Sensitivity Analysis (taken from Capurro et al 1998)

We conducted a series of simulations for a system with two types of neighbourhoods. We assumed a population of 12,000,000 inhabitants, similar to the population in the city of Buenos Aires,  and assumed that 30% of the individuals were members of the group of low-income population, that frequently take buses, riding the buses for a long time. The remaining 60% were of the medium-to-high-income population. This population takes the bus less frequently as it has access to other means of transportation (for example their own car, train or metro). If they take buses, their trips are shorter.

For the simulations, we used values of parameters which are most frequently found in the literature (Table S1). We made a sensitivity analysis for a set of them regarding the time spent in a bus and to the number of contacts that can favour the transmission of the disease.

We assumed that the mortality rate due to the disease was equal to zero.

Methodology

We analyzed the combined effect of a series of parameters, using a factorial design. For this type of analysis, the necessary amount of runs to study the effect of k parameters for two levels of each parameter is equal to  2^k. The number of runs increases exponentially with the number of parameters included in the analysis. Many of these runs are necessary to estimate higher levels interactions. These interactions, for high values of k, are negligible compared to main effects or the double or triple interactions (Box et al., 1978). The fractioned factorial analysis proposes to run only a fraction of all the possible runs, since the main effects would anyway be maintained (Box et al., 1978). Rose (1983) proved that this alternative is superior to other ways of estimating sensitivity.

We used a fractioned factorial design of resolution 4, resulting from analysing two levels of factors for 7 parameters (6 degrees of freedom). We made 2^(6-2) = 16 runs. The levels used correspond to variations in ± 25% of the parameter values. The levels corresponding to each one of the parameters are shown in Table S2, + or - (+25% or -25%, respectively).

The parameters that we tested were  ß, r₁, r₂, a₁, a₂, b₁ and b₂, with respect to the variables that represent the number of latent and infected individuals of the communicable and non-communicable groups of the two socio-economic levels. In the case of r₁and r₂, the value that was changed was the time of travel, and based on this change we calculated the corresponding r_i  and s_i . We analysed the sensitivity in 1- and 10-year runs, with a pre-iteration of 100 days to reduce the effect of initial conditions. In Table S3, we show the corresponding values for each one of the parameters for the different runs.

The effect of the variation of the parameters was measured over 8 output variables:

E1: non-communicable latent individuals of group 1 (lower socio-economic level);
I1: non-communicable infected individuals of group 1;
X1: communicable latent individuals of group 1;
Y1: communicable infected individuals of group 1;
E2: non-communicable latent individuals of group 2 (higher socio-economic level);
I2: non-communicable infected individuals of group 2;
X2: communicable latent individuals of group 2;
Y2: communicable infected individuals of group 2;

The effect was calculated following Box et al. (1978), that is, using the formula:
             n
          S  VO_i x LF
              i
EP =  -----------
            VON

Where:
EP: effect of the parameter
VO_i: value of the output variable in run
VON: value of the output variable when the parameters havenot been altered
LF: level of the factor (+1 or -1) .

Results

In Table S4, we show the effect of variations on each one of the parameters over the different variables mentioned above as calculated using the formula described in the previous section, for 1-year runs.

In Table S5, we show the results of the calculation of the effect on the corresponding parameters, for 10-year runs.

Analysis of the results

1-year runs:

For 1-year runs, ß is the parameter with the most important effect in all cases, except for communicable individuals of group 1 (lower socio-economic level). For these individuals, b₁ is the parameter to which the variables are more sensitive. Following in importance are ß  and a₁; r₂ is in fourth place and r₁ in sixth, with much lower values than the previous parameters. For non-communicable individuals of group 1, after  ß are b₁, a₁,; r ₁ is in fourth place, but r₂ is last.

For non-communicable individuals of group 2 (higher socio-economic level), after ß  are b_1, b₂ and a₂. Only after them are r₁ and r₂, alternating places for latent and infected. For communicable individuals of group 2,  ß is still the most important parameter, followed by b1 and b2; r₂ and r₁ are in fourth and fifth place.

10-year runs:

The first thing we observed was that, for group 1, the effect of the parameters was not so important, whereas for group 2, the effect was more notorious. This shows that group 1 has already reached an equilibrium 10 years after infection has been introduced. On the other hand, group 2 has not yet reached that state.

For these runs, ß is also the parameter with greatest sensitivity in all cases, except for communicable individuals of group 1, where b₁ is the most important parameter, and the parameters that follow in importance are in the same order than for 1-year runs, i.e., ß  and a₁;  r₂ is fourth and r₁ is sixth. In the case of non-bus takers individuals of the same group, ß is followed by a₁ and b₁ ; r₁ and r₂ are fourth and last in importance, respectively.

For non-bus takers individuals in group 2, ß is the most important parameter, followed by a₂ , b₁ and b₂; r₂ and r₁ are fifth and seventh, respectively. For bus-takers individuals of the same group, we found the same order of importance for the parameters, except for b₂ and a₂, which exchange places.

Discussion

In 1-year runs, the parameters with significant sensitivity for the communicable and non-communicable infected individuals of the different socio-economic groups are the following, in order, the following:
 

I1                         Y1                          I2                                          Y2
 ß, b₁ and a₁        b₁ , ß and a₁          ß, b₁, b₂ , a₂ , r₂  and r₁           ß, b₁ ,b₂  , r₂ , r₁  and a₂

In almost all cases, and as expected,  ß is the parameter with greatest effect on the infection of the population, since the transmission per contact is a factor present in all cases of infection. Regarding the remaining parameters, for the group with lower socio-economic level (group 1), the number of contacts of bus-takers ( b₁ ) and non-bus-takers (a₁ )individuals in the same group is more important; whereas for the group with higher socio-economic level (group 2), it is the number of contacts of the bus-takers individuals in both groups that is more important ( b₁and b₂), as well as the rates of being on the bus (indicator of time of travel) combined with these. The effect of the time of travel of group 2 is more notorious over the infectious individuals of the same group, than the time of travel of group 1. The number of contacts of non-bus-takers individuals of group 2 (a₂ ) has an important effect on the number of non-bus-takers infectious of group 2; not as much on the bus-takers infectious of the same group.

Therefore, we can conclude that the time of travel of all the population has mainly an effect in the infection of individuals of higher socio-economic level, associated to the number of contacts of the bus-takers individuals of both groups. We cannot see a significant effect on the individuals of lower socio-economic level.

In the 10-year runs, we observe the following significant effects of the parameters on the different variables:

I1                        Y1                        I2                        Y2
 ß, a₁ and  b₁      b₁ , ß and a₁        ß, a₂ , b_1, b₂         ß, b₂ ,b₁    and a₂

In general, ß is still the parameter with greater significance for all variables. As was previously mentioned, we observe that while infection in group 1 would be near an equilibrium, this does not occur in group 2, where the effects of the parameters are more significant. In any way, we can see a very similar influence to the one observed in 1-year runs for group 1. In the group 2, nevertheless, already the effect of the time of travel is not so significant. On the other hand, the number of contacts of non-bus-takers individuals of group 2 has a greater effect on the infection of non-bus-takers individuals of this group. For bus-takers individuals of the same group, the number of contacts of bus-takers individuals has a greater effect.

Conclusions

In 10-year runs, the group with higher socio-economic level has not yet reached and equilibrium. We can expect that in the same period of time, the dynamics of public transportation in the city suffer some changes, that might impact on the time of travel. The importance of these changes is the continued variation and, consequently, no possibility of reaching and endemic equilibrium of tuberculosis. Our results, may change significantly in centres like Mexico City who are experiencing significant demographic growth, particularly in the lower socio-economic levels (high migration to urban centres).
 

Box , G.E.P., Hunter, W.G., Hunter, J.S. (1978).b Statistics for experimenters : an introduction to design, data analysis, and model building. J. Wiley & Sons, NY.

Capurro, A F, Zellner, M L, Castillo-Chavez C . Public Transportation and the Transmission of air-borne communicable diseases. Documneto de trabajo N° 20 - Departamento de investigaciones Universidad de Belgrano -Buenos Aires, Argentina(1998)

Rose, K. A. (1983). A simulation comparison and evaluation of parameter sensitivity methods applicable to large models. In : Lauenroth, W.K, Skogerboe, G.V. , Flug, M. (Eds.). Analysis of Ecological Systems : State-Of-The-Arts in Ecological Modelling. Elsevier, Amsterdam pp129-140.