Biomathematics
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This document presents a mathematical model analyzing the genetic effects of pneumococcal transmission and how vaccination influences the presence of different serotypes in children. It outlines key variables and parameters, providing differential equations to describe disease progression and equilibrium solutions, alongside stability analysis. The findings suggest that simpler models serve as foundational elements for developing more intricate mathematical frameworks for understanding pneumococcal disease dynamics.
Biomathematics
- 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!!