CMDS Presentation
-
Upload
harsh-patel -
Category
Documents
-
view
239 -
download
0
Transcript of CMDS Presentation
Modelling Cholera Dynamics with Allee Effect
Hersh PatelDr Jin Wang
Modelling Cholera Dynamics with AlleeEffect
• Cholera – introduction, stats
• Allee Effect – definition, introduction
• Dynamical System – defining variables, parameters and feasibility
• Basic Reproduction Number
• Disease Free Equilibrium
• Endemic Equilibrium
• Numerical Simulation
• Stability
Cholera• Water borne disease• Vibrio cholerae• Affecting 47 countries around the world• 129064 cases reported in 2013• 2102 (1.63%) cases resulted in deaths
Allee Effect• Warder Clyde Allee• Population correlation with density
Dynamical System
Domain and Feasibility Assumptions
• Domain: Ω = {𝑆, 𝐼, 𝑅, 𝐵: 𝑆 ≥ 0, 𝐼 ≥ 0, 𝑅 ≥ 0, 𝐵 ≥ 0}• 𝑆 + 𝐼 + 𝑅 = 𝑁
• 𝑓 𝐼, 𝐵 = 𝛼𝐵 + 𝛾𝐼; ℎ 𝐼, 𝐵 = −𝑟
𝑘𝐵 𝐵 − 𝑏 𝐵 − 𝑘 + 𝜉𝐼
• 𝑓 0,0 = ℎ 0,0 = 0 (Disease Free condition)• 𝑓𝐼 𝐼, 𝐵 ≥ 0, 𝑓𝐵 ≥ 0 for 𝐼, 𝐵 ≥ 0• 𝐷2𝑓 ≤ 0
Threshold Population• Threshold (limit) beyond which a susceptible
population is likely to get a widespread disease.
• 𝑁0 =𝑟𝑏(𝜇+𝛿)
𝛾𝑟𝑏+𝛼𝜉
Basic Reproduction Number• Number of secondary infections caused by an
infected individual in a population susceptible to the disease.
• 𝑅0 =𝛾𝑟𝑏+𝛼𝜉
𝑟𝑏(𝜇+𝛿)𝑁
Jacobian Matrix
• 𝐽 =𝜕𝑋′
𝜕𝑋=
𝜕(𝑆′,𝐼′,𝑅′,𝐵′)
𝜕(𝑆,𝐼,𝑅,𝐵)
Jacobian Matrix Evaluated at DFE
• DFE: 𝐼 = 𝐵 = 𝑅 = 0, 𝑆 = 𝑁
Eigenvalues of 𝐽0• Characteristic Equation:
𝜆 + 𝜇 2 𝜆2 + 𝜇 + 𝛿 + 𝑟𝑏 − 𝛾𝑁 𝜆 + 𝜇 + 𝛿 + 𝛾𝑁 𝑟𝑏 − 𝛼𝜉𝑁 = 0• Eigenvalues with negative real parts require:
𝑁 <𝜇 + 𝛿
𝛾𝑟𝑏 + 𝛼𝜉𝑟𝑏 = 𝑁0
• Define 𝑅0 =𝑁
𝑁0=
𝛾𝑟𝑏+𝛼𝜉
𝑟𝑏 𝜇+𝛿𝑁
• 𝑅0 < 1 at the DFE
Next Generation Matrix
• Van Den Driessche and Watmough• 𝑅0 = spectral radius of the next generation matrix• Next Generation Matrix
• Matrix 𝐴 is negative semidefinite• Characteristic equation of 𝐴
𝜆 𝜆 +𝛾𝑟𝑏 + 𝛼𝜉
𝑟𝑏 𝜇 + 𝛿𝑁 = 0
• 𝑅0 =𝛾𝑟𝑏+𝛼𝜉
𝑟𝑏 𝜇+𝛿𝑁
Stability of DFE
• Theorem: The DFE of the given dynamical system is locally asymptotically stable if 𝑅0 < 1. Otherwise, if 𝑅0 > 1, the DFE is unstable.
• Global asymptotic stability?
Endemic Equilibrium
•𝑑𝑆
𝑑𝑡=
𝑑𝐼
𝑑𝑡=
𝑑𝑅
𝑑𝑡=
𝑑𝐵
𝑑𝑡= 0
Non-trivial
•𝑑𝑆
𝑑𝑡=
𝑑𝐼
𝑑𝑡= 0
𝐵 =𝜂𝜇𝐼
𝛼(𝜇𝑁−𝜂𝐼)−
𝛾
𝛼𝐼 ; 𝜂 = 𝜇 + 𝛿
•𝑑𝐵
𝑑𝑡= 0
𝐼 =𝑟
𝑘𝜉𝐵(𝐵 − 𝑏)(𝐵 − 𝑘)
• Intersection of the two curves
Endemic Equilibrium
• Additional assumption for unique endemic equilibrium:
𝑁 >𝑟2𝑝2 + 2𝜉𝑘𝑟𝑝(𝜂𝜇 + 𝛼𝜂𝐵1)
4𝜉2𝑘2𝛼𝜇𝛾𝜂𝐵1 + 2𝜉𝑘𝑟𝑝𝜇𝛾
𝐵1 =𝑏 + 𝑘 − 𝑏2 − 𝑏𝑘 + 𝑘2
3𝑝 = 2𝜂𝛾(𝐵1
3 − 𝑏 + 𝑘 𝐵12 + 𝑏𝑘𝐵1)
Numerical Simulation
Results
Stability of the Endemic Equilibrium
• Theorem: Let 𝑋 𝑡 be a non-trivial solution to the given dynamical system, and assume 𝑅0 > 1. If lim
𝑡→∞𝑋 𝑡 = 𝑋∞
exists, then 𝑋∞ = 𝑋∗ is the positive endemic equilibrium.
Jacobian at 𝑋∗ = (𝑆∗, 𝐼∗, 𝑅∗, 𝐵∗)
• All four eigenvalues of 𝐽∗ are real and negative.• 𝐽∗ is negative definite.• 𝑋∗ is locally asymptotically stable.• 𝑋 𝑡 → 𝑋∗ as 𝑡 → ∞.• 𝑋∗ is globally asymptotically stable
(numerically).
• WHO/Department of Control of Epidemic Diseases.• S. Liao and J. Wang, Stability analysis and application of a mathematical cholera
model, Math. Biosci. Eng. 8 (2011), pp. 733–752.• P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold
endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), pp. 29–48.
• C.T. Codeço, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infect. Dis. 1 (2001), p. 1.
References