We have presented next-generation matrices for both T. b. gambiense and T. b. rhodesiense, and global sensitivity analyses of these models using R0 and a binary form of R0 (a value of 1 if R0 is greater than 1 and 0 otherwise) as the model outputs of interest. The global sensitivity analyses have suggested that the parameters most likely to explain the distribution of West African sleeping sickness are the proportion of bloodmeals taken from humans and, perhaps more intriguingly, the proportion of tsetse refractory to infection with the parasites. The importance of the proportion of bloodmeals taken from humans could be explained by transmission from vertebrate host to insect vector and back to vertebrate host representing completion of the parasite’s life cycle. That is, for a loop of the infection graph to be completed (there are only 2 loops in the graph and both require this) a tsetse fly must take a first bloodmeal from a human (in order to become infected), and then another feed from a human to complete the chain of transmission. The probability of a tsetse fly taking 2 or more bloodmeals from humans is determined directly by the proportion of bloodmeals taken from humans. For East African sleeping sickness this is not highly relevant because humans are more of an accidental host than a reservoir. The most important parameters for East African sleeping sickness were instead the proportion of flies refractory, followed by population parameters determining the abundance, composition and life span of tsetse flies.
Global sensitivity analysis of a model for R0
is an approach to sensitivity that is distinct from the perturbation methods used by Rogers (1988)
or the plots of the effective reproduction number used by Welburn et al. (2001)
. Both of these approaches take fixed-point estimates of all parameters except the one of immediate interest and quantify the behaviour of the model around that point. Such methods inevitably measure the local behaviour of the model around a single point in parameter space. Global approaches avoid this pitfall (Saltelli, 2002
) and require at the very least that a range of values for each parameter be specified, which can be a useful exercise in its own right.
The application of the next-generation matrix method to sleeping sickness adds to a growing list of multi-host, vector-borne pathogens for which next-generation matrices have now been applied (Roberts and Heesterbeek, 2003
; Hartemink et al. 2008
; Matser et al. 2009
). We point out though, that in the case of sleeping sickness the mechanics of transmission are simple enough for types at infection to be defined by host species. This means that a reproduction number for sleeping sickness can be obtained relatively easily, as was done by Rogers (1988)
who derived equations for the average number of infectious tsetse flies produced from 1 infected tsetse fly. This is not precisely the same as the R0
given here – the expression from Rogers (1988)
does not average over all host types – but it is closely related (one need only take the square root) and has the desired threshold properties. This arguably more direct approach works because the parasite is obligated to complete its life cycle by transmitting from tsetse fly to vertebrate host and back again (Roberts and Heesterbeek, 2003
). Further expressions for R0
based on the work of Rogers (1988)
were proposed by Welburn and others (Welburn et al. 2001
) to compare the efficacy of chemoprophylaxis of domestic livestock, vector control and treatment of humans.
For sleeping sickness, then, a next-generation matrix is not strictly necessary
to calculate an R0
, as it is for tick-borne diseases for example (Hartemink et al. 2008
values calculated from next-generation matrices do, however, have the property that regardless of the number of vector or host species they always have the same interpretation of per generation growth in infected, they are always calculated in the same way and they always gives values
consistent with the single-host definition of R0
(Diekmann and Heesterbeek, 2000
). An added advantage is that the type reproduction number, which is a measure of the control effort required to eliminate a pathogen when a single host species is targeted by the control, quickly follows (Roberts and Heesterbeek, 2003
), as do the elasticity and sensitivity matrices (Matser et al. 2009
). In the case of sleeping sickness, an elasticity matrix would indicate (simply by summing its rows or columns) the contributions to R0
from each Glossina
species and for each vertebrate host type. This approach may be attractive if, for example, there is strong interest in quantifying the role of a particular host type because it is of growing importance in agriculture, or a tsetse vector species that is reinvading a region.
The use of global sensitivity indices is a relatively novel approach for models of infectious disease and we would always argue that using the literature to determine a range of values for a parameter is more appropriate than attempting to arrive at single point values. However, one limitation is that the range of values represents 2 types of variation; uncertainty about the parameter due to little or no information available and natural geographical variation arising from real differences in host and vector populations, climate, or the parasite strain. Ideally, one would like to separate these two types of variation, particularly when asking which parameters might be important for predicting the geographical distributions of a pathogen.
Care must be taken too when interpreting Sobol’s indices, or interpreting rankings of biological parameters based on these indices, to comment on the likely efficacy of control methods that target different biological parameters. For example, we note that the main and total effects of the infectious period of humans are, in the case of West African sleeping sickness rather low. Yet early detection and treatment of West African sleeping sickness, which targets this parameter, came close to eliminating the disease during the decades that these programmes ran. A dramatic reduction in the infectious period in the single vertebrate host population, to values outside the range of values considered in the global analysis, would logically have a large impact on R0. However, Sobol’s indices do not reflect this. Hence, low values of Sobol’s indices hence do not necessarily imply that a control strategy that targets that parameter should be dismissed because so much depends on the ease with which change to a parameter can be affected. In the case of early detection and treatment of West African sleeping sickness, there are also immediate benefits to patients and the wider community, which are quite separate from the prevention of future transmission.
values for T. b. gambiense
and T. b. rhodesiense
indicate that T. b. gambiense
is expected to spread for only a very small part of the parameter space, whereas for T. b. rhodesiense
, the median value of R0
is greater than 1. This result somewhat contradicts the magnitude of the public health problem posed by T. b. gambiense
in comparison to the public health problem posed by T. b. rhodesiense
, as 90% of sleeping sickness cases reported are caused by T. b. gambiense
. The model may underestimate the amount of transmission between hosts and vectors for T. b. gambiense
, or overestimate this for T. b. rhodesiense
. The treatment of livestock with trypanocidal drugs, which is incorporated into the model for T. b. rhodesiense
, is likely an important part of controlling East African sleeping sickness (Welburn et al. 2006
; Fèvre et al. 2006b
). However, we note too that humans, wildlife and livestock are not as mixed as the model assumes. That is, the opportunities for tsetse flies to take feeds from both
wildlife and humans may be relatively rare in present day Uganda, which would also explain why human cases of East African sleeping sickness are less common than our results might suggest.
Recent experimental work has shown that starvation (nutritional stress) affects the susceptibility of tsetse flies to trypanosome infection (Kubi et al
. 2006), and this is true for teneral and non-teneral flies (Akoda et al. 2009a
), as well as for the offspring of nutritionally stressed female tsetse (Akoda et al. 2009b
). The results of the global sensitivity analysis effectively underline the possible implications of these findings for the epidemiology of human Trypanosomiasis, because the susceptibility of tsetse was consistently identified as an important factor for R0
. That is, periods of nutritional stress that increase susceptibility of flies above what is normally reported would be predicted to have a large effect on R0
. These same results also give broad support for control strategies aimed at further increasing refractoriness in flies (Durvasula et al. 1997
; Rio et al. 2004
), suggesting that changes to this parameter may in fact be enough to bring R0
below 1. Although such methods may be a long way from being applied in the field, they are attractive to consider because the statement applies to both East and West African sleeping sickness. Finally, we note that flies carrying mature infections (infective to humans) have been found to have reduced survival such that infectivity to humans comes at a high cost to the parasite (Welburn and Maudlin, 1999
). Such a relationship between infectivity and maturation exacerbates the already narrow window of infectiousness that is a weakness in the transmission cycle. This may explain the moderate sensitivity values in the model results for mean survival of tsetse. These results suggest that sustained tsetse control (that lowers the mean survival of flies over a period of years) should have strong effects on sleeping sickness. Lowering the mean survival narrows the window of opportunity for infectious flies and decreases the likelihood that a fly will live long enough to become infectious at all. It also does not require that tsetse be eliminated altogether, effectively sidestepping the problems associated with achieving eradication.