Dynamics of a Networked Connectivity Model of Waterborne Disease Epidemics
|
|
- Darrell Baldwin
- 6 years ago
- Views:
Transcription
1 Dynamics of a Networked Connectivity Model of Waterborne Disease Epidemics A. Edwards, D. Mercadante, C. Retamoza REU Final Presentation July 31, 214
2 Overview Background on waterborne diseases Introduction to epidemic model and parameters Local stability analysis Lyapunov functions and global stability analysis Bifurcation analysis (Non)Existence of periodic solutions Our contribution
3 Waterborne Diseases Mainly caused by protozoa or bacteria present in water. Waterborne diseases are one of the leading causes of death in low-income countries, particularly affecting infants and children in those areas. The models we have been studying use mainly V. cholera as an example. Vast majority of models in the literature consider one single community. First networked connectivity model introduced in late 212.
4 Why Are We Studying This Model? Learn about dynamical systems To gain a better understanding of: Onset conditions for outbreak The spread of diseases by hydrological means Insight Emergency management New health-care resources
5 Reproduction Matrices/Values R is known as the basic reproduction number in SIR and related models. This stands only when n = 1 (most research articles study only this case) Outbreak occurs when R > 1 With networked connectivity, this statement does not hold. For connectivity models, we study G, or the generalized reproduction matrix. When the dominant eigenvalue of this matrix crosses the value one, an outbreak will occur. This value is independent from the values of R i.
6 Bifurcations Bifurcation occurs when small changes in parameters imply drastic changes in the solutions, number of equilibrium points, or their stability properties. Disease-free equilibrium Transcritical Bifurcation
7 The Model Consider n communities (nodes). For i = 1,..., n, let S i = number of susceptibles in node i I i = number of infectives in node i B i = concentration of bacteria in water at node i Spatially explicit nonlinear differential model Communities are connected by hydrological and human mobility networks through which a disease can spread.
8 Strongly Connected Graph Γ(P Q) is strongly connected.
9 The Epidemic Model (Gatto et al.) For i = 1,..., n ds i dt di i dt db i dt (3n differential equations) [ n ] = µ(h i S i ) (1 m s )β i f (B i ) + m s Q ij β j f (B j ) S i = j=1 [ n ] (1 m s )β i f (B i ) + m s Q ij β j f (B j ) S i φi i ( n = µ B B i + l where f (B i ) = B i K + B i. j=1 j=1 P ji W j W i B j B i ) + p [ i (1 m I )I i + W i n ] m I Q ji I j j=1
10 (Local)Stability Analysis of a Single Community For n = 1, the model is ds dt di dt db dt = µ(h S) βf (B)S = βf (B)S φi = (nb mb)b + pi where f (B) = B K + B.
11 Linearization Definition A set S R n is said to be invariant with respect to x = f (x) if x() S = x(t) S for all t. Linearization Technique: Consider the nonlinear system x = f (x) in R n Let x be a hyperbolic equilibrium point (f (x ) = ) Local stability analysis: Study the linear system f 1 x 1 A = Df (x ), Df (x) =. f n x 1 f 1 x n. f n x n x = Ax, where
12 (Local)Stability Analysis of a Single Community Theorem Stable Manifold Theorem: Let E be an open subset of R n containing the equilibrium point x of x = f (x), and let f C 1 (E). Suppose that Df (x ) 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 x = Ax at x such that S is invariant, and solutions approach x as t. And there exists an n k dimensional differentiable manifold U tangent to the unstable subspace E u of x = Ax at x such that U is invariant and solutions move away from x as t. Theorem Hartman - Grobman Theorem: Let E be and open subset of R n containing a hyperbolic equilibrium point x of x = f (x), and let f C 1 (E). Then there exists a homeomorphism H of an open set U containing x into an open set V containing x such that H maps trajectories of x = f (x) near x onto the trajectories of x = Ax near x.
13 (Local)Stability Analysis of a Single Community.4 (a).5 (b) E s E u.2.1 S U
14 (Local)Stability Analysis of a Single Community For this model, the general Jacobian is J(S, I, B) = µ βb K + B βb K + B βsk (K + B) 2 φ βsk (K + B) 2 p nb mb
15 Disease-Free Equilibrium S1 = H, I 1 =, B 1 = A = J(S1, I1, B1 ) = µ βh K φ βh K p nb mb Characteristic Equation: P(λ) = λ 3 ( + (µ + φ nb + mb)λ µφ + (µ + φ)(mb nb) βhp K +µφmb µφnb µβhp K ) λ+
16 Routh-Hurwitz Criteria Theorem (Routh-Hurwitz) Given the polynomial, P(λ) = λ n + a 1 λ n a n 1 λ + a n, where the coefficients a i are real constants, i=1,...,n. For polynomials of degree n=3, the Routh-Hurwitz criteria are summarized by a 1 >, a 3 >, and a 1 a 2 > a 3. These are necessary and sufficient conditions for all of the roots of the characteristic polynomial (with real coefficients) to lie in the left half of the complex plane (implying stability).
17 Disease-Free Equilibrium By applying the Routh Hurwitz Criteria to the characteristic equation, it was found that a 1 >, a 3 >, and a 1 a 2 > a 3, were true for the following inequalities, mb > nb, S c = φk(mb nb) βp > H = S. Let R = S S c, then R < 1 and our equilibrium is stable.
18 Basic Reproduction Number R In the previous model, R = determined in general? Theorem (Castillo-Chavez et al.) β ps φk(mb nb) = S S c. How is R Consider the (n 1 + n 2 + n 3 )-dimensional system x = f (x, E, I ) E = g(x, E, I ) I = h(x, E, I ). Let (x,, ) be the disease-free equilibrium. Assume the equation g(x, E, I ) = implicitly determines a function E = g(x, I ), and let A = D I h(x, g(x, ), ). Assume further that A can be written as A = M D, where M and D > is diagonal. Then, R = ρ(md 1 ).
19 Lyapunov Functions and Global Stability Definition Let E be an open set in R n, and let x E. A function Q : E R n R is called a Lyapunov function for x = f (x) if Q C 1 (E) and satisfies Q(x ) =, and Q(x) > if x x. Theorem Let R 1, mb > nb, and let w be a left eigenvector of the matrix V 1 F corresponding to R = ρ(v 1 F), then the function Q(x) = w T V 1 x is a Lyapunov function of the system of equations for n=1 satisfying Q (x(t)).
20 Lyapunov Functions and Global Stability Our system can be written as compartmental model x = F (x, y) V (x, y), y = g(x, y) where x = [I, B] T R 2 is the disease compartment and y = S R is the disease-free compartment, respectively. And [ F ] [ F = i V ] (, y ) and V = i (, y ), 1 i, j n x j x j
21 Lyapunov Functions and Global Stability [ ] T βsb Let F = p I and V = [ φi (mb nb)b] T, then the K + B matrices F and V are given by F = [ BH K p ], V = [ φ mb nb ]. The matrix leads us to V 1 F = p mb nb βh Kφ ρ(v 1 βhp F) = R = Kφ(mb nb)
22 Lyapunov Functions and Global Stability The left eigenvector is found to be w T = Q(x) = w T V 1 x, we can now write it as [ ] 1 mb nb Q = 1 R ( ) φ p [ ] mb nb 1 R ( ). As p 1 mb nb I B = I φ + R B p. Now let f(x, y) = (F V)x x and Q = w T V 1 x = (R 1)w T x w T V 1 f(x, y), where βhb βsb f(x, y) = k K + B. ( Q = (R 1) I + R ) B (mb nb) βb p φ ( H K S ). K + B
23 Lyapunov Functions and Global Stability Theorem (LaSalle s Invariance Principle) Let Ω D R n be a compact invariant set with respect to x = f (x). Let Q : D R be a C 1 function such that Q (x(t)) in Ω. Let E Ω be the set of all points in Ω where Q (x) =. Let M E be the largest invariant set in E. Then lim t [ ] inf x(t) y y M =. That is, every solution starting in Ω approaches M as t. Theorem Let Ω be any compact invariant set containing the disease-free equilibrium point. Let f, F, and V be defined as above. Suppose R < 1, mb > nb, and f(x, y), F, V 1, f(, y) =. Also, assume the disease-free system y = g(, y) has a unique equilibrium y = y > that is globally asymptotically stable in R. Then, the disease-free equilibrium is globally asymptotically stable in Ω.
24 Lyapunov Functions and Global Stability The disease-free system y = g(, y) is equivalent to S = µh µs, whose solution is S = H + e µt C. y = H is globally asymptotically stable in the disease-free system. Q = (R 1)w T x w T V 1 f(x, y). Assuming Q = implies x =. Then, the set of all points where Q = is E = {(I, B, S) : I = B = }. The largest and only invariant set in E is (,, H). Therefore, the disease-free equilibrium is globally asymptotically stable in Ω.
25 Endemic Equilibrium The endemic equilibrium represents a single isolated community where there is interaction between susceptibles, infectives, and bacteria in water. S 2 = µh(k + B) µ(k + B) + βb I 2 = B 2 = βbµh φ(µ(k + B) + βb) µ(pβh + Kφ(nb mb)) φ(µ + β)(mb nb)
26 Endemic Equilibrium Let A = βb K + B and C = J(S2, I2, B2 ) = βsk (K + B) 2, (µ + A) C A φ C p nb mb Using Routh-Hurwitz criteria: As long as H > S c (or R > 1), all real parts of the eigenvalues are negative, making the endemic equilibrium locally stable. Therefore, endemic equilibrium is stable exactly when disease-free equilibrium is unstable.
27 Phase Portrait β = 1, φ =.2, µ =.1, p = 1, mb =.4, nb =.67, H = 1, K = 1 and S c = B I S
28 Bifurcations Theorem (Sotomayor) Consider x = f(x, α), where x R n and α is a parameter. Let (x, α ) be an equilibrium point and assume A = Df(x, α ) has a simple eigenvalue λ = with eigenvector v, and left eigenvector w. If w T f α (x, α ) =, w T [Df α (x, α )v], w T [D 2 f(x, α )(v, v)] then a transcritical bifurcation occurs at (x, α ). n n Note: D 2 2 f(x ) f(x )(u, v) = u j1 v j2 x j1 x j2 j 1 =1 j 2 =1
29 Bifurcations Theorem The n = 1 system undergoes a transcritical bifurcation at the disease-free equilibrium point (S1, I 1, B 1 ) = (H,, ) when φ(mb nb)k p =, and mb > nb. βh Proof: Stability of the system at the equilibrium point depends on the bottom right 2 2 matrix of J, given by J = µ βh K φ βh K p nb mb [ φ βh J = K p nb mb ]. When the detj =, p = p = φ(mb nb)k βh.
30 Sotomayor s Theorem To satisfy the first condition of the theorem, w T f p (x, p ) =, we [ ] have w T = 1 φ p, and f p =. Then, I w T f p (x, p ) = [ 1 φ p ] =.
31 Sotomayor s Theorem To satisfy the second condition of the theorem, w T [Df p (x, p )v], we have Df p =. Then w T [Df p (x, p )v] = [ 1 1 ]( φ p 1 βh Kµ mb nb p 1 ) = φ(nb mb) p 2.
32 Sotomayor s Theorem To satisfy the third condition of the theorem, w T [D 2 f(x, p )(v, v)], we have 2βH K ( β 2 µ + 1) D 2 f(x, p )(v, v) = 2βH K ( β 2 µ + 1). Then w T [D 2 f(x, p )(v, v)] = [ 1 φ p ] 2βH K 2 ( β µ + 1) 2βH K 2 ( β µ + 1) = 2βH K 2 (β + 1). µ
33 Bifurcations Figure : p < p 2 x B x I S x 1 6 4
34 Bifurcations Figure : p = p B I S x 1 5 4
35 Bifurcations Figure : p > p 8 x B x I S x
36 Hopf Bifurcation A stable focus (solutions spiral toward the equilibrium) corresponds to λ = a ± bi where a <. When a =, stability may be lost, periodic orbits may appear, and a Hopf Bifurcation could occur. The main the condition to prove the existence of a Hopf Bifurcation is λ = ±bi. Proposition: A Hopf Bifurcation at the DFE does not exist. Proof. For the lower right block of Jacobian at the equilibrium, we have: T = φ + (nb mb) and D = ( φ(nb mb)) (pβh/k). To obtain purely imaginary eigenvalues, we must have T = and D >. But T = is only possible if nb > mb. But then D <.
37 Periodic Solutions Theorem (Bendixon s Criterion in R n ) A simple closed rectifiable curve which is invariant with respect to the differential equation, dx = f (x), cannot dt exist if any one of the following conditions is satisfied on R n : { fr (i) sup + fs + ( f q x r x s q r,s x r + f q x s ) } : 1 r < s n <, { fr (ii) sup + fs + ( f r x r x s q r,s x q + f ) } s : 1 r < s n <, x q (iii) λ 1 + λ 2 <, { fr iv) inf + fs + ( f q x r x s q r,s x r + f q x s ) } : 1 r < s n >, { fr (v) sup + fs + ( f r x r x s q r,s x q + f ) } s : 1 r < s n >, x q (vi) λ n 1 + λ n >. where the λ i are the ordered eigenvalues of M = 1 2 [ Df (x) + Df (x) T ].
38 Periodic Solutions { } sup µ β( B ) φ+p, φ+(nb mb)+ βsk K+B (K+B) 2, µ+(nb mb)+ βsk (K+B) 2 < p µ + β( B K+B ) + φ p (nb mb) + βsk (K+B) 2 + µ + β( B K+B ) p < (nb mb) + βsk (K+B) 2 + φ + β( B K+B ) µ nb + mb + βsk (K+B) 2 βb K+B + p φ µ φ > nb mb + βsk (K+B) 2 φ nb + mb + βsk (K+B) 2 βb K+B + p φ µ φ > nb mb + βsk (K+B) 2
39 The Networked Epidemic Model First consider n = 3 (i = 1, 2, 3). This gives 9 differential equations: [ ds i n ] = µ(h i S i ) (1 m s )β i f (B i ) + m s Q ij β j f (B j ) S i dt di i dt db i dt = j=1 [ n ] (1 m s )β i f (B i ) + m s Q ij β j f (B j ) S i φi i ( n = µ B B i + l j=1 j=1 P ji W j W i B j B i ) + p [ i (1 m I )I i + W i n ] m I Q ji I j j=1
40 (Local) Stability Analysis of Three Communities Jacobian J at disease-free equilibrium is µ H 1 (1 m S )β 1 H 2 m S Q 12 β 2 H 3 m S Q 13 β 3 µ H 1 m S Q 21 β 1 H 2 (1 m S )β 2 H 3 m S Q 23 β 3 µ H 1 m S Q 31 β 1 H 2 m S Q 32 β 2 H 3 (1 m S )β 3 φ H 1 (1 m S )β 1 H 2 m S Q 12 β 2 H 3 m S Q 13 β 3 φ H 1 m S Q 21 β 1 H 2 (1 m S )β 2 H 3 m S Q 23 β 3 φ H 1 m S Q 31 β 1 H 2 m S Q 32 β 2 H 3 (1 m S )β 3 p 1 (1 m I ) p 1 m I Q 21 p 1 m I Q 31 W 2 W 3 µ B l lp 21 lp 31 W 1 K W 1 K W 1 K W 1 W 1 p 2 m I Q 12 p 2 (1 m I ) p 2 m I Q 32 W 1 W 3 lp 12 µ B l lp 32 W 2 K W 2 K W 2 K W 2 W 2 p 3 m I Q 13 p 3 m I Q 23 p 3 (1 m I ) W 1 W 2 lp 13 lp 23 µ B l W 3 K W 3 K W 3 K W 3 W 3 Note: B i = B i /K
41 (Local) Stability Analysis of Three Communities We need to find general conditions for (local) stability. We can block the Jacobian J as J 11 J 13 [ J = J 22 J 23 J22 J λ(j) = λ(j 11 ) λ 23 J J 32 J 32 J ] [ J J22 J = 23 J 32 J 33 ] Each block J ij is a 3 3 matrix, and J 11 = diag( µ).
42 Irreducibility Definition A matrix A nxn is reducible if it is similar to a block triangular matrix: There is a permutation matrix P such that P T AP = [ B C D ], R r x (n r), (1 r n 1) A matrix is irreducible if it is not reducible.
43 (Local) Stability Analysis of Three Communities J is given by φ H 1 (1 m S )β 1 H 2 m S Q 12 β 2 H 3 m S Q 13 β 3 φ H 1 m S Q 21 β 1 H 2 (1 m S )β 2 H 3 m S Q 23 β 3 φ H 1 m S Q 31 β 1 H 2 m S Q 32 β 2 H 3 (1 m S )β 3 p 1 (1 m I ) W 1 K p 2 m I Q 12 W 2 K p 3 m I Q 13 W 3 K p 1 m I Q 21 W 1 K p 2 (1 m I ) W 2 K p 3 m I Q 23 W 3 K p 1 m I Q 31 W 1 K p 2 m I Q 32 W 2 K p 3 (1 m I ) W 3 K µ B l lp 21 W 2 W 1 lp 31 W 3 W 1 lp 12 W 1 W 2 µ B l lp 32 W 3 W 2 lp 13 W 1 W 3 lp 23 W 2 W 3 µ B l J is irreducible.
44 Metzler Matrix Definition A matrix M is said to be Metzler if all of the off-diagonal entries are nonnegative. 1 If you have an M nxn Metzler matrix, then the eigenvalue with maximum real part is real. α(m) = max i Reλ i (M), i = 1, 2,..., n 2 The Metzler matrix, M, is asymptotically stable iff α (M) < Therefore J is a Metzler matrix.
45 (Local) Stability Analysis of Three Communities A very useful variation of Perron-Frobenius theorem: Theorem Let M R nxn be a Metzler matrix. If M is irreducible, then λ = α(m) is a simple eigenvalue, and its associated eigenvector is positive. When λ reaches zero, the det(j ) = and the disease free equilibrium point will lose stability.
46 Determinant of J [ ] J J22 J = 23, det(j J 32 J ) = (J 22 J 33 ) (J 23 J 32 ) = 33 J 22 = φu 3 J 23 = m S HQβ + (1 m S )Hβ J 23 = m I K pw 1 Q T + 1 m I K pw 1 J 33 = (µ B + l)u 3 + l W 1 P T W [ det(j ) = det φ(µ B + l)u 3 φl W 1 P T W m S m I K m I (1 m S ) K pw 1 Q T HQβ pw 1 Q T Hβ (1 m I )m S pw 1 HQβ ] K = (1 m I )(1 m S ) pw 1 Hβ K where p, H, β, W, W 1 are diagonal, hence commute with each other.
47 Determinant of J One can prove that det(j ) = det(u 3 G ) = Thus, the the disease-free equilibrium loses stability when det(u 3 G ) =, where G = l µ B + l PT + µ B µ B + l T T = (1 m I )(1 m S )R + m S m I R IS + m I (1 m S )R I + (1 m I )m S R S. Is a transcritical bifurcation happening here? Stay put!
48 Theorems Theorem (Perron-Frobenius) If A nxn is non-negative and irreducible, then: 1 ρ(a) is a positive eigenvalue of A 2 The eigenvector associated to ρ(a) is positive 3 The algebraic and geometric multiplicity of ρ(a) is 1 Theorem (Berman, Plemmons) Let A = si B, where B is an nxn nonnegative matrix. If there exists a vector x >, such that Ax, then ρ(b) s.
49 Dominant Eigenvalue G > and irreducible ρ(g ) is a simple eigenvalue (Perron-Frobenius) det(u 3 G ) = implies 1 is an eigenvalue of G ρ(g ) 1 (Berman, Plemmons) Therefore, ρ(g ) = 1 and 1 is the dominant eigenvalue. This proves: The disease free equilibrium loses stability when ρ(g ) crosses one, and an outbreak occurs. ρ(g ) is called the Generalized Reproduction Number.
50 Lyapunov Functions and Global Asymptotic Stability First write equations as a compartmental system: x = F (x, y) V (x, y), y = g(x, y) where x = [I 1 I 2 I 3 B 1 B 2 B 3 ] T and y = [S 1 S 2 S 3 ] T. R 1 does not determine stability for connected communities. Cannot use ρ(g ) Need to find a new condition. Exploited the fact that J = F V to prove: λ (J ) λ (V 1 J ) λ F (V 1 F) 1 The three matrices above are Metzler!
51 Lyapunov Functions and Global Asymptotic Stability Theorem Let < λ F (V 1 F) 1, and let w be a left eigenvector of V 1 F corresponding to λ F (V 1 F). Then the function Q(x) = w T V 1 x is a Lyapunov function satisfying Q. ( 1 p Q = Kφ(µ B +l)λ [I 1(1 m I ) 1 F W 1 + p2m I Q 12 W 2 + p3m I Q 13 W 3 )+ + I 2 ( p 1m I Q 21 W 1 + p2(1 m I W 2 + p3m I Q 23 W 3 )+ + I 3 ( p 1m I Q 31 W 1 + p2m I Q 32 W 2 + p3(1 m I ) W 3 )]+ + 1 µ B +l (B 1 + B 2 + B 3 ) Q(x ) = Q(,, H) = and Q(x) > when x x. Q = (λ F 1)wT x w T V 1 f(x, y)
52 Lyapunov Functions and Global Asymptotic Stability Theorem (LaSalle s Invariance Principle) Let Ω D R n be a compact invariant set with respect to x = f (x). Let Q : D R be a C 1 function such that Q (x(t)) in Ω. Let E Ω be the set of all points in Ω where Q (x) =. Let M E be the largest invariant set in E. Then [ ] lim t inf x(t) y y M =. That is, every solution starting in Ω approaches M as t.
53 Lyapunov Functions and Global Asymptotic Stability Using LaSalle s invariance principle we can now prove global asymptotic stability. Theorem Let Ω be any compact invariant set containing the disease-free equilibrium point of the compartmental model and let F, V, V 1, f(x, y), and f(, y) =. Assume < λ F (V 1 F) < 1 and assume the disease-free system y = g(, y) has a unique equilibrium y = y > that is globally asymptotically stable in the disease-free system. Then, the disease-free equilibrium of the nine-dimensional system is globally asymptotically stable in Ω.
54 Lyapunov Functions and Global Asymptotic Stability Solving for S explicitly gives S = H 1 + e µt C 1 H 2 + e µt C 2 H 3 + e µt C 3 Q = (λ F 1)wT x w T V 1 f(x, y). Assuming Q = implies x =. Then, the set of all points where Q = is E = {(I i, B i, S i ) : I i = B i = }. Applying LaSalle s invariance principle shows that the disease-free equilibrium is globally asymptotically stable..
55 Existence of Endemic Equilibrium Recall that λ (J ) λ F (V 1 F) 1. Theorem Let Ω be any compact invariant set containing the disease-free equilibrium point of compartmental model. Let F, V, V 1, f(x, y), and f(, y) =. If λ (J ) >, then there exists at least one endemic equilibrium.
56 Hydrological Transportation and Human Mobility P ij = P out d out (i)p out + d in (i)p in P in d out (i)p out + d in (i)p in if i j if i j otherwise Q ij = H j e ( d ij /D) n H k e ( d ik/d) k i
57 Geography of Disease Onset ] ] J [ i b = λ [ i b [ J A B = C D ] Ai + Bb = λ i Ci + Db = λ b i = A 1 Bb and we can re-write the above expression as CA 1 Bb + Dd = (AD CB)b =
58 Geography of Disease Onset After MUCH simplification, we get [ ( l AD CB = φ(µ B + l) U 3 W 1 µ B + l PT + µ ) ] B µ B + l T W Substituting G, we get AD CB = φ(µ B + l)(u 3 W 1 G W). We can now write (U 3 W 1 G W)b =, then Wb = G Wb Thus, Wb is the eigenvector of G associated to the eigenvalue λ = 1.
59 Geography of Disease Onset i = A 1 Bb After substitution and simplification we get, i = m shqβ + (1 m s )Hβ (W 1 g ) φ
60 Bifurcations Recall that J determines stability of disease-free equilibrium, and is a Metzler matrix. When λ = stability is lost. Proved existence of transcritical bifurcation using Sotomayor s theorem. Choose p 1 to be our varying parameter.
61 Sotomayor s Theorem Let f = [f 1 f 2 f 9 ] T, and x = (S 1 S 2 S 3 I 1 I 2 I 3 B 1 B 2 B 3 ) T so x = (H 1 H 2 H 3 ) T. Take partial derivatives of f with respect to p 1 to arrive at the vector f p1 = [(1 m I )I 1 + m I (Q 21 I 2 + Q 31 I 3 )] W 1 1. First condition : w T f p1 (x, p 1 ) =
62 Sotomayor s Theorem Second condition: w T [Df p1 (x, p1 )v] = H 1 φ [(1 m S)β 1 + m S Q 12 β 2 + m S Q 13 β 3 ] (1 m I )W 1 H φ [m SQ 21 β 1 + (1 m S )β 2 + m S Q 23 β 3 ] m I Q 21 W1 1 + H 3 φ [m SQ 31 β 1 + m S Q 32 β 2 + (1 m S )β 3 ] m I Q 31 W1 1
63 Sotomayor s Theorem D 2 f(x, p1 )(v, v) is found. Third condition: w T [D 2 f(x, p1 )(v, v)] = w 4 (2(1 m S )β 1 v 1 + 2m S Q 12 β 2 v 1 + 2m S Q 13 β 3 v 1 2H 1 (1 m S )β 1 2H 1 m S Q 12 β 2 2H 1 m S Q 13 β 3 ) + w 5 (2m S Q 21 β 1 v 2 + 2(1 m S )β 2 v 2 + 2m S Q 23 β 3 v 2 2H 2 m S Q 21 β 1 2H 2 (1 m S )β 2 2H 2 m S Q 23 β 3 ) + w 6 (2m S Q 31 β 1 v 3 + 2m S Q 32 β 2 v 3 + 2(1 m S )β 3 2H 3 m S Q 31 β 1 2H 3 m S Q 32 β 2 2H 3 (1 m S )β 3 )
64 p < p DFE =(1, 13, 11,,,,,, ) B x I S 1.5
65 p = p B x I S 1
66 p > p EE = (11.195, 14.78, , , 28.68, ,.135,.97,.157) B x 1 4 I S 8
67 n Number of Communities The Jacobian at the disease-free equilibrium x = (H 1, H 2,..., H n,,..., ) is given by J 11 J 13 J(x ) = J 22 J 23, J 32 J 33 where each J ij block is an n n matrix, and J 11 = µu n J 13 = m S HQβ (1 m S )Hβ J 22 = φu n J 23 = m S HQβ + (1 m S )Hβ J 32 = m I K pw 1 Q T + 1 m I K pw 1 J 33 = (µ B + l)u n + lw 1 P T W
68 Math Commandments 1 The disease-free equilibrium loses stability when λ (J ) crosses zero. 2 The system undergoes a transcritical bifurcation when λ (J ) crosses zero. 3 The condition det(j ) = is equivalent to det(u n G ) =. 4 λ = 1 is the dominant eigenvalue of G. 5 If < λ F (V 1 F) 1, then Q = w T V 1 x is a Lyapunov function satisfying Q. 6 If < λ F (V 1 F) < 1 and the disease-free equilibrium is globally asymptotically stable in the disease-free system, then it is a globally asymptotically stable equilibrium of the general system.
69 Math Commandments Cont. 7 If λ (J ) >, then the system has at least one endemic equilibrium point. 8 The dominant eigenvector of G, once projected onto the subspace of infectives, provides us with an effective way to forecast the geographical spread of the disease. 9 Under some conditions, no periodic orbits can exist (n=1). 1 The system has no Hopf bifurcations at DFE (n=1).
70 Our Contribution n = 1 Found correct endemic equilibrium point Proved global stability of DFE Proved existence of transcritical bifurcation (Non)existence of periodic solutions n > 1 (Networked Connectivity Model) Established a new condition for outbreak of epidemics Introduced an appropriate Lyapunov function Proved global stability of DFE Proved existence of transcritical bifurcation Proved existence of endemic equilibrium Numerical evidence of homoclinic orbit
71 Back to the Future: Proving the existence of the homoclinic orbit. Conditions for global stability of endemic equilibrium. Hopf bifurcation at endemic equilibrium. Get help from XPPAUT to find some other possible bifurcations. Include seasonal behavior of some diseases. Include water treatment/sanitation and vaccination in the model Lunch at Union Club: Chicken Parmesan Come back to MSU for MAKO conference!!!!
72
Modeling and Global Stability Analysis of Ebola Models
Modeling and Global Stability Analysis of Ebola Models T. Stoller Department of Mathematics, Statistics, and Physics Wichita State University REU July 27, 2016 T. Stoller REU 1 / 95 Outline Background
More informationDynamics and Bifurcations in Predator-Prey Models with Refuge, Dispersal and Threshold Harvesting
Dynamics and Bifurcations in Predator-Prey Models with Refuge, Dispersal and Threshold Harvesting August 2012 Overview Last Model ẋ = αx(1 x) a(1 m)xy 1+c(1 m)x H(x) ẏ = dy + b(1 m)xy 1+c(1 m)x (1) where
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationSummary of topics relevant for the final. p. 1
Summary of topics relevant for the final p. 1 Outline Scalar difference equations General theory of ODEs Linear ODEs Linear maps Analysis near fixed points (linearization) Bifurcations How to analyze a
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre
More informationAPPPHYS217 Tuesday 25 May 2010
APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationMATH 215/255 Solutions to Additional Practice Problems April dy dt
. For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the
More informationDepartment of Mathematics IIT Guwahati
Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,
More informationNonlinear Autonomous Dynamical systems of two dimensions. Part A
Nonlinear Autonomous Dynamical systems of two dimensions Part A Nonlinear Autonomous Dynamical systems of two dimensions x f ( x, y), x(0) x vector field y g( xy, ), y(0) y F ( f, g) 0 0 f, g are continuous
More informationIntermediate Differential Equations. Stability and Bifurcation II. John A. Burns
Intermediate Differential Equations Stability and Bifurcation II John A. Burns Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and
More informationExamples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling
1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationSection 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d
Section 5.4 (Systems of Linear Differential Equation); 9.5 Eigenvalues and Eigenvectors, cont d July 6, 2009 Today s Session Today s Session A Summary of This Session: Today s Session A Summary of This
More information2.10 Saddles, Nodes, Foci and Centers
2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one
More informationThe goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T
1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.
More informationNonlinear Control. Nonlinear Control Lecture # 3 Stability of Equilibrium Points
Nonlinear Control Lecture # 3 Stability of Equilibrium Points The Invariance Principle Definitions Let x(t) be a solution of ẋ = f(x) A point p is a positive limit point of x(t) if there is a sequence
More informationStochastic modelling of epidemic spread
Stochastic modelling of epidemic spread Julien Arino Centre for Research on Inner City Health St Michael s Hospital Toronto On leave from Department of Mathematics University of Manitoba Julien Arino@umanitoba.ca
More informationInvariant Manifolds of Dynamical Systems and an application to Space Exploration
Invariant Manifolds of Dynamical Systems and an application to Space Exploration Mateo Wirth January 13, 2014 1 Abstract In this paper we go over the basics of stable and unstable manifolds associated
More informationConstruction of Lyapunov functions by validated computation
Construction of Lyapunov functions by validated computation Nobito Yamamoto 1, Kaname Matsue 2, and Tomohiro Hiwaki 1 1 The University of Electro-Communications, Tokyo, Japan yamamoto@im.uec.ac.jp 2 The
More information1 Introduction Definitons Markov... 2
Compact course notes Dynamic systems Fall 2011 Professor: Y. Kudryashov transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Introduction 2 1.1 Definitons...............................................
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture 9: Diagonalization Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./9 Section. Diagonalization The goal here is to develop a useful
More informationGlobal Stability of a Computer Virus Model with Cure and Vertical Transmission
International Journal of Research Studies in Computer Science and Engineering (IJRSCSE) Volume 3, Issue 1, January 016, PP 16-4 ISSN 349-4840 (Print) & ISSN 349-4859 (Online) www.arcjournals.org Global
More informationBifurcation Analysis of Neuronal Bursters Models
Bifurcation Analysis of Neuronal Bursters Models B. Knowlton, W. McClure, N. Vu REU Final Presentation August 3, 2017 Overview Introduction Stability Bifurcations The Hindmarsh-Rose Model Our Contributions
More informationMathematical Modeling I
Mathematical Modeling I Dr. Zachariah Sinkala Department of Mathematical Sciences Middle Tennessee State University Murfreesboro Tennessee 37132, USA November 5, 2011 1d systems To understand more complex
More informationModeling and Stability Analysis of a Zika Virus Dynamics
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
More informationEpidemics in Two Competing Species
Epidemics in Two Competing Species Litao Han 1 School of Information, Renmin University of China, Beijing, 100872 P. R. China Andrea Pugliese 2 Department of Mathematics, University of Trento, Trento,
More informationIntroduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium
Introduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium point and the model reproduction number Prove the stability
More information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations
More informationDynamics of a Population Model Controlling the Spread of Plague in Prairie Dogs
Dynamics of a opulation Model Controlling the Spread of lague in rairie Dogs Catalin Georgescu The University of South Dakota Department of Mathematical Sciences 414 East Clark Street, Vermillion, SD USA
More informationMath Introduction to Numerical Analysis - Class Notes. Fernando Guevara Vasquez. Version Date: January 17, 2012.
Math 5620 - Introduction to Numerical Analysis - Class Notes Fernando Guevara Vasquez Version 1990. Date: January 17, 2012. 3 Contents 1. Disclaimer 4 Chapter 1. Iterative methods for solving linear systems
More informationLotka Volterra Predator-Prey Model with a Predating Scavenger
Lotka Volterra Predator-Prey Model with a Predating Scavenger Monica Pescitelli Georgia College December 13, 2013 Abstract The classic Lotka Volterra equations are used to model the population dynamics
More informationBifurcation Analysis in Simple SIS Epidemic Model Involving Immigrations with Treatment
Appl. Math. Inf. Sci. Lett. 3, No. 3, 97-10 015) 97 Applied Mathematics & Information Sciences Letters An International Journal http://dx.doi.org/10.1785/amisl/03030 Bifurcation Analysis in Simple SIS
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationw T 1 w T 2. w T n 0 if i j 1 if i = j
Lyapunov Operator Let A F n n be given, and define a linear operator L A : C n n C n n as L A (X) := A X + XA Suppose A is diagonalizable (what follows can be generalized even if this is not possible -
More informationStochastic modelling of epidemic spread
Stochastic modelling of epidemic spread Julien Arino Department of Mathematics University of Manitoba Winnipeg Julien Arino@umanitoba.ca 19 May 2012 1 Introduction 2 Stochastic processes 3 The SIS model
More informationLyapunov Stability Theory
Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous
More informationSection 1.7: Properties of the Leslie Matrix
Section 1.7: Properties of the Leslie Matrix Definition: A matrix A whose entries are nonnegative (positive) is called a nonnegative (positive) matrix, denoted as A 0 (A > 0). Definition: A square m m
More information154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.
54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and
More informationTHE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS
THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS GUILLAUME LAJOIE Contents 1. Introduction 2 2. The Hartman-Grobman Theorem 2 2.1. Preliminaries 2 2.2. The discrete-time Case 4 2.3. The
More information3 Stability and Lyapunov Functions
CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such
More informationMathematical Analysis of Epidemiological Models: Introduction
Mathematical Analysis of Epidemiological Models: Introduction Jan Medlock Clemson University Department of Mathematical Sciences 8 February 2010 1. Introduction. The effectiveness of improved sanitation,
More informationOn the simultaneous diagonal stability of a pair of positive linear systems
On the simultaneous diagonal stability of a pair of positive linear systems Oliver Mason Hamilton Institute NUI Maynooth Ireland Robert Shorten Hamilton Institute NUI Maynooth Ireland Abstract In this
More informationLinearization of Differential Equation Models
Linearization of Differential Equation Models 1 Motivation We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking
More informationGLOBAL STABILITY OF SIR MODELS WITH NONLINEAR INCIDENCE AND DISCONTINUOUS TREATMENT
Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 304, pp. 1 8. SSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GLOBAL STABLTY
More informationMA5510 Ordinary Differential Equation, Fall, 2014
MA551 Ordinary Differential Equation, Fall, 214 9/3/214 Linear systems of ordinary differential equations: Consider the first order linear equation for some constant a. The solution is given by such that
More informationLecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.
Lecture 4 Chapter 4: Lyapunov Stability Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 4 p. 1/86 Autonomous Systems Consider the autonomous system ẋ
More informationEN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015
EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions
More informationLecture 15 Perron-Frobenius Theory
EE363 Winter 2005-06 Lecture 15 Perron-Frobenius Theory Positive and nonnegative matrices and vectors Perron-Frobenius theorems Markov chains Economic growth Population dynamics Max-min and min-max characterization
More informationWe have two possible solutions (intersections of null-clines. dt = bv + muv = g(u, v). du = au nuv = f (u, v),
Let us apply the approach presented above to the analysis of population dynamics models. 9. Lotka-Volterra predator-prey model: phase plane analysis. Earlier we introduced the system of equations for prey
More information= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F :
1 Bifurcations Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 A bifurcation is a qualitative change
More informationDelay SIR Model with Nonlinear Incident Rate and Varying Total Population
Delay SIR Model with Nonlinear Incident Rate Varying Total Population Rujira Ouncharoen, Salinthip Daengkongkho, Thongchai Dumrongpokaphan, Yongwimon Lenbury Abstract Recently, models describing the behavior
More informationOutline. Learning Objectives. References. Lecture 2: Second-order Systems
Outline Lecture 2: Second-order Systems! Techniques based on linear systems analysis! Phase-plane analysis! Example: Neanderthal / Early man competition! Hartman-Grobman theorem -- validity of linearizations!
More informationStability of SEIR Model of Infectious Diseases with Human Immunity
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1811 1819 Research India Publications http://www.ripublication.com/gjpam.htm Stability of SEIR Model of Infectious
More informationNonlinear dynamics & chaos BECS
Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes
More informationGlobal Analysis of an Epidemic Model with Nonmonotone Incidence Rate
Global Analysis of an Epidemic Model with Nonmonotone Incidence Rate Dongmei Xiao Department of Mathematics, Shanghai Jiaotong University, Shanghai 00030, China E-mail: xiaodm@sjtu.edu.cn and Shigui Ruan
More information10 Back to planar nonlinear systems
10 Back to planar nonlinear sstems 10.1 Near the equilibria Recall that I started talking about the Lotka Volterra model as a motivation to stud sstems of two first order autonomous equations of the form
More informationDifferential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
More informationDynamical Systems & Lyapunov Stability
Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence
More informationAlgebra C Numerical Linear Algebra Sample Exam Problems
Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric
More informationBifurcation Analysis of a SIRS Epidemic Model with a Generalized Nonmonotone and Saturated Incidence Rate
Bifurcation Analysis of a SIRS Epidemic Model with a Generalized Nonmonotone and Saturated Incidence Rate Min Lu a, Jicai Huang a, Shigui Ruan b and Pei Yu c a School of Mathematics and Statistics, Central
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 1. Linear systems Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Linear systems 1.1 Differential Equations 1.2 Linear flows 1.3 Linear maps
More informationMath 273 (51) - Final
Name: Id #: Math 273 (5) - Final Autumn Quarter 26 Thursday, December 8, 26-6: to 8: Instructions: Prob. Points Score possible 25 2 25 3 25 TOTAL 75 Read each problem carefully. Write legibly. Show all
More informationCourse 216: Ordinary Differential Equations
Course 16: Ordinary Differential Equations Notes by Chris Blair These notes cover the ODEs course given in 7-8 by Dr. John Stalker. Contents I Solving Linear ODEs 1 Reduction of Order Computing Matrix
More informationStability lectures. Stability of Linear Systems. Stability of Linear Systems. Stability of Continuous Systems. EECE 571M/491M, Spring 2008 Lecture 5
EECE 571M/491M, Spring 2008 Lecture 5 Stability of Continuous Systems http://courses.ece.ubc.ca/491m moishi@ece.ubc.ca Dr. Meeko Oishi Electrical and Computer Engineering University of British Columbia,
More informationLinear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems
Linear ODEs p. 1 Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Linear ODEs Definition (Linear ODE) A linear ODE is a differential equation taking the form
More informationGLOBAL STABILITY OF A VACCINATION MODEL WITH IMMIGRATION
Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 92, pp. 1 10. SSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GLOBAL STABLTY
More informationMATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation
MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation 1 Bifurcations in Multiple Dimensions When we were considering one-dimensional systems, we saw that subtle changes in parameter
More informationThursday. Threshold and Sensitivity Analysis
Thursday Threshold and Sensitivity Analysis SIR Model without Demography ds dt di dt dr dt = βsi (2.1) = βsi γi (2.2) = γi (2.3) With initial conditions S(0) > 0, I(0) > 0, and R(0) = 0. This model can
More informationModeling and Analysis of Dynamic Systems
Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability
More informationNotes on Linear Algebra and Matrix Theory
Massimo Franceschet featuring Enrico Bozzo Scalar product The scalar product (a.k.a. dot product or inner product) of two real vectors x = (x 1,..., x n ) and y = (y 1,..., y n ) is not a vector but a
More informationIntroduction to Applied Nonlinear Dynamical Systems and Chaos
Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures 4jj Springer I Series Preface v L I Preface to the Second Edition vii Introduction 1 1 Equilibrium
More informationBIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs
BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits. 2. Equilibria. 3. Periodic orbits and limit cycles. 4. Homoclinic orbits.
More informationMath 331 Homework Assignment Chapter 7 Page 1 of 9
Math Homework Assignment Chapter 7 Page of 9 Instructions: Please make sure to demonstrate every step in your calculations. Return your answers including this homework sheet back to the instructor as a
More informationStabilization and Passivity-Based Control
DISC Systems and Control Theory of Nonlinear Systems, 2010 1 Stabilization and Passivity-Based Control Lecture 8 Nonlinear Dynamical Control Systems, Chapter 10, plus handout from R. Sepulchre, Constructive
More informationZ-Pencils. November 20, Abstract
Z-Pencils J. J. McDonald D. D. Olesky H. Schneider M. J. Tsatsomeros P. van den Driessche November 20, 2006 Abstract The matrix pencil (A, B) = {tb A t C} is considered under the assumptions that A is
More informationAn Undergraduate s Guide to the Hartman-Grobman and Poincaré-Bendixon Theorems
An Undergraduate s Guide to the Hartman-Grobman and Poincaré-Bendixon Theorems Scott Zimmerman MATH181HM: Dynamical Systems Spring 2008 1 Introduction The Hartman-Grobman and Poincaré-Bendixon Theorems
More informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
More informationPerron Frobenius Theory
Perron Frobenius Theory Oskar Perron Georg Frobenius (1880 1975) (1849 1917) Stefan Güttel Perron Frobenius Theory 1 / 10 Positive and Nonnegative Matrices Let A, B R m n. A B if a ij b ij i, j, A > B
More informationStability Analysis of a SIS Epidemic Model with Standard Incidence
tability Analysis of a I Epidemic Model with tandard Incidence Cruz Vargas-De-León Received 19 April 2011; Accepted 19 Octuber 2011 leoncruz82@yahoo.com.mx Abstract In this paper, we study the global properties
More informationMATH 614 Dynamical Systems and Chaos Lecture 24: Bifurcation theory in higher dimensions. The Hopf bifurcation.
MATH 614 Dynamical Systems and Chaos Lecture 24: Bifurcation theory in higher dimensions. The Hopf bifurcation. Bifurcation theory The object of bifurcation theory is to study changes that maps undergo
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationCompetitive and Cooperative Differential Equations
Chapter 3 Competitive and Cooperative Differential Equations 0. Introduction This chapter and the next one focus on ordinary differential equations in IR n. A natural partial ordering on IR n is generated
More informationFIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland
FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland 4 May 2012 Because the presentation of this material
More informationCalculus and Differential Equations II
MATH 250 B Second order autonomous linear systems We are mostly interested with 2 2 first order autonomous systems of the form { x = a x + b y y = c x + d y where x and y are functions of t and a, b, c,
More informationSolutions to Final Exam Sample Problems, Math 246, Spring 2011
Solutions to Final Exam Sample Problems, Math 246, Spring 2 () Consider the differential equation dy dt = (9 y2 )y 2 (a) Identify its equilibrium (stationary) points and classify their stability (b) Sketch
More informationMathematical Foundations of Neuroscience - Lecture 7. Bifurcations II.
Mathematical Foundations of Neuroscience - Lecture 7. Bifurcations II. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010 Filip
More informationChapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12
Chapter 6 Nonlinear Systems and Phenomena 6.1 Stability and the Phase Plane We now move to nonlinear systems Begin with the first-order system for x(t) d dt x = f(x,t), x(0) = x 0 In particular, consider
More informationFrom Lay, 5.4. If we always treat a matrix as defining a linear transformation, what role does diagonalisation play?
Overview Last week introduced the important Diagonalisation Theorem: An n n matrix A is diagonalisable if and only if there is a basis for R n consisting of eigenvectors of A. This week we ll continue
More informationLAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC
LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic
More informationSTABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL
VFAST Transactions on Mathematics http://vfast.org/index.php/vtm@ 2013 ISSN: 2309-0022 Volume 1, Number 1, May-June, 2013 pp. 16 20 STABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL Roman Ullah 1, Gul
More information6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities
6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities 1 Outline Outline Dynamical systems. Linear and Non-linear. Convergence. Linear algebra and Lyapunov functions. Markov
More informationStable Coexistence of a Predator-Prey model with a Scavenger
ANTON DE KOM UNIVERSITEIT VAN SURINAME INSTITUTE FOR GRADUATE STUDIES AND RESEARCH Stable Coexistence of a Predator-Prey model with a Scavenger Thesis in partial fulfillment of the requirements for the
More informationGLOBAL ASYMPTOTIC STABILITY OF A DIFFUSIVE SVIR EPIDEMIC MODEL WITH IMMIGRATION OF INDIVIDUALS
Electronic Journal of Differential Equations, Vol. 16 (16), No. 8, pp. 1 1. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GLOBAL ASYMPTOTIC STABILITY OF A DIFFUSIVE SVIR
More informationGLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS
CANADIAN APPIED MATHEMATICS QUARTERY Volume 13, Number 4, Winter 2005 GOBA DYNAMICS OF A MATHEMATICA MODE OF TUBERCUOSIS HONGBIN GUO ABSTRACT. Mathematical analysis is carried out for a mathematical model
More informationLinear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form
Linear Planar Systems Math 246, Spring 2009, Professor David Levermore We now consider linear systems of the form d x x 1 = A, where A = dt y y a11 a 12 a 21 a 22 Here the entries of the coefficient matrix
More informationNonnegative and spectral matrix theory Lecture notes
Nonnegative and spectral matrix theory Lecture notes Dario Fasino, University of Udine (Italy) Lecture notes for the first part of the course Nonnegative and spectral matrix theory with applications to
More informationLINEAR ALGEBRA QUESTION BANK
LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,
More informationProblem set 7 Math 207A, Fall 2011 Solutions
Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase
More informationControl Systems. Internal Stability - LTI systems. L. Lanari
Control Systems Internal Stability - LTI systems L. Lanari outline LTI systems: definitions conditions South stability criterion equilibrium points Nonlinear systems: equilibrium points examples stable
More information