Dynamics of a Networked Connectivity Model of Waterborne Disease Epidemics

Dynamics of a Networked Connectivity Model of Waterborne Disease Epidemics A. Edwards, D. Mercadante, C. Retamoza REU Final Presentation July 31, 214

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

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.

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

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.

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

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.

Strongly Connected Graph Γ(P Q) is strongly connected.

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

(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.

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

(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.

(Local)Stability Analysis of a Single Community.4 (a).5 (b).3.4.2.3.1.1 E s E u.2.1 S U.2.1.3.2.4.5.5.3.5.5

(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

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 2 + + µφ + (µ + φ)(mb nb) βhp K +µφmb µφnb µβhp K ) λ+

Routh-Hurwitz Criteria Theorem (Routh-Hurwitz) Given the polynomial, P(λ) = λ n + a 1 λ n 1 +... + 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).

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.

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 ).

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)).

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

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)

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

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.

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.

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)

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.

Phase Portrait β = 1, φ =.2, µ =.1, p = 1, mb =.4, nb =.67, H = 1, K = 1 and S c = 666 15 1 B 5 35 3 25 2 15 1 5 5 6 7 8 9 1 4 I S

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

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.

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 ] =.

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.

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

Bifurcations Figure : p < p 2 x 1 4 1.5 1.5 B.5 1 1.5 2 1 x 1 5 1 2 I 3 4 5 6 7.5 1 1.5 2 S 2.5 3 3.5 x 1 6 4

Bifurcations Figure : p = p 7 6 5 4 B 3 2 1 14 12 1 8 I 6 4 2.5 1 1.5 2 S 2.5 3 3.5 x 1 5 4

Bifurcations Figure : p > p 8 x 1 4 7 6 5 B 4 3 2 1 9 8 x 1 4 7 6 5 I 4 3 2 1.5 1 1.5 2 S 2.5 3 x 1 5 3.5

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 <.

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 ].

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

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

(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

(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 33 33 ] [ J J22 J = 23 J 32 J 33 ] Each block J ij is a 3 3 matrix, and J 11 = diag( µ).

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.

(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.

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.

(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.

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.

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!

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.

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.

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!

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)

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.

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.

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..

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.

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

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 =

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.

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

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.

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 ) =

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 2 1 + 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

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 )

p < p DFE =(1, 13, 11,,,,,, ) 2 15 1 B 5 5 3 2.5 2 15 x 1 4 1.5 I 1.5 5 S 1.5

p = p 1 9 8 7 6 B 5 4 3 2 1 3 2.5 2 12 14 x 1 4 1.5 I 1.5 2 4 6 8 S 1

p > p EE = (11.195, 14.78, 12.952, 215.439, 28.68, 236.969,.135,.97,.157) 1 9 8 7 6 B 5 4 3 2 1 3.5 3 2.5 2 1 12 14 x 1 4 I 1.5 1.5 2 4 6 S 8

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

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.

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).

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

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!!!!