From the previous look at fatality rates a simple linear model shows that increasing use of transport modes that pose the least threat to others (namely walking and cycling) would increase the total road toll as they as so vulnerable to the disproportionately dangerous motorised modes. The limitations of a first order model make it fail to predict the obvious extremely low road toll if all motorised traffic were to be eliminated and only non motorised modes remained. Thus a second order model that includes the relative rates of modes interactions on each other is needed and presented here.

Data from the Australian Bureau of Infrastructure, Transport and Regional Economics (BITRE) breaks down the road toll into many dimensions, and along with their analysis they distribute a raw data set of all road fatalities from 1989 to the present. Categories of vehicles are able to be extracted from the data and its possible to break down the fatalities of each transport mode with the other vehicle type involved (although data is available, rail transport is not included in this analysis). Single vehicle accidents are also reported and including all of this data into a table we can form a fatality matrix *L* as such

To reduce noise in the data it was noted that the ratio of deaths in Victoria for each mode tracked ratiometrically with the national figures, so their fraction of the national figures were used to populate the table. Further the trends in fatalities have been clear so the data was further averaged by interpolating a linear fit across the January 2001 to July 2016 period of each value. BITRE also provides mode share data measured in passenger•km, this was compared to Census data for mode share and correlated well nationally, so calculated correction factors between the Census data national and Victorian mode shares were used to estimate the Victorian mode share from the national figure provided by BITRE. This then filled a table *D*, where *d _{i}* is the passenger•km share of each mode:

The simple weighting of each modes relative exposure to all modes of transport by their passenger•km share fails to capture the real world. Each mode has some degree of separation (segregation in transport parlance) from the others and will interact with its same mode more than with others, busses have bus lanes, trucks have truck routes, pedestrians and cyclists have exclusive paths/lanes. These reductions in exposure are symmetric and the table requires a symmetric matrix where each column m sums to the same as row *n*, and equals the the value *d _{i}*. To model the unbundling of transport modes a simple overall weighting was applied equally to all transport modes.

Here a single parameter λ is used to increase the self interaction share of each mode with its self while maintaining the required symmetry. A better description of λ can be found by expressing it as the ratio which each mode is exclusively interacting with its self *r*:

The excess mode share that is not exclusively with its own mode is then ratio metrically split by each modes passenger•km share. For comparative analysis a figure of 0.5 for *r* is used, although determining more accurate exposure ratios would be desirable no suitable data sets have been found. The intermodal shares are the hidden factor of the analysis. A use of mode share measured in time as passenger•hrs might be more appropriate for this weighting but walking and cycling are already well unbundled and would require individual corrections, the split on passenger•km distance already reduces their interaction shares with other modes and is a basis until more accurate data can be obtained.

From these computed relative exposures in passenger•km/year and the reported fatalities/year we can find a matrix *A*:

where:

Related by the hadamard product (elementwise multiplication) this new table *A* is the risk of death measured in fatalities/passenger•km commonly referred to simply as deaths per distance travelled. Contemporary analysis uses mixed figures such a fatalities per 100 million km, or sometimes billion km irksome and unnesseacry complications of the metric system adding multiple scaling factors when the succinct representation would be simply Tm (terrameters).

Once the fatality rate is computed for the current situation it is now possible to explore more accurately the effects of changes in mode share, changes in levels of unbundling, and changes in risk by intentionally changing *A* or *D* to see the effects on overall road toll (sum of *L*). This second order model correctly captures the expected low road tolls of corner cases such as exclusive mass transit or unmotorised transport and will be used in future posts to assess options for improving Victorias road toll.