# Mathematical modeling of infectious childhood diseases

Authors: J. Zibolenová 1;  D. Ševčovič 2;  T. Baška 1;  D. Rošková 1;  E. Malobická 1;  V. Szabóová 1;  V. Švihrová 1;  H. Hudečková 1
Authors‘ workplace: Ústav verejného zdravotníctva, Jesseniova lekárska fakulta v Martine, Univerzita Komenského v Bratislave, vedúca prof. MUDr. H. Hudečková, PhD., MPH 1;  Katedra aplikovanej matematiky a štatistiky, Fakulta matematiky, fyziky a informatiky, Univerzita Komenského, Bratislava, vedúci prof. RNDr. D. Ševčovič, CSc. 2
Published in: Čes-slov Pediat 2015; 70 (4): 210-214.
Category: Original Papers

## Overview

The article deals with mathematical modeling in epidemiology and analyses principles of deterministic SIR model (susceptible – infected – resistant) which is used particularly to describe spread of infectious childhood diseases and represents a concept of herd immunity in association with a basic reproduction number and vaccination.

Using the open SIR model, we can explain basic features of spread of diseases causing permanent immunity such as mumps, varicella, measles and rubella.

Development of proportions of susceptible, infected and resistant individuals in non-vaccinated population shows a character of damped oscillations. Oscillations of proportions of individual groups are mutually interconnected. As an effect of a birth rate, the number of susceptible individuals increases up to a critical level, when the epidemic outbreak emerges followed by increase of proportion of infected individuals. This leads to a dramatic decrease of proportion of susceptible individuals resulting in deceleration of spread of the infection.

The vaccination substantially influences occurrence of the disease. If the vaccination rate is below of a threshold of the herd immunity, spread of infection is limited. However, mean values of proportions of susceptible individuals are not significantly changed. If the vaccination rate exceeds the level needed for the herd immunity, according to the model, spread of the infection is halted and the proportion of susceptible persons continuously decreases.

In Slovakia, within the above mentioned diseases, mass vaccination against measles, rubella and mumps is provided. Situation regarding occurrence of these disease is relatively favorable and except minor local outbreaks of mumps they almost do not occur in Slovakia. However, we should take into account possible changes of epidemiological situation, particularly considering increase of antivaccination activities. As seen in our contribution, accumulation of susceptible individuals above certain level constitutes a risk of reemerging of epidemic outbreaks.

Key words:
mathematical modeling, infectious childhood diseases, basic reproduction number, vaccination

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##### Labels
Neonatology Paediatrics General practitioner for children and adolescents

Article was published in

### Czech-Slovak Pediatrics

2015 Issue 4

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