A mathematical model of Streptococcal bacteria in young children. Analysing the effect of vaccination in a 2-serotype and 1-sequence type system.

1. Introduction Variables The Model Conclusion
A simple mathematical model for genetic
effects in pneumococcal carriage and
transmission
Francis Mponda
February 12, 2013

2. Introduction Variables The Model Conclusion
Outline
1 Introduction
2 Variables
3 The Model
Formulation
Analysis
4 Conclusion

3. Introduction Variables The Model Conclusion
Introduction
Streptococcus pneumoniae (Pneumococcus) is a bacterium
commonly found in the throat of young children.
Pneumococcal serotypes can cause a variety of diseases such
as meningitis and pneumonia
In 2000, a vaccine was introduced to not only prevent vaccine
type disease but also to eliminate carriage of the vaccine
serotypes.
A key problem with vaccination is that the same sequence types
are able to manifest in more than one serotype.
We present a 4-D mathematical model for exploring the
relationship between sequence types and serotypes where a
sequence type is able to manifest itself in one vaccine serotype
and one non-vaccine serotype.

4. Introduction Variables The Model Conclusion
Variables and Parameters
Variables
X : Unvaccinated susceptible to carriage of sequence type 1
T1 : Unvaccinated carrying sequence type 1
V : Vaccinated susceptible to carriage of sequence type 1
VT1 : Vaccinated carrying sequence type 1
Parameters
L : Constant recruitment rate of susceptible children into X or V
f : A proportion of children who receive the vaccine
u : per capita exit rate of children from the population
β1 : A mass action transmission coefficient
γ : per capita rate at which carriers become susceptible again

5. Introduction Variables The Model Conclusion
Formulation
Compartmental model
L(1 − f )
? β1 X (T1 + VT1 )
- Y1
X T1
Y2
γT1
? ?
uX uT1
Lf
? β1 V (T1 + VT1 )
-
V VT1 Y2
γVT1
? ?
uV uVT1
Fig1: Flow diagram for model equations.

6. Introduction Variables The Model Conclusion
Formulation
Equations of Change
Differential equations which describe the progress of the disease are
as follow:
dX
= L(1 − f ) − uX − β1 X (T1 + VT1 ) + γT1
dt
dT1
= β1 X (T1 + VT1 ) − (γ + u)T1
dt
dV
= Lf − uV − β1 V (T1 + VT1 ) + γVT1
dt
dVT1
= β1 V (T1 + VT1 ) − (γ + u)VT1
dt

7. Introduction Variables The Model Conclusion
Analysis
Equilibrium Solutions
dF
We now solve the system at = 0 to find the equilibrium solutions
dt
X , T1 , V , Vˆ . For simplicity, let
ˆ ˆ ˆ T
1
x1 = X + V , x2 = T1 + VT1
so that our 4-D system reduces to the 2-D system;
x˙1 = L − ux1 − β1 x1 x2 + γx2
x˙2 = β1 x1 x2 − (γ + u)x2
which can be easily solved to give
∗ ∗ L γ+u L γ+u
(x1 , x2 ) = ,0 and , −
u β1 u β1

8. Introduction Variables The Model Conclusion
Analysis
Equilibrium Solutions cont’d
With further manipulation, we get the following two equilibrium states:
L L
X , T1 , V , Vˆ 1 =
ˆ ˆ ˆ T (1 − f ) , 0, f , 0 , the CFE and the CE given by
u u
γ+u L γ+u γ+u L γ+u
(1 − f ) , (1 − f ) − , f , f −
β1 u β1 β1 u β1
β1 L
Physically we require u ≥ γ+u so that R = u(γ+u) ≥ 1. Hence
L
β1
if R ≤ 1 there is only the CFE whereas for R 1 there is the CFE
and a unique CE. At the CE we have;
ˆ ˆ L γ+u
Y1 = pT1 = p (1 − f ) −
u β1
ˆ L γ+u
Y2 = (1 − p)T1 + Vˆ 1 = (1 − p(1 − f ))
ˆ T −
u β1

9. Introduction Variables The Model Conclusion
Analysis
Effective reproduction number, Re
The basic reproduction number, R0 : The expected number of
secondary cases caused by a ‘typical’ infected individual entering a
completely susceptible population at equilibrium. Notice, here we are
calling it Re because vaccination has been included in the model. We
define the next generation matrix M = (mij ), where mij is the expected
number of type i infected individuals caused by a single type j
infected individual entering the CFE during the entire infectious
1
period γ+u . We have
β1 L(1−f ) β1 L(1−f )
u(γ+u) u(γ+u)
M= β1 Lf β1 Lf
u(γ+u) u(γ+u)
whose spectral radius, Re , is given as
β1 L
Re = R =
u(γ + u)

10. Introduction Variables The Model Conclusion
Analysis
Global stability analysis
Stability analysis of the model leads to the following observations;
if Re ≤ 1 the system approaches the CFE as t −→ ∞
if Re 1 the system approaches the CE as t −→ ∞
which leads to the theorem:
1 For Re ≤ 1 the system has only the CFE which is G.A.S. as time
becomes large.
2 For Re 1 there are two equilibria, the CFE and a unique CE. If
there is no disease initially present the system tends to the CFE.
If there is any disease initially present the system goes to the CE
as time becomes large.

11. Introduction Variables The Model Conclusion
Remarks
Conclusion
We have discussed a basic mathematical model for the transmission
of pneumococcal disease. Simple models such as this provide a
building block on which more complex and realistic mathematical
models can be based.

12. Introduction Variables The Model Conclusion
Remarks
Conclusion
We have discussed a basic mathematical model for the transmission
of pneumococcal disease. Simple models such as this provide a
building block on which more complex and realistic mathematical
models can be based.
Acknowledgement
I am very grateful to the following people:

13. Introduction Variables The Model Conclusion
Remarks
Conclusion
We have discussed a basic mathematical model for the transmission
of pneumococcal disease. Simple models such as this provide a
building block on which more complex and realistic mathematical
models can be based.
Acknowledgement
I am very grateful to the following people:
Prof. Wilson Lamb

14. Introduction Variables The Model Conclusion
Remarks
Conclusion
We have discussed a basic mathematical model for the transmission
of pneumococcal disease. Simple models such as this provide a
building block on which more complex and realistic mathematical
models can be based.
Acknowledgement
I am very grateful to the following people:
Prof. Wilson Lamb
Dr. Karen Lamb

15. Introduction Variables The Model Conclusion
Remarks
Conclusion
We have discussed a basic mathematical model for the transmission
of pneumococcal disease. Simple models such as this provide a
building block on which more complex and realistic mathematical
models can be based.
Acknowledgement
I am very grateful to the following people:
Prof. Wilson Lamb
Dr. Karen Lamb
Thanks for your attention, questions are very welcome!!