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A. Zechner, C. Urach:
"Tracing Herd Immunity after Vaccinations in SIS-Models using Cellular Automata";
Talk: MATHMOD 2012 - 7th Vienna Conference on Mathematical Modelling, Wien; 2012-02-14 - 2012-02-17; in: "Preprints Mathmod 2012 Vienna - Full Paper Volume", F. Breitenecker, I. Troch (ed.); Argesim / Asim, 38 (2012), 420 - 421.

Introduction. The aim of our work was to examine the effect of herd immunity in a model for pneumococcal
infections. By herd immunity we understand the effect that after the introduction of a vaccination the portion of
infected individuals decreases more strongly than just by the potion of vaccinated individuals.
This work was undertaken as an extension to the work of the cooperation between HVB (Hauptverband der österreichischen
Sozialversicherungsträger) and Vienna UT (see e.g. [3]) where the decrease of the infection rate under
the above conditions was modelled using differential equations and an agent-based approach. We were considering
a similar model using a cellular automaton (CA). Making use of this simplified model, we had the advantages of
easier implementation as well as lower computational complexity.
Implementation and Choice of Parameters. We implemented an FHP-lattice-gas CA in MATLAB (see [1] and
[2]). As we represented the hexagonal structure using a matrix, part of the symmetry was lost and we thus had to
distinguish between even and odd rows of the matrix. We described the state of the CA at any point in time with
an n n 6-matrix. During each time step the FHP-I-transition algorithm (see [2]) was applied.
The model had to be chosen in such a way that it could be used to describe the infection with Streptococcus
pneumoniae. For this purpose we used the SIS-model: Each particle of the CA represents a person which is either
susceptible, infected or vaccinated. After recovery infected individuals return to a susceptible state. By means of
the infection probability, immunisation rate and random matrices, we used the number of infected particles in a
cell at each time step to calculate the number of infected particles in the subsequent time step. Each time step was
presumed to represent a day and the length of infections to be normally distributed with mean 12 days.
Due to the lack of data for infection probabilities, in the initial step we set the rate of vaccination to be zero and
chose the infection probability such that the initially assumed proportion of infected individuals within the overall
population remained approximately stable over time. We computed the potion of infected individuals (0:1646) by
averaging the rates of infections for Streptococcus pneumoniae for the different age groups and the population data
for Austria as of 2007 (according to Statistik Austria, http://www.statistik.at/).
At the next step we introduced a vaccination rate of 0:1. This means that 10% of the population are vaccinated.
As we only simulated the CA over the duration of 1 to 2 years, we did not consider the expiration of the protection
provided by the vaccination or the influence of restoring immunity by re-vaccination. In order to circumvent effect
caused by the great fluctuations in the number of infected individuals in the beginning, we ran the CA for 200 time
steps without any vaccinations taking place and took the values thereby obtained as the starting point for the actual
simulation incorporating vaccinations.
Results. For a 300 300 6-CA which is (statistically) occupied with half of particles possible (which correspond
to individuals), an average duration of infection of 12 days and a running time of 500 days, we varied the
parameters r (infection rate) and v (proportion of vaccinated individuals within the overall population) and averaged
over 10 runs each: We used five different values for the infection rate r: r = 0.0385, 0.039, 0.0395, 0.04
and 0.0405. For all these values we achieved approximately the same portion of infected individuals within the
population. Once herd immunity occurred, the effects of different infection rates were different, though. For lower
infection rates, the portion of infected individuals decreased more strongly given equal vaccination rates. For the
vaccination rate v we used the four realistic values v = 0:05;0:1;0:15 and 0:2, as well as the value v = 0:9 which
would correspond to an almost complete immunisation of the population. Unsurprisingly, this value induced the
extinction of the infection. We learned that for a relatively low vaccination rate of 5% the portion of infected
individuals decreases by approximately 30%. For v = 0:2 that decrease amounted to even about 90%. However,
the effect of the herd immunity was proportionately diminished for larger vaccination rates.