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The Simulation of Epidemics by Two Dimensional Model
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KMID : 0353019750120020307
Abstract
This study was performed to find the epidemic pattern dependent the infective location and various infectious probabilities. The author described one of the simplest possible models for a two dimensional epidemic and presented a number of results obtained by straight-forward Monte Carlo simulations carried out on an electronic computer (CYBER 73-18).
The author assumed that community size was finite and divided into (2k+1)©÷. Each unit was occupied by a susceptible individual. The author supposed that an epidemic was initiated by susceptible individual at center unit becoming infected. The model of chain-binomial type was adopted. There were two forms, one was a simple epidemic with no recovery and the other was a general epidemic with infectives undergoing removal. There was infectious probability p of any susceptible at risk becoming infective. A given susceptible may be exposed to several infectives (r). The maximum r was different depending on its position, that is, in the internal part the maximum r is 8, if susceptible was on boundary r is 5, and in the corner r is 3 (Fig.1). The probability of the susceptible becoming infected was 1-(1-p)r.
The general scheme of the computation might be briefly outlined as follows. An array Aij, with i,j=-k, -k+1, ...., 0, ....k-1, k was used to prosent the individual. If the individual at the (i,j) was susceptible, Aij=0 and if infectious, Aij=1. At time g all parts of array that could be reached by the epidemic the square formed by the lines x=k-1¡¾g, y=k+1¡¾g were inspected. If Aij=1, There was no change. But if Aij=0, all eight (or 5,or 3) must be examined and summed up by the number of adjacent infectives. A probability and pseudo-random number was calculated. If probability was larger than pseudo-random number the status of Aij was changed. The auther proceeded in this way until there were no susceptibles or under certain conditions, that is, density of susceptibles was satisfied by the Kermack-McKendrick Theorem. In the investigations whose results were described below, 20 seperate simulations were performed for each epidemic set up with a different value of p. Results were shown for the cases k=5-an 11¡¿11 square-in tables 1, 2, 3, 4 and 5 with p=0.2, 0.4, 0.6 and 0.8 respectively.
The results in simple epidemics were as followings:
1) As the probability increased the epidemic duration period was decreased.
2) As the probability increased, the number of new infectives per unit time was increased.
3) When p was small the curve was relatively flatter.
The results in general epidemics are as follows: This results are grouped into three patterns. The first pattern shows that infection which has a longer infectious period has larger epidemic size and longer epidemic duration time (Fig.3). The second pattern shows that infection having longer infectious period has larger epidemic size but its completion time is shorter (Fig.4). In the third pattern the infection which has longer infectious period presented larger epidemic size, but its completion time does not depent on their infectious period (Fig.5).
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