Modeling and Stability Analysis of a Zika Virus Dynamics

Transcription

1 Modeling and Stability Analysis of a Zika Virus Dynamics S. Bates 1 H. Hutson 2 1 Department of Mathematics Jacksonville University 2 Department of Mathematics REU July 28, 2016 S. Bates, H. Hutson REU 1 / 71

2 Outline Original Model: Local Stability S. Bates, H. Hutson REU 2 / 71

3 Dynamical Systems A dynamical system may be defined as ż = f(z, λ) z R n, λ R m, f : R n R n. A system has a equilibrium point when there exists a z 0 R n such that f(z 0, λ) = 0 S. Bates, H. Hutson REU 3 / 71

4 Equilibria Types and Stability Analysis Non-hyperbolic Equilibrium Center Manifold Theorem Liapunov Functions Hyperbolic Equilibrium Linearization Stable Manifold Theorem and Hartman-Grobman Theorem S. Bates, H. Hutson REU 4 / 71

5 Crash Course on Zika Virus Zika Virus (ZIKV) was first isolated in humans in The current ZIKV epidemic ongoing in the Americas began in Brazil in Apr ZIKV is transmitted from mosquito bites. Evidence suggests it may also be transmitted sexually. 1 out of 5 infected people develop symptoms. Symptoms include mild fever, rash, conjunctivitis and joint pain. Evidence also suggest ZIKV may also increase the chance of microcephaly in newborns and cause Guillain-Barr syndrome. The current epidemic is expected to worsen due to Olympics and increasing mosquito population. S. Bates, H. Hutson REU 5 / 71

6 The Original Zika Model The Gao et. al. model for ZIKV is represented by: S h = ab Iv N h S h β κe h+i h1 +τi h2 N h S h ( ) E h = θ ab Iv N h S h + β κe h+i h1 +τi h2 N h S h ν h E h I h1 = ν h E h γ h1 I h1 I h2 = γ h1 I h1 γ h2 I h2 A h = (1 θ) R h = γ h2 I h2 + γ h A h ( ab Iv N h S h + β κe h+i h1 + h2 N h S h ) γ h A h S v = µ v N v ac ηe h+i h1 S v µ v S v N h E v = ac ηe h+i h1 S v (ν v + µ v )E v N h I v = ν v E v µ v I v S. Bates, H. Hutson REU 6 / 71

7 Original Model: Equilibrium Points The original model has 7 Disease-Free equilibrium points of the form DFE = (s h, 0, 0, 0, 0, r h, s v, 0, 0). They are all non-hyperbolic. There are no Endemic equilibrium points (EE). S. Bates, H. Hutson REU 7 / 71

8 Local Center Manifold Theorem If f(0) = 0 and Df(0) has c eigenvalues with zero real parts and s eigenvalues with negative real parts, where c + s = n, and F(0) = G(0) = 0 and DF(0) = DG(0) = [0], Then the system can be written in diagonal form ẋ = Cx + F(x, y) and ẏ = Py + G(x, y), there exists an h that defines the local center manifold and satisfies Dh(x)[Cx + F(x, h(x))] Ph(x) G(x, h(x)) = 0, and the flow on the center manifold is defined by the system of differential equations ẋ = Cx + F(x, h(x)) S. Bates, H. Hutson REU 8 / 71

9 Original Model: Jacobian J(DFE) = abs h s h + r h s hβκ s h + r h s hβ s h + r h s hβτ s h + r h γ h γ h µ v µ v acsv η s h + r h acsv s h + r h γ h 0 abs h (1 θ) s h + r h µ v ν v 0 s h β(1 θ)κ s h + r h s h β(1 θ) s h + r h s h β(1 θ)τ s h + r h acs v η acs v s h + r h s h + r h ν v µ v abs h θ s h + r h s h βθκ s h + r h ν h s h βθ s h + r h ν h γ h1 0 s h βθτ s h + r h γ h1 γ h2 S. Bates, H. Hutson REU 9 / 71

10 Diagonal Form Original Model: Local Stability Models can be written in diagonal form, [ ẋ ż = f(z) ż = ẏ where C has c eigenvalues with zero real parts and P has s eigenvalues with negative real part, to yield ẋ = Cx + F(x, y) ẏ = Py + G(x, y). ], S. Bates, H. Hutson REU 10 / 71

11 Original Model: F(x, y) F(x, y) = ab y 9 N xy (x 1 + s h ) β κy 2 + y 3 + τy 4 N xy (x 1 + s h ) γ h2 y 4 + γ h y 5 µ v (y 8 + y 9 ) ac ηy 2 + y 3 N xy (x 7 + s v ) S. Bates, H. Hutson REU 11 / 71

12 Original Model: G(x, y) G(x, y) = ( (1 θ) ab y 9 (x 1 + s h ) + β κy 2 + y 3 + τy 4 y 9 (x 1 + s h ) ab s h β κy ) 2 + y 3 + τy 4 s h N xy N xy s h + r h s h + r h ac ηy 2 + y 3 (x 7 + s v ) ac ηy 2 + y 3 s v N xy s h + r h ( 0 θ ab y 9 (x 1 + s h ) + β κy 2 + y 3 + τy 4 y 9 (x 1 + s h ) ab s h β κy ) 2 + y 3 + τy 4 s h N xy N xy s h + r h s h + r h 0 0 S. Bates, H. Hutson REU 12 / 71

13 Local Center Manifold Theorem If! f(0) = 0 and Df(0) has c eigenvalues with zero real parts and s eigenvalues with negative real parts, where c + s = n, and F(0) = G(0) = 0 and DF(0) = DG(0) = [0], Then! the system can be written in diagonal form ẋ = Cx + F(x, y) and ẏ = Py + G(x, y), there exists an h that defines the local center manifold and satisfies Dh(x)[Cx + F(x, h(x))] Ph(x) G(x, h(x)) = 0, and the flow on the center manifold is defined by the system of differential equations ẋ = Cx + F(x, h(x)) S. Bates, H. Hutson REU 13 / 71

14 Outline Original Model: Local Stability S. Bates, H. Hutson REU 14 / 71

15 Matrix Theoretic Method for Global Stability Epidemiology models can be written in a compartmentalized form [ ] ẋ ż = f(z) ż = ẏ The disease compartment is further split into F, which represents the transmission terms, and V, which is the transition terms ẋ = F(x, y) V(x, y) x R p, ẏ = g(x, y) y R q, n = p + q S. Bates, H. Hutson REU 15 / 71

16 Matrix Theoretic Method for Global Stability Then we split the disease compartment, ẋ, into its linear and non-linear components. where F(x, y) = ẋ = (F V)x f(x, y) [ δfi δx j (0, y 0 ) ] [ δvi, V(x, y) = (0, y 0 ) δx j ] and f(x, y) = (F V)x F(x, y) + V(x, y) S. Bates, H. Hutson REU 16 / 71

17 Theorem 2.1 [Shuai and Van Den Driessche] If Goal is to construct a Liapunov function, Q. f(x, y) 0 in Γ R p+q +, F 0, V 1 0, and R 0 1. Then the function Q = ω T V 1 x where ω is the left eigenvector of FV 1 associated with R 0 is a Liapunov function for the system on Γ. S. Bates, H. Hutson REU 17 / 71

18 Liapunov Functions If E is an open subset of R n containing z 0, f C 1 (E) and f(z 0 ) = 0, and there exists V C 1 (E) satisfying V (z 0 ) = 0 and V (z) > 0 if z z 0. Then (a) if V (z) 0 for all z E, z 0 is stable, (b) if V (z) < 0 for all z E, z 0 is asymptotically stable, (c) if V (z) > 0 for all z E, z 0 is unstable. S. Bates, H. Hutson REU 18 / 71

19 Basic Reproduction Number The reproduction number, R 0, is defined biologically as the average number of individuals infected by a single infected individual entering the susceptible population. R 0 is defined mathematically as the spectral radius, ρ(a), of matrix A which is the Next Generation Matrix. S. Bates, H. Hutson REU 19 / 71

20 Theorem 2.2 [Shuai and Van Den Driessche] If Γ R p+q + is compact such that (0, y 0 ) Γ, Γ is positively invariant with respect to the system, ẏ = g(0, y) has a unique equilibrium y = y 0 > 0 that is GAS in R q +, f(x, y) 0 with f(x, y 0 ) = 0 in Γ, and F 0, V 1 0, V 1 F be irreducible. Then If R 0 < 1, then the DFE is GAS in Γ. If R 0 > 1, then the DFE is unstable, the system is uniformly persistent, and there exists at least one endemic equilibrium point. S. Bates, H. Hutson REU 20 / 71

21 Original Model: Feasible Region Γ N h = 0 N v = 0 N h = N h (0) N v = N v (0) Thus, Γ = {S h, E h, I h1, I h2, A h, R h, S v, E v, I v R 9 + S h + E h + I h1 + I h2 + A h + R h N h (0), S v + E v + I v N v (0)} S. Bates, H. Hutson REU 21 / 71

22 Original Model: Equilibrium Points in Γ The Disease-Free equilibrium, (s h, 0, 0, 0, 0, r h, s v, 0, 0) Γ. The DFE is GAS in the disease-free subsystem because we let N h (0) = s h + r h and N v (0) = s v. S. Bates, H. Hutson REU 22 / 71

23 Theorem 2.2 [Shuai and Van Den Driessche] If! Γ R p+q + is compact such that (0, y 0 ) Γ,! Γ is positively invariant with respect to the system,! ẏ = g(0, y) has a unique equilibrium y = y 0 > 0 that is GAS in R q +, f(x, y) 0 with f(x, y 0 ) = 0 in Γ, and F 0, V 1 0, V 1 F be irreducible. Then If R 0 < 1, then the DFE is GAS in Γ. If R 0 > 1, then the DFE is unstable, the system is uniformly persistent, and there exists at least one endemic equilibrium point. S. Bates, H. Hutson REU 23 / 71

24 Original Model: F and V and F = ( θ ab I v S h + β κe h + I h1 + τi h2 N h V = 0 0 ac ηe h + I h1 N h 0 N h S v ν h E h γ h1 I h1 ν h E h γ h2 I h2 γ h1 I h1 (ν v + µ v )E v µ v I v ν v E v S h ) S. Bates, H. Hutson REU 24 / 71

25 Original Model: F(x,y) and V(x,y) and s h βθκ s h βθ s h βθτ s h abθ 0 s h + r h s h + r h s h + r h s h + r h F = acηs v acs v s h + r h s h + r h V = ν h ν h γ h γ h1 γ h µ v + ν v ν v µ v S. Bates, H. Hutson REU 25 / 71

26 Original Model: f(x,y) f(x, y) = (F V)x F(x, y) + V(x, y) f = θ(n h s h (s h + r h )S h )(abi v + β(κe h + I h1 + τi h2 )) (s h + r h )N h 0 0 ac(n h s v (s h + r h )S v )(ηe h + I h1 ) (s h + r h )N h 0 f(s h, E h, I h1, I h2, 0, r h, s v, E v, I v ) 0 0 S. Bates, H. Hutson REU 26 / 71

27 Theorem 2.2 [Shuai and Van Den Driessche] If! Γ R p+q + is compact such that (0, y 0 ) Γ,! Γ is positively invariant with respect to the system,! ẏ = g(0, y) has a unique equilibrium y = y 0 > 0 that is GAS in R q +,! f(x, y) 0 with f(x, y 0 ) = 0 in Γ, and F 0, V 1 0, V 1 F be irreducible. Then If R 0 < 1, then the DFE is GAS in Γ. If R 0 > 1, then the DFE is unstable, the system is uniformly persistent, and there exists at least one endemic equilibrium point. S. Bates, H. Hutson REU 27 / 71

28 Original Model: V 1 (x, y) V 1 = ν 1 h γ 1 h1 γ 1 h2 γ 1 h γ 1 h2 γ 1 h (µ v + ν v ) ν v (µ v (µ v + ν v )) 1 µ 1 v 0 S. Bates, H. Hutson REU 28 / 71

29 Original Model: Next Generation Matrix FV 1 = s hβθ(κγ h1 γ h2 + γ h2 ν h + γ h1 ν hτ) (s h + r h)γ h1 γ h2 ν h s hβθ(κγ h2 + γ h1 τ) (s h + r h)γ h1 γ h2 s hβθτ (s h + r h)γ h2 s h abθν v (s h + r h)µ v (µ v + ν v ) s h abθ (s h + r h)µ v acs v (ηγ h1 + ν h) (s h + r h)γ h1 ν h acs v (s h + r h)γ h S. Bates, H. Hutson REU 29 / 71

30 Theorem 2.2 [Shuai and Van Den Driessche] If! Γ R p+q + is compact such that (0, y 0 ) Γ,! Γ is positively invariant with respect to the system,! ẏ = g(0, y) has a unique equilibrium y = y 0 > 0 that is GAS in R q +,! f(x, y) 0 with f(x, y 0 ) = 0 in Γ, and! F 0, V 1 0, V 1 F be irreducible. Then If R 0 < 1, then the DFE is GAS in Γ. If R 0 > 1, then the DFE is unstable, the system is uniformly persistent, and there exists at least one endemic equilibrium point. S. Bates, H. Hutson REU 30 / 71

31 Outline Original Model: Local Stability S. Bates, H. Hutson REU 31 / 71

32 The Modified Zika Model The Gao et. al. model was modified according to Manore et. al. S h = µ h (H h S h ) ab Iv N h S h β κe h+i h1 +τi h2 N h S h ( ) E h = θ ab Iv N h S h + β κe h+i h1 +τi h2 N h S h (ν h + µ h )E h I h1 = ν h E h (γ h1 + µ h )I h1 I h2 = γ h1 I h1 (γ h2 + µ h )I h2 A h = (1 θ) ( ab Iv N h S h + β κe h+i h1 +τi h2 N h S h ) (γ h + µ h )A h R h = γ h2 I h2 + γ h A h µ h R h ( ) S v = Ψ v rv K v N v N v ac ηe h+i h1 N h S v µ v S v E v = ac ηe h+i h1 S v (ν v + µ v )E v N h I v = ν v E v µ v I v S. Bates, H. Hutson REU 32 / 71

33 Modified Model: Equilibrium Points The modified model has one Disease-Free equilibrium point, DFE = (H h, 0, 0, 0, 0, 0, K v, 0, 0). Use Routh-Hurwitz to ensure that they are hyperbolic and locally stable equilibrium points. We have numerically confirmed the existence of at least one Endemic equilibrium point (EE). S. Bates, H. Hutson REU 33 / 71

34 Linearization Original Model: Local Stability Study linear systems ż = Az. A system is linearized using the Jacobian δf 1 δf 1 (z 0 )... (z 0 ) δz 1 δz n A = Df(z 0 ) =..... δf n δf n (z 0 )... (z 0 ) δz 1 δz n. Local stability of an equilibrium point of a linear system is determined by looking at the real part of the eigenvalues of A = Df(z 0 ). S. Bates, H. Hutson REU 34 / 71

35 Modified Model: Jacobian J(DFE) = µ h ab βκ β βτ 0 µ h 0 γ h γ h2 0 0 µ v Ψ v 0 2µ v Ψ v 2µ v Ψ v ack v η H h ackv H h γ h µ h 0 ab(1 θ) βκ(1 θ) β(1 θ) βτ(1 θ) µ v ν v 0 ack v η H h ack v H h ν v µ v abθ βθκ µ h ν h βθ βθτ ν h γ h1 µ h γ h1 γ h2 µ h S. Bates, H. Hutson REU 35 / 71

36 We have found the following sufficient conditions for DFE to be locally stable: R + T + V + W + µ v > Bκ RT + RV + TV + RW + TW + VW + (R + T + V + W )µ v > B(κ(R + V + W + µ v ) + ν h ) RTV + RTW + RVW + TVW + µ v (RT + RV + TV + RW + TW + VW ) > ASην v + B(κ(RV + RW + VW + µ v (R + V + W )) + ν h (R + V + µ v + γ h1 τ)) RTVW + µ v (RTV + RTW + RVW + TVW ) > ASν v (V η + W η + ν h ) + B(γ h1 µ v + κ(rvw + µ v (RV + RW + VW )) + ν h (RV + Rµ v + V µ v Rγ h1 τ)) RTVW µ v > ASV ν v (W η + ν h ) + Bµ v (RVW κ + ν h (RV + Rγ h1 τ)) where R = µ v + ν v W = γ h1 + µ h T = µ h + ν h A = ack v H h S = abθ V = γ h2 + µ h B = βθ S. Bates, H. Hutson REU 36 / 71

37 Outline Original Model: Local Stability S. Bates, H. Hutson REU 37 / 71

38 Modified Model: Feasible Region Γ N h = µ h H h µ h N h N h (t) = H h + k 3 e µ ht lim t N h(t) = H h ( N v = r v N v 1 N ) v K v ) 1 N v (t) = K v (1 + e rv t+k4kv 1 lim t N v (t) = K v Thus, our feasible region is defined as: Γ = {S h, E h, I h1, I h2, A h, R h, S v, E v, I v R 9 + S h + E h + I h1 + I h2 + A h + R h H h, S v + E v + I v K v } S. Bates, H. Hutson REU 38 / 71

39 Modified Model: Equilibrium Points in Γ The Disease-Free equilibrium, DFE = (H h, 0, 0, 0, 0, 0, K v, 0, 0) Γ. The DFE = (H h, 0, 0, 0, 0, 0, K v, 0, 0) is GAS in the disease-free subsystem because we have already seen that lim N h(t) = H h and lim N v (t) = K v. t t S. Bates, H. Hutson REU 39 / 71

40 Theorem 2.2 [Shuai and Van Den Driessche] If! Γ R p+q + is compact such that (0, y 0 ) Γ,! Γ is positively invariant with respect to the system,! ẏ = g(0, y) has a unique equilibrium y = y 0 > 0 that is GAS in R q +, f(x, y) 0 with f(x, y 0 ) = 0 in Γ, and F 0, V 1 0, V 1 F be irreducible. Then If R 0 < 1, then the DFE is GAS in Γ. If R 0 > 1, then the DFE is unstable, the system is uniformly persistent, and there exists at least one endemic equilibrium point. S. Bates, H. Hutson REU 40 / 71

41 Modified Model: F and V and F = ( θ ab I v S h + β κe h + I h1 + τi h2 N h V = 0 0 ac ηe h + I h1 N h 0 N h S v (ν h + µ h )E h (γ h1 + µ h )I h1 ν h E h (γ h2 + µ h )I h2 γ h1 I h1 (ν v + µ v )E v µ v I v ν v E v S h ) S. Bates, H. Hutson REU 41 / 71

42 Modified Model: F(x,y) and V(x,y) and V = F = βθκ βθ βθτ 0 abθ acηk v ack v H h H h µ h + ν h ν h γ h1 + µ h γ h1 γ h2 + µ h µ v + ν v ν v µ v S. Bates, H. Hutson REU 42 / 71

43 Modified Model: f(x,y) f(x, y) = (F V)x F(x, y) + V(x, y) f = θ(n h S h )(abi v + β(κe h + I h1 + τi h2 )) N h 0 0 ac(k v N h H h S v )(ηe h + I h1 ) H h N h 0 f(h h, E h, I h1, I h2, 0, 0, K v, E v, I v ) 0 0 S. Bates, H. Hutson REU 43 / 71

44 Numerical Simulation: R 0 < S h E h I h1 I h2 Population A h R h S v E v I v Time (days) R 0 = S. Bates, H. Hutson REU 44 / 71

45 Numerical Simulation: R 0 > S h E h I h1 I h2 A h Population R h S v E v I v Time (days) R 0 = S. Bates, H. Hutson REU 45 / 71

46 Theorem 2.2 [Shuai and Van Den Driessche] If! Γ R p+q + is compact such that (0, y 0 ) Γ,! Γ is positively invariant with respect to the system,! ẏ = g(0, y) has a unique equilibrium y = y 0 > 0 that is GAS in R q +,! f(x, y) 0 with f(x, y 0 ) = 0 in Γ, and F 0, V 1 0, V 1 F be irreducible. Then If R 0 < 1, then the DFE is GAS in Γ. If R 0 > 1, then the DFE is unstable, the system is uniformly persistent, and there exists at least one endemic equilibrium point. S. Bates, H. Hutson REU 46 / 71

47 Modified Model: V 1 (x, y) V 1 = µ h + ν h ν h (γ h1 + µ h )(µ h + ν h ) γ h1 + µ h γ h1 ν h γ h1 1 0 (γ h1 + µ h )(γ h2 + µ h )(µ h + ν h ) (γ h1 + µ h )(γ h2 + µ h ) γ h2 + µ h µ v 0 S. Bates, H. Hutson REU 47 / 71

48 Modified Model: Next Generation Matrix FV 1 = βθ((γ h2 + µ h)(κ(γ h1 + µ h) + ν h) + γ h1 ν hτ) βθ(γ h2 + µ h + γ h1 τ) βθτ abν v θ abθ (γ h1 + µ h)(γ h2 + µ h)(µ h + ν h) (γ h1 + µ h)(γ h2 + µ h) γ h2 + µ h µ v (µ v + ν v ) ack v (η(γ h1 + µ h) + ν h) ack v H h(γ h1 + µ h)(µ h + ν h) H h(γ h1 + µ h) µ v S. Bates, H. Hutson REU 48 / 71

49 Theorem 2.2 [Shuai and Van Den Driessche] If! Γ R p+q + is compact such that (0, y 0 ) Γ,! Γ is positively invariant with respect to the system,! ẏ = g(0, y) has a unique equilibrium y = y 0 > 0 that is GAS in R q +,! f(x, y) 0 with f(x, y 0 ) = 0 in Γ, and! F 0, V 1 0, V 1 F be irreducible. Then If R 0 < 1, then the DFE is GAS in Γ. If R 0 > 1, then the DFE is unstable, the system is uniformly persistent, and there exists at least one endemic equilibrium point. S. Bates, H. Hutson REU 49 / 71

50 Theorem 3.5 [Shuai and Van Den Driessche] If Goal is to construct a Liapunov function, D(z). There exist functions D i : U R and G ij : U R and constants a ij 0 such that for every 1 i n, D i n j=1 a ijg ij (z). And for A = [a ij ], each directed cycle C of G has (s,r) ε(c) G rs(z) 0 where ε(c) denotes the arc set of the directed cycle C. Then the function D(z) = n c i D i (z) i=1 is a Liapunov function for the system. S. Bates, H. Hutson REU 50 / 71

51 Modified Model: D i s D i = S h S h S h ln S h S h for i = 1, 2, 3, 4 D 5 = E h E h E h ln E h E h D 6 = I h1 I h1 I h1 ln I h1 I h1 D 7 = I h2 I h2 I h2 ln I h2 I h2 D j = S v S v S v ln S v S v for j = 8, 9 D 10 = E v E v E v ln E v E v D 11 = I v I v I v ln I v I v S. Bates, H. Hutson REU 51 / 71

52 Modified Model: D i s continued Differentiation yields D i ab I v S h N h + βκ S h E h + β S h I h1 N h ( Iv Iv ( Eh Nh Eh ( Ih1 ln I v Ih1 ( Ih2 I v ln E h E h I v S h I v S h + ln I ) v S h Iv Sh E hs h E h S h ln I h1 Ih1 I h1s h Ih1 S h ln I h2 Ih2 I h2s h Ih2 S h + ln E ) hs h Eh S h + ln I ) h1s h Ih1 S h + ln I h2s h I h2 S h + βτ S h I h2 Nh Ih2 := a 4,11 G 4,11 + a 15 G 15 + a 36 G 36 + a 27 G 27 ) S. Bates, H. Hutson REU 52 / 71

53 Modified Model: D i s continued D 5 abθ I v S h N h ( Iv S h Iv Sh + βθκ S h E h ( Eh S h Nh Eh S h + βθ S h I h1 ( Ih1 S h Nh Ih1 S h + βθτ S h I h2 N h ( Ih2 S h I h2 S h ln I v S h I v S h E h E h ln E hs h E h S h ln I h1s h I h1 S h ln I h2s h I h2 S h + ln E ) h Eh E h E h E h E h E h E h := a 54 G 54 + a 51 G 51 + a 53 G 53 + a 52 G 52 D 6 ν h E h ( Eh E h ln E h E h + ln E ) h E h + ln E h E h ) + ln E h E h ) I h1 Ih1 + ln I ) h1 Ih1 := a 65 G 65 S. Bates, H. Hutson REU 53 / 71

54 Modified Model: D i s continued D 7 γ h1 I h1 ( Ih1 D j acη E h S v N h + ac I h1 S v N h Ih1 ( Eh ln I h1 Eh ( Ih1 I h1 := a 95 G 95 + a 86 G 86 I h1 ln E h E h I h2 Ih2 + ln I ) h2 Ih2 := a 76 G 76 ) E hs v E h S v ln I h1 Ih1 I h1s v Ih1 S v + ln E hs v E h S v + ln I h1s v I h1 S v ) S. Bates, H. Hutson REU 54 / 71

55 Modified Model: D i s continued D 10 acη E h S v N h + acη I h1 S v N h ( Eh S v Eh S v ( Ih1 S v I h1 S v := a 10,9 G 10,9 + a 10,8 G 10,8 D 11 ν v E v ( Ev E v ln E v E v ln E hs v E h S v ln I h1s v I h1 S v I v I v E v E v E v E v + ln E ) v Ev + ln E v E v ) + ln I ) v Iv := a 11,10 G 11,10 S. Bates, H. Hutson REU 55 / 71

56 Modified Model: The Sydney Graph S. Bates, H. Hutson REU 56 / 71

57 Modified Model: Directed Cycles Consider: 1 the cycle beginning at node 1, going to node 5, and returning to node 1, 2 the cycle beginning at node 5, going to nodes 6 then 3, and returning to node 5, 3 the cycle beginning at node 5, going to nodes 6, 7, then 2, and returning to node 5, 4 the cycle beginning at node 5, going to nodes 9, 10, 11, then 4, and returning to node 5, 5 the cycle beginning at node 5, going to nodes 6, 8, 10, 11, then 4, and returning to node 5. S. Bates, H. Hutson REU 57 / 71

58 Theorem 3.5 [Shuai and Van Den Driessche] If! There exist functions Di : U R and G ij : U R and! constants a ij 0 such that! for every 1 i n, D i n j=1 a ijg ij (z). And! for A = [a ij ], each directed cycle C of G has (s,r) ε(c) G rs(z) 0 where ε(c) denotes the arc set of the directed cycle C. Then the function D(z) = n c i D i (z) i=1 is a Liapunov function for the system. S. Bates, H. Hutson REU 58 / 71

59 Modified Model: Constructing Constants c i c i = a 51 a 15 = a 52 a 27 = a 53 a 36 = a 54 a 4,11 = θ c 5 = 1 c 6 = a 53 + a 52 a 54 a 10,8 + a 65 a 65 (a 10,8 + a 10,9 ) = S h θ(abi v Ih1 + β(ηe h + I h1 )(I h1 + τi h2 )) ν h Eh N h (ηe h + I h1 ) c 7 = a 52 = βθτi h2 S h a 76 γ h1 Ih1 N h a 54 a 10,8 c j = a 86 (a 10,8 + a 10,9 ) = a 54 a 10,9 a 95 (a 10,8 + a 10,9 ) = bθiv Sh csv (ηeh + I h1 ) a 54 bθiv Sh c 10 = = a 10,8 + a 10,9 csv (ηeh + I h1 ) c 11 = a 54 = abθi v Sh a 11,10 ν v Ev Nh. S. Bates, H. Hutson REU 59 / 71

60 Modified Model: Constructing the Liapunov Function D D = c i D i + c 5 D 5 + c 6 D 6 + c 7 D 8 + c j D j + c 10 D 10 + c 11 D 11 ( = c i S h Sh S h ln S ) ( h Sh + c 5 E h Eh E h ln E ) h Eh + c 6 ( I h1 I h1 I h1 ln I h1 I h1 + c j ( S v S v S v ln S v S v + c 11 ( I v I v I v ln I v I v ) ( + c 7 I h2 Ih2 I h2 ln I ) h2 Ih2 ) ( + c 10 E v Ev Ev ln E ) v Ev ). S. Bates, H. Hutson REU 60 / 71

61 LaSalle s Invariance Principle If! Γ D R n be a compact positively invariant set,! V : D R be a continuously differentiable function,! V (x(t)) 0 in Γ, E Γ be the set of all points in Γ where V (x) = 0, and M E be the largest invariant set in E. Then every solution starting in Γ approaches M as t, that is, lim t inf x(t) z z M = 0. }{{} dist (x(t), M) S. Bates, H. Hutson REU 61 / 71

62 Modified Model: LaSalle s Invariance Principle Note that Ḋ = c i ( Sh S h S h + c j ( Sv S v S v ) ( Eh E S ) ( h Ih1 I h + c 5 E ) ( h1 Ih2 I h + c 6 I h1 ) h2 + c 7 I h2 E h I h1 I h2 ) ( S Ev E ) ( v v + c 10 Iv I ) v E v + c 11 I v E v I v which only equals zero at the EE. Since D is a Liapunov function and the EE is the only set where Ḋ = 0, we can conclude that the EE is the largest invariant set M E. Thus, under numerical simulations supporting Theorem 2.2 and LaSalle s Invariance Principle, we can conclude that the EE is GAS in Γ when R 0 > 1. S. Bates, H. Hutson REU 62 / 71

63 Numerical Simulation 400 S h E h 300 I h1 I h2 Population 200 A h R h S v E v I v Time (days) S. Bates, H. Hutson REU 63 / 71

64 Appendix References Gao, Daozhou et. al. Prevention and Control of Zika as a Mosquito-Borne and Sexually Transmitted Disease: A Mathematical Modeling Analysis. Scientific Reports. Nature Publishing Group 6 (2016). n. pag. Web. Garba, S.M. et. al. Backward bifurcations in dengue transmission dynamics. Mathematical Biosciences 215 (2008): Web. Kucharski, Adam et. al. Transmission Dynamics of Zika Virus in IslandPopulations: A Modelling Analysis of the French Polynesia Outbreak. PLoS Neglected Tropical Diseases 10.5 (2016): n. pag. Web. Manore, Carrie A. et. al. Comparing Dengue and Chikungunya Emergence and Endemic Transmission in A. Aegypti and A. Albopictus. Journal of Theoretical Biology 356 (2014). Web. Mitkowski, W. Dynamical Properties of Metzler Systems. Bulletin of the Polish Academy of Sciences 56.4 (2008). Web. Mohney, Gillian, and Quart, Justine. Fighting Zika in the US: The Battle Over GMO Mosquitoes. ABC News 9 July Web. Perko, Lawrence. Differential Equations and Dynamical Systems. New York: Springer, Web. Rebaza, Jorge. A First Course in Applied Mathematics. New Jersey: John Wiley Sons, Inc Print. Shuai, Z. and Van Den Driessche, P. Global Stability of Infectious Diesase Models Using Lyaponov Functions. SIAM 73:4 (2013): Web. Van Den Driessche, P. and Watmough, James. Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences 180 (2002): Web. S. Bates, H. Hutson REU 64 / 71

65 Appendix Stable Manifold Theorem Let E be an open subset of R n containing the origin, let f C 1 (E), and let φ t be the flow of the nonlinear system. Suppose that f (0) = 0 and that Df (0) = 0 has k eigenvalues with negative real part and n k eigenvalues with positive real part. Then there exists a k-dimensional differentiable manifold S tangent to the stable subspace E s of the linear system at 0 such that for all t 0, φ t (S) S and for all x 0 S. lim φ t(x 0 ) = 0 t and there exists an n k dimensional differentiable manifold U tangent to the unstable subspace E u of the corresponding linear system at 0 such that for all t 0, φ t (U) U and for all x 0 U, lim φ t(x 0 ) = 0 t S. Bates, H. Hutson REU 65 / 71

66 Appendix Hartman-Grobman Theorem Let E be an open subset of R n containing the origin, let f C 1 (E), and let φ t be the flow of the nonlinear system. Suppose that f (0) = 0 and that the matrix A = DF (0) has no eigenvalue with zero real part. Then there exists a homeomorphism H of an open set U containing the origin onto an open set V containing the origin such that for x 0 U, there is an open interval I 0 R containing zero such that for all x 0 U and t I 0 H φ t (x 0 ) = e At H(x 0 ) i.e., H maps trajectories of the nonlinear system near the origin onto trajectories of the linear system near the origin and preserves the parametrization of time. S. Bates, H. Hutson REU 66 / 71

67 Appendix Local Center Manifold Theorem Let f C r (E) where E is an open subset of R n containing the origin and r 1. Suppose that f(0) = 0 and that Df(0) has c eigenvalues with zero real parts and s eigenvalues with negative real parts, where c + s = n. The nonlinear system x = f(x) can then be written in diagonal form ẋ = Cx + F(x, y) ẏ = Py + G(x, y). where (x, y) R c R s, C is a square matrix with c eigenvalues having zero real part, P is a square matrix with s eigenvalues having negative real part, and F(0) = G(0) = 0, DF(0) = DG(0) = 0; furthermore, there exists a δ > 0 and function h C r (N δ (0)) that defines the local center manifold and satisfies S. Bates, H. Hutson REU 67 / 71

68 Appendix Local Center Manifold Theorem Dh(x)[Cx + F(x, h(x))] Ph(x) G(x, h(x)) = 0 for x < δ; and the flow on the center manifold W c is defined by the system of differential equations x = Cx + F(x, h(x)) for all x R c with x < δ. S. Bates, H. Hutson REU 68 / 71

69 Appendix Routh-Hurwitz Criterion Consider a three dimensional system. If the characteristic polynomial λ 3 + a 1 λ 2 + a 2 λ + a 3 satisfies the following: a 1 > 0, a 3 > 0, a 1 a 2 a 3 > 0 then the equilibrium point is locally stable. S. Bates, H. Hutson REU 69 / 71

70 Spectral Radius Appendix The spectral radius of a matrix or bounded linear operator is the supremum among the absolute values of the elements in its spectrum. The spectrum of such a matrix is its set of eingenvalues, but the spectrum of a linear operator T : V V is the set of scalars λ such that T λi is non-invertible. S. Bates, H. Hutson REU 70 / 71

71 Appendix Partitioned Inversion (Using Shur Complements) Given a partitioned matrix M: [ ] 1 M 1 A B = C D [ (M/D) = 1 D 1 C(M/D) 1 (M/D) 1 BA 1 D 1 + D 1 C(M/D) 1 BD 1 ] Where (M/D) 1 = A 1 + A 1 B(M/A) 1 CA 1 if A and D are non-singular S. Bates, H. Hutson REU 71 / 71