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Equations: Road Traffic Accidents

In forecasting mortality related to road traffic accidents, IFs replaces the GBD regression model with a structural formulation designed to better capture relevant drivers for this cause group.  Specifically, IFs projects deaths due to traffic accidents (DEATHCAT, Traffic) as a function of deaths in traffic per vehicle (DEATHTRPV) and vehicle numbers (VEHICLESTOT), both computed in the automobile module of IFs.   We first need to compute the total size of the vehicle fleet.

Total vehicles per capita (VEHICFLPC) is based on a formula proposed in a paper by Dargay et al (2007) in which fleet size per capita is a function of GDP per capita at PPP (GDPPCP).  Translating the Dargay et al (2007) equation into one using IFs variable names yields: [1]

traffic accidents equation 1

The parameter vehicfpcm allows scenario intervention. RF is an adjustment factor that compensates for different land densities, that is the ratio of population (POP) to land area (LANDAREA), taking the U.S. as the base: 

traffic accidents equation 2

The computation was only used when country R had higher density than the US. The paper also describes another adjustment factor related to urbanization as percentage of total population, but we did not use this additional adjustment factor in our model.

Given fleet size per capita and the population, we compute the total size of the fleet.

traffic accidents equation 3

The number of deaths per vehicle is based on Smeed’s Law [2] , an empirical rule originally proposed by R.J. Smeed, which relates deaths to vehicle ownership.  In the original conceptual form Smeed’s Law is:

traffic accident equation 4

D is annual road deaths

n is number of vehicles

p is population

In terms of IFs variable names this would translate literally (ignoring some unit issues) as:

traffic accidents equation 5

The actual representation in IFs involves two steps.  First we calculate the death rates per vehicle, adding a division by a multiplicative term that is equivalent to total vehicle numbers VEHICLESTOT.  One of the virtues of this first step is that we can add an exogenous multiplier for death rates per vehicle, deathtrpvm .

traffic accidents equation 6

The second step is to use the death rate per vehicle, the vehicle fleet size per capita, and information on the age and sex distribution of deaths from vehicles to compute the mortality rate from vehicle accidents by age and sex, putting the results into a variable internal to model named modmordstdet.  In a third step that variable is used with population by age and sex to compute the total deaths from vehicle accidents (DEATHCAT).  These second and third steps stylistically yield

traffic accidents equation 7

After initialization in the base year (using GBD estimates of road traffic-related mortality and total vehicles from the automobile module in IFs), IFs calculates a multiplicative shift factor that is kept constant for the entire forecast horizon. If this initialization value is greater than 40 deaths per 1000 vehicles, we adjust the number of vehicles per capita to set 40 as our initialization value. We started using this limit after finding inconsistencies between estimates derived from Smeed’s Law and those from initial estimates. [3]

IFs also computes a ratio (in a variable internal to the model) of traffic accident mortality for males compared to females.   The model converges that ratio to 1.5 over 100 years by preserving the total mortality for each age category but adjusting the distribution between males and females.


[1] Dargay, Gately, and Sommer 2007. “Vehicle Ownership and Income Growth, Worldwide: 1960-2030”. Joyce Dargay, Dermot Gately and Martin Sommer, January 2007.


Smeed, RJ 1949. "Some statistical aspects of road safety research".  Royal Statistical Society , Journal (A) CXII (Part I, series 4). 1-24.

Adams 1987. "Smeed's Law: some further thoughts." Traffic Engineering and Control (Feb) 70-73

[3] The case of Bangladesh is illustrative, where the forecast calculation of 141 deaths/thousand vehicles contrasts with an expectation of 30 deaths/thousand vehicles  using Smeed’s Law.  We concluded that our mortality figures were consistent with WHO estimates, but sometimes the total number of vehicles was too low.  For example, for Bangladesh our data showed 1 vehicle per thousand people, which meant about 141,000 vehicles, when several reports indicate the real number is much higher (850,000) (