16042 Medley et al Supplmentary Information · 2015-12-02 · Microsoft Word - 16042 Medley et al...
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1 Visceral leishmaniasis control: healthseeking, diagnostics and transmission Graham F Medley 1 , T Déirdre Hollingsworth 2 , Piero L Olliaro 3,4 , Emily R Adams 2,5* Supplementary information 1. Healthcare seeking and diagnostic model The healthcare seeking model describes the pathways from onset of nonspecific symptoms to diagnosis. The full model and parameters are described by the flow diagram below (Figure 1) and the following equations. Fever, nonhealthseeking: !! ! !" = − ! − ! Fever, healthseeking: !! ! !" = ! − ! KalaAzar, nonhealthseeking: !! ! !" = ! − ! KalaAzar, healthseeking: !! ! !" = ! + ! − ! The model describes the behaviour from the point of onset of nonspecific fever, symptoms, and nonhealth seeking behaviour. From this point individuals can either seek health care or progress to KA. If they seek health care, they cannot be diagnosed until they progress to KA. If they progress to KA prior to seeking health care, they can only be diagnosed after seeking health care. The parameters in the model were estimated by fitting to the data in Figure 1a, which are the mean times to healthseeking and to diagnosis in patients who were subsequently diagnosed with KA. The progression from fever to KA, which occurs with rate a, is assumed to be a biological ‘constant’ across settings, whereas the other parameters were setting specific. We assume that c≥b, i.e. individuals with full KA are more likely to seek healthcare than those with nonspecific symptoms. There were no priors available for the parameters, even the progression from fever to full KA, and the individual data was not available. We therefore created a grid of all possible integer values for durations in each class (the reciprocal of the rate), and thus generated locationspecific combinations of parameters which produced results which were consistent with the data. On the basis of results we additionally presume that 1/c > 5d, i.e. that on average the time spent health seeking is less than 5 days if they have full KA, and that 33d < 1/a < 55d, i.e. that the average period of nonsymptomatic illness, before KA, lies between these limits. Note that this period has to be at least 14d based on the diagnostic criteria for KA. Using these criteria, there are 526 unique parameter combinations (a, b, c, d) for Bihar, and 2320 and 312 for Nepal and Bangladesh respectively based on integer day periods which are shown in Figure 2. In Bihar, the minimum and maximum proportions of individuals seeking healthcare before progressing to full KA (going to F h before K n ) are 76% and 87%; the min and max are 24%49% and 88%93% for Nepal and Bangladesh respectively. 2. Transmission dynamic model
Transcript of 16042 Medley et al Supplmentary Information · 2015-12-02 · Microsoft Word - 16042 Medley et al...
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Visceral leishmaniasis control: health-‐seeking, diagnostics and transmission
Graham F Medley1, T Déirdre Hollingsworth2, Piero L Olliaro3,4, Emily R Adams2,5*
Supplementary information
1. Health-‐care seeking and diagnostic model The health-‐care seeking model describes the pathways from onset of non-‐specific symptoms to diagnosis. The full model and parameters are described by the flow diagram below (Figure 1) and the following equations. Fever, non-‐health-‐seeking: !!!
!"= −𝑎𝐹! − 𝑏𝐹!
Fever, health-‐seeking: !!!!"
= 𝑏𝐹! − 𝑎𝐹!
Kala-‐Azar, non-‐health-‐seeking: !!!!"
= 𝑎𝐹! − 𝑐𝐾!
Kala-‐Azar, health-‐seeking: !!!!"
= 𝑎𝐹! + 𝑐𝐾! − 𝑑𝐾!
The model describes the behaviour from the point of onset of non-‐specific fever, symptoms, and non-‐health seeking behaviour. From this point individuals can either seek health care or progress to KA. If they seek health care, they cannot be diagnosed until they progress to KA. If they progress to KA prior to seeking health care, they can only be diagnosed after seeking health care. The parameters in the model were estimated by fitting to the data in Figure 1a, which are the mean times to health-‐seeking and to diagnosis in patients who were subsequently diagnosed with KA. The progression from fever to KA, which occurs with rate a, is assumed to be a biological ‘constant’ across settings, whereas the other parameters were setting specific. We assume that c ≥ b, i.e. individuals with full KA are more likely to seek health-‐care than those with non-‐specific symptoms. There were no priors available for the parameters, even the progression from fever to full KA, and the individual data was not available. We therefore created a grid of all possible integer values for durations in each class (the reciprocal of the rate), and thus generated location-‐specific combinations of parameters which produced results which were consistent with the data. On the basis of results we additionally presume that 1/c > 5d, i.e. that on average the time spent health-‐seeking is less than 5 days if they have full KA, and that 33d < 1/a < 55d, i.e. that the average period of non-‐symptomatic illness, before KA, lies between these limits. Note that this period has to be at least 14d based on the diagnostic criteria for KA. Using these criteria, there are 526 unique parameter combinations (a, b, c, d) for Bihar, and 2320 and 312 for Nepal and Bangladesh respectively based on integer day periods which are shown in Figure 2. In Bihar, the minimum and maximum proportions of individuals seeking health-‐care before progressing to full KA (going to Fh before Kn) are 76% and 87%; the min and max are 24%-‐49% and 88%-‐93% for Nepal and Bangladesh respectively. 2. Transmission dynamic model
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The transmission dynamic model was formulated by extending the health behaviour model to include the stages prior to and after the onset of symptoms, and by including the feedback from infections to transmission. Following exposure and latent infection, approximately 1-‐p=4% of individuals will progress to have symptomatic disease 11. These individuals follow the healthcare model described above. On diagnosis individuals are treated and are assumed to progress to a dormant state. This dormant state is a simplification of the dynamics of treatment success and failure and the potential development of PKDL, each of which are currently poorly quantified and outside the scope of the current model. The duration of the latent stage is unknown, and we have selected an average of 80 days based on being slightly longer than the duration between onset of symptoms and full KA and that for most infections the rate of progression quickens as the infection progresses. Anecdotally, in more seasonal environments (but with different Leishmania species), the peak incidence in cases lags the peak transmission season by 2 – 3 months. The relative infectiousness of the L stage is also unknown, and we have chosen a value suggesting that they are 1/1000 as infectious as KA patients. We expect that infectiousness will increase as disease progresses, but there is no data for humans. We have chosen the relative infectiousness of the dormant class on the basis of the dynamics of infection. For larger values of βD , the dynamics are essentially SI, with infection resulting in a carrier state. However, VL has been characterised as having intrinsic SIR type dynamics 22, which imposes an upper limit on infectiousness post-‐KA. Note that for both latent and dormant classes our presumed infectiousness parameters are averages, so that there might be a minority of PKDL patients (subsumed into our class D) who are highly infectious and a majority who are not infectious. Whilst such a distribution would impact on the dynamics we believe it is relatively unimportant in the short-‐term. The equations which describe the transmission dynamic model are as follows. Susceptible: !"
!"= 𝜇𝑁 − 𝜆𝑆
Latently infected: !"!"= 𝜆𝑆 − 𝜎 + 𝜇 𝐿
Fever, non-‐health-‐seeking: !!!!"
= 1 − 𝑝 𝜎𝐿 − 𝑎 + 𝑏 + 𝜇 𝐹!
Fever, health-‐seeking: !!!!"
= 𝑏𝐹! − 𝑎 + 𝛼 + 𝜇 𝐹!
Kala-‐azar, non-‐health-‐seeking: !!!!"
= 𝑎𝐹! − 𝑐 + 𝜇 𝐾!
Kala-‐azar, health-‐seeking: !!!!"
= 𝑎𝐹! + 𝑐𝐾! − 𝑑 + 𝜇 𝐾!
Dormant: !"!"= 𝑝𝜎𝐿 + 𝑑𝐾! + 𝛼𝐹! − 𝜇𝐷
Force of infection: 𝜆 = 𝛽! 𝛽!𝐿 + 𝛽! 𝐹! + 𝐹! + 𝐾! + 𝐾! + 𝛽!𝐷 /𝑁
Total population 𝑁 = 𝑆 + 𝐿 + 𝐹! + 𝐹! + 𝐾! + 𝐾! + 𝐷
The basic reproduction number and equilibrium state of this system can be found algebraically. The basic reproduction number is:
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𝑅! = 𝛽!𝛽!
𝜎 + 𝜇+ 1 − 𝑝
𝜎𝜎 + 𝜇
𝛽!𝑎 + 𝑏 + 𝜇
1 +𝑏
𝑎 + 𝛼 + 𝜇
+𝑎
𝑎 + 𝑏 + 𝜇1
𝑐 + 𝜇+
𝑎𝑎 + 𝑏 + 𝜇
𝑐𝑐 + 𝜇
+𝑏
𝑎 + 𝑏 + 𝜇𝑎
𝑎 + 𝛼 + 𝜇
1𝑑 + 𝜇
+ 𝛽!𝜎
𝜎 + 𝜇𝑝
+ 1 − 𝑝𝑎
𝑎 + 𝑏 + 𝜇𝑐
𝑐 + 𝜇𝑑
𝑑 + 𝜇
+𝑏
𝑎 + 𝑏 + 𝜇1
𝑎 + 𝛼 + 𝜇𝑎
𝑑𝑑 + 𝜇
+ 𝛼1𝜇
The parameter values are given in Table S1. Table S1 Parameter descriptions and values
Parameter Symbol
Description Source Value
a Rate of progression from F to K
Inferred from patient histories
constant across locations; the duration is >33 days, <55days, depending on parameter set
b Rate of progression Fn to Fh
Inferred from patient histories
location and parameter set specific
c Rate of progression Kn to Kh
Inferred from patient histories
location and parameter set specific, >1/5days
d Rate of diagnosis, progression from Kh to D
Inferred from patient histories
location and parameter set specific
μ Human life expectancy
Assumed 1/50yrs
N Population size Assumed 10,000
σ Rate of progression from Latent to symptoms (F)
Assumed, based on duration of fever and observed delays between sandfly activity and disease peaks in seasonal environments
1/80d
1-‐p Proportion of infections that result in KA
Literature 4% [Ref 13]
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βO Overall transmission coefficient
Assumed Either varied (Figure 3), or fixed at 1.0 per day (Figures 4, 5, 6)
βL Relative infectiousness of Latently infected (L)
Assumed, based on estimates of Stauch [6]
Either varied (Figure 3), or fixed at 0.001 (Figures 4, 5, 6)
βF Relative infectiousness of non-‐specific symptoms (F)
Assumed, based on estimates of Stauch [6]
Either varied (Figure 3), or fixed at 0.1 (Figures 4, 5, 6)
βD Relative infectiousness of Dormant infected (D)
Assumed, based on estimates of Stauch [6], and propensity of the model to reproduce ~15yr oscillations
Fixed at 0.000001 (see text and SI)
All model simulations are performed in Matlab (version 2015a).
