We want to model the effect of recent criminal activity on house . First of all, let’s assume that every time a house is burgled, increases by an amount This increase isn’t going to last forever though: it’s going to diminish over time. Write for the time scale over which repeat burglaries are likely to occur in a neighbourhood.
Now write for the length of a time step. So what we’re after is a formula for in terms of . The formula used by Bertozzi and her colleagues is
where is the number of burglaries that happened at house at the time . You can see that each of these burglaries contributes an amount of Burglaries that happened before that time are accounted for by but the amount by which they contribute is determined by the relationship of and the length of the time step. The larger the smaller the contribution of (we assume that is less than ).
This formula only reflects how recent burglaries at house affect But we also need to take account of the effect of recent burglaries at neighbouring houses. To do this, replace in the equation above by
where is a number between 0 and 1 that measures how significant the effect of crimes in adjacent houses is. The parameter is the number of houses that directly neighbour (so in a square lattice that number is four). is the sum of the for all the houses adjacent to The formula now becomes