination coverage).

Dr. Chao elaborates on these findings, "mathematical modelers often assume that epidemics burn out when enough individuals are infected then become immune. This is true much of the time and produces a characteristic epidemic curve: the disease initially spreads exponentially quickly, slows down, peaks, then declines. We believe that some epidemics can end abruptly when their season ends. Many diseases occur during particular seasons of the year. Disease seasonality can be linked to human behavior, like increased influenza during the school year, or weather, such as waterborne disease during the rainy season. If a disease has a short season, the epidemic curve could be truncated. Maybe you would only have the initial exponential phase before the season is over and the epidemic stops abruptly. Vaccinating part of the population should slow down an epidemic, which both reduces and delays the epidemic peak. If that disease has a short season and you could vaccinate enough people to push the peak into the off-season, then you would prevent a disproportionate number of cases. If a disease has a long season, then it is harder to push the peak into the off-season and you would prevent fewer cases."

In summary, the length of a pathogen’s epidemic season may need to be considered when evaluating the costs and benefits of vaccination programs.

Funding for this study was provided by the National Institutes of Health and the Bill and Melinda Gates Foundation.

Chao DL, Dimitrov DT. 2016. Seasonality and the effectiveness of mass vaccination. Math Biosci Eng. 13(2): 249-59.


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