The effect of recent criminal activity

We want to model the effect $B_ H(t)$ of recent criminal activity on house $H$. First of all, let’s assume that every time a house $H$ is burgled, $B_ H(t)$ increases by an amount $\theta .$ This increase isn’t going to last forever though: it’s going to diminish over time. Write $\omega $ for the time scale over which repeat burglaries are likely to occur in a neighbourhood.

Now write $\delta t$ for the length of a time step. So what we’re after is a formula for $B_ H(t + \delta t)$ in terms of $B_ H(t)$. The formula used by Bertozzi and her colleagues is

  \[ B_ H(t + \delta t) = B_ H(t)(1-\omega \delta t) + \theta E_ H(t), \]    

where $E_ H(t)$ is the number of burglaries that happened at house $H$ at the time $t$. You can see that each of these burglaries contributes an amount of $\theta .$ Burglaries that happened before that time are accounted for by $B_ H(t),$ but the amount by which they contribute is determined by the relationship of $\omega $ and the length of the time step. The larger $\omega \delta t,$ the smaller the contribution of $B_ H(t)$ (we assume that $\omega \delta t$ is less than $1$).

This formula only reflects how recent burglaries at house $H$ affect $B_ H(t).$ But we also need to take account of the effect of recent burglaries at neighbouring houses. To do this, replace $B_ H(t)$ in the equation above by

  \[ (1-\eta )B_ H(t) + \frac{\eta }{z} S_ B(t), \]    

where $\eta $ is a number between 0 and 1 that measures how significant the effect of crimes in adjacent houses is. The parameter $z$ is the number of houses that directly neighbour $H$ (so in a square lattice that number is four). $S_ B(t)$ is the sum of the $B_ J(t)$ for all the houses $J$ adjacent to $H.$ The formula now becomes

  \[ B_ H(t + \delta t) = \left( (1-\eta )B_ H(t) + \frac{\eta }{z} S_ B(t)\right)(1-\omega \delta t)+\theta E_ H(t). \]    

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