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