Modeling of the effect of pneumococcal conjugate
vaccination on carriage and transmission of
Streptococcus pneumoniae in Kenyan children
John Ojal
KEMRI-Wellcome Trust Research Programme
Background
• Conjugate pneumococcal vaccines have been
shown to protect very young vaccinated
children and older unvaccinated children.
• A pneumococcal conjugate vaccine (PCV) has
recently been introduced into the Kenyan
immunization schedule.
• However, we do not know the long term
population effects of these vaccines in a
setting like Kenya
Strathmore University International Mathematics Research Meeting
Objective
The study aims to model the impact of PCV on
pneumococcal transmission and disease using
mathematical modelling frameworks.
Strathmore University International Mathematics Research Meeting
Unvaccinated
dSi  t 
dt
Vaccinated
 rVi *Vi  t   rNsi * N si  t   rNwi * N wi  t   Si  t  *  Vi  t   Nsi  t   Nwi  t    i  a, t  * Si  t    * Si
Strathmore University International Mathematics Research Meeting
v
t   b * P t   i * Si t 
Methods: Model calibration
• The model will be fitted to pre-vaccination
carriage prevalence data to estimate some
parameters.
• The waning rate of vaccine induced protection
against carriage and the vaccine efficacy
against carriage will act as control parameters
• Clearance rates to be obtained from studies
investigating the nasopharyngeal carriage and
clearance rates among Kenyan study subjects
Strathmore University International Mathematics Research Meeting
Methods: Likelihood
• Denote the number of individuals in the ith
age group and jth pre-vaccination carriage
status in the empirical calibration data by
xij , i  1, 2,..., k  6, j  1, 2,..., z  7
• Denote the vectors containing the model
output of the serotype distribution in the ith
age group by pi     pi1 , pi 2 ,..., piz 
Strathmore University International Mathematics Research Meeting
Methods: Likelihood
• The counts, xij , in the ith age group will follow
a multinomial distribution with the loglikelihood function given by:
 
l  xi , j 1,2,.., z | pi      log p
z
j 1
xij
ij
(4)
• The combined log-likelihood for all the age
groups is then given by:
 log  p 
k
z
i 1 j 1
xij
ij
(5)
Strathmore University International Mathematics Research Meeting
Methods: Estimation
• Given any set of parameters the model
generates the steady-state serotype
distribution.
• The likelihood for any given set of parameters
is can then be computed using equation 5.
• The likelihoods are used within a MetropolisHastings MCMC algorithm to estimate the
best set of parameters that fit the observed
data.
Strathmore University International Mathematics Research Meeting
 
Strathmore University International Mathematics Research Meeting
dSi  t 
dt
dN si  t 
dt
Model equations
 rVi *Vi  t   rNsi * N si  t   rNwi * N wi  t   Si  t  *  Vi  t   Nsi  t   Nwi  t    i  a, t  * Si  t    * Si
v
t   b * P t   i * Si t 
 rNwi * N swi  t   rVi * Bsi  t   rNsi * N si  t   Nsi  t  * Si  t   N si  t  *  cv * Vi  t   cw * Nwi  t    i  a, t  * N si t    * N si  t   i * N si t 
dVi  t 
v
 rNsi * Bsi  t   rNwi * Bwi  t   rVi *Vi  t   Vi  t  * Si  t   Vi  t  *  cN * Nsi  t   cN * Nwi  t    i  a, t  *Vi t    *Vi 
dt
dN wi  t 
dt
dN swi  t 
dt
dBsi  t 
v
t   i *Vi t 
 rNsi * N swi  t   rVi * Bwi  t   rNwi N wi  t   Nwi  t  * Si  t   N wi  t  *  cv * Vi  t   cs * Nsi t    i  a, t  * N wi t    * N wi  t   i * N wi t 
v
 
 cs * Nsi  t  * N wi  t   cw * Nwi  t  * N si  t    rNsi  rNwi  * N swi  t   i  a, t  * N swi  t    * N swi
t   i * N swi t 
v
 cN * Nsi  t  *Vi  t   cv * Vi  t  * N si  t    rNsi  rVi  * Bsi  t   i  a, t  * Bsi  t    * Bsi   t   i * Bsi t 
dt
dBwi  t 
v
 cN * Nwi  t  *Vi  t   cv * Vi  t  * N wi  t    rNwi  rVi  * Bwi  t   i  a, t  * Bwi  t    * Bwi   t   i * Bwi t 
dt
v
dSi v  t 
v
v
v
v
v
v
 rVi *Vi    t   rNsi * N si   t   rNwi * N wi   t   Si   t  *  1    * Vi t   Nsi t   Nwi  t    i  a, t  * Si t    * Si  t   i * Si  t 
dt
dVi  v  t 
v
v
v
v
v
v
v
 rNsi * Bsi   t   rNwi * Bwi   t   rVi *Vi    t   1    * Vi  t  * Si   t   Vi    t  *  cN * Nsi t   cN * Nwi t    i  a, t  *Vi t    *Vi   t   i *Vi   t 
dt
dN siv  t 
v
 rNwi * N swi
t   rVi * Bsiv t   rNsi * Nsiv t   Nsi t  * Siv t   Nsiv t  * 1    cv * Vi t   cw * Nwi t   i  a, t  * N si t    * N siv t   i * N siv t 
dt
dNwiv  t 
v
 rNsi * N swi
t   rVi * Bwiv t   rNwi * N wiv t   Nwi t  * Siv t   N wiv t  * 1    * cv * Vi t   cs * Nsi t  i  a, t  * N wi t    * N wiv t   i * N wiv t 
dt
v
dN swi
 t   c *  t * N  v  t  c *  t * N  v  t  r  r * N  v  t   a, t * N t   * N  v  t   * N  v  t
 swi  
s
Nsi  
wi  
w
Nwi  
si    Nsi
Nwi 
swi  
i
swi  
i
swi  
dt
dBsi v  t 
v
v
v
v
v
 cN * Nsi  t  *Vi    t   1    * cv * Vi  t  * N si   t    rNsi  rVi  * Bsi   t   i  a, t  * Bsi t    * Bsi  t   i * Bsi  t 
dt
dBwi v  t 
v
v
v
v
v
 cN * Nwi  t  *Vi    t   1    * cv * Vi  t  * N wi   t    rNwi  rVi  * Bwi  t   i  a, t  * Bwi t    * Bwi  t   i * Bwi  t 
dt
Strathmore University International Mathematics Research Meeting
Force of infection

Vi (t )   ij * V j  t   Bsj  t   Bwj  t   V j v   t   Bsj v   t   Bwj v   t 
j

(1)

v
Nwi (t )   ij * N wj  t   N swj  t   Bwj  t   N wj v   t   N swj
 t   Bwjv t 
j

v
Nsi (t )   ij * N sj  t   N swj  t   Bsj  t   N sj v   t   N swj
t   Bsjv t 
j
Strathmore University International Mathematics Research Meeting


(2)
(3)