An epidemic model with post-contact prophylaxis of distributed - PowerPoint PPT Presentation

An epidemic model with post-contact prophylaxis of distributed length Horst R. Thieme ⋄ , Abdessamad Tridane*, Yang Kuang* School of Mathematical and Statistical Sciences Arizona State University Tempe, Arizona USA partially supported by NSF grants ⋄ DMS-0314529 *DMS-0436341 and *DMS/NIGMS-0342388 (ASU) Tsing-Hua Univ, May 2011 1 / 31

Prophylactic use of antimicrobials influenza in addition to vaccination antiviral agents Neurominidase inhibitors (NAI) Post-contact prophylaxis: treat exposed (but not necessarily infected) individuals Treatment of uninfected exposed individuals lowers their susceptibility to infection by a second exposure Simplification: they are not susceptible at all Treatment of infected individuals lowers their infectivity Simplification: they are not infective at all (ASU) Tsing-Hua Univ, May 2011 2 / 31

Notation S susceptibles U treated uninfected exposed individuals I (untreated) infectious individuals J treated infected individuals R recovered and immune individuals Assumptions: No disease fatalities, births balance deaths 1 = S + U + I + J + R (ASU) Tsing-Hua Univ, May 2011 3 / 31

The model ✯ ✟ J ✟✟✟✟✟✟ ❅ ✻ ❅ ❅ ❘ ❅ ✲ ✲ ✛ S U R S ❍❍❍❍❍❍ � ✒ � ❄ � ❥ ❍ � I I ′ = (1 − τ ) pκIS − ( µ + γ 1 ) I J ′ = τpκIS − ( µ + γ 2 ) I R ′ = γ 1 I + γ 2 J − ( µ + ρ ) R S = 1 − U − I − J − R (ASU) Tsing-Hua Univ, May 2011 4 / 31

distributed length of treatment � ∞ Contact age a , U ( t ) = u ( t, a ) da. 0  τ (1 − p ) κS ( t − a ) I ( t − a ) G ( a ) , t > a   u ( t, a ) = G ( a ) u 0 ( a − t ) G ( a − t ) , t < a   G ( a ) = e − µa F ( a ) . 0 < p, τ < 1 , Ferguson, Mallett, Jackson, Roberts, Ward (2003) (ASU) Tsing-Hua Univ, May 2011 5 / 31

probability to be still treated, F ( a ) Gamma distribution ν = 1 ν =2 1 ν =8 ν =100 ν =2 ν =8 0.8 ν =100 ν =1 0.6 Prob 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 age (ASU) Tsing-Hua Univ, May 2011 6 / 31

treatment termination rate � ∞ F ′ ( a ) U ( t ) = u ( t, a ) da. F ( a ) = − ζ ( a ) . 0 ( ∂ t + ∂ a ) u ( t, a ) = − ( µ + ζ ( a )) u ( t, a ) , t � = a, u ( t, 0) = τ (1 − p ) κS ( t ) I ( t ) , u (0 , a ) = u 0 ( a ) . U ′ = τ (1 − p ) κSI − ( µ + ζ ) U . ζ constant (ASU) Tsing-Hua Univ, May 2011 7 / 31

Existence of solutions three ODEs, one integral equation I ′ = (1 − τ ) pκIS − ( µ + γ 1 ) I J ′ = τpκIS − ( µ + γ 2 ) J R ′ = γ 1 I + γ 2 J − ( µ + ρ ) R S = 1 − U − I − J − R � t U ( t ) = τ (1 − p ) κS ( r ) I ( r ) G ( t − r ) dr + U 1 ( t ) 0 � ∞ u 0 ( a ) G ( a + t ) U 1 ( t ) = da. G ( a ) 0 Volterra integral equations, contraction mapping theorem. (ASU) Tsing-Hua Univ, May 2011 8 / 31

Disease extinction R 0 = (1 − τ ) pκ basic replacement ratio of the disease µ + γ 1 average number of secondary infections caused by one infectious individual that is introduced into an otherwise completely susceptible population Theorem If R 0 ≤ 1, then I ( t ) → 0 and R ( t ) → 0 as t → ∞ . I ′ ( t ) ≤ (1 − τ ) pκI (1 − I ) − ( µ + γ 1 ) I � � = ( µ + γ 1 ) I R 0 (1 − I ) − 1 ≤ 0 . (ASU) Tsing-Hua Univ, May 2011 9 / 31

Disease persistence Uniform (strong) disease persistence: There exists some ǫ > 0 such that lim inf I ( t ) ≥ ǫ t →∞ for all solutions with I (0) > 0 . Uniform weak disease persistence: There exists some ǫ > 0 such that lim sup I ( t ) ≥ ǫ t →∞ for all solutions with I (0) > 0 . (ASU) Tsing-Hua Univ, May 2011 10 / 31

uniform weak disease persistence Theorem R 0 > 1 = ⇒ uniform weak disease persistence. suppose I ∞ = lim sup t →∞ I ( t ) < ǫ . Proof by contradiction: κ J ′ ≤ κI − ( µ + γ 2 ) J J ∞ ≤ I ∞ = ⇒ µ + γ 2 R ∞ ≤ γ 1 I ∞ + γ 2 J ∞ R ′ = γ 1 I + γ 2 J − ( µ + ρ ) R = ⇒ . µ + ρ � t � ∞ u 0 ( a ) G ( a + t ) U ( t ) ≤ κ I ( t − a ) G ( a ) da + da. G ( a ) 0 0 � ∞ U ∞ ≤ I ∞ G ( a ) da < ǫ ˆ Fatou’s lemma: G (0) . 0 (ASU) Tsing-Hua Univ, May 2011 11 / 31

the contradiction I ′ ( t ) I ( t ) ≥ (1 − τ ) pκ (1 − U ∞ − I ∞ − J ∞ − R ∞ ) − µ − γ 1 lim inf t →∞ � � ≥ ( µ + γ 1 ) R 0 (1 − Mǫ ) − 1 > 0 , if ǫ is sufficiently small. Then I ( t ) → ∞ exponentially fast. (ASU) Tsing-Hua Univ, May 2011 12 / 31

Uniform strong persistence  Uniform weak persistence   uniform persistence semiflow on metric space = ⇒  (T 1993, 2003; Smith T 2011) compact attractor of points  state space X ⊆ L 1 ( R + ) × R 3   I, J, R ≥ 0 , u ∈ L 1 + ( R + )  u,      � ∞   I,     X =  ; metric space I + J + R + u ( a ) da ≤ 1   J,   0    R   G ( a ) = 0 ⇒ u ( a ) = 0   (ASU) Tsing-Hua Univ, May 2011 13 / 31

semiflow, compact attractor Φ : R + × X → X Φ( t, x 0 ) = ( u ( t, · ) , I ( t ) , J ( t ) , R ( t )) , x 0 = ( u 0 , I 0 , J 0 , R 0 ) . semiflow: Φ( t + r, x 0 ) = Φ( t, Φ( r, x 0 )) compact attractor: there exists a compact set A in X such that d (Φ( t, x 0 ) , A ) → 0 , t → ∞ , x 0 ∈ X. The convergence is even uniform in x 0 ∈ X . A is stable. (ASU) Tsing-Hua Univ, May 2011 14 / 31

transform to a scalar integro-diff. eqn I ′ = (1 − τ ) pκIS − ( µ + γ 1 ) I J ′ = τpκIS − ( µ + γ 2 ) J R ′ = γ 1 I + γ 2 J − ( µ + ρ ) R � t U ( t ) = τ (1 − p ) κS ( t − a ) I ( t − a ) G ( a ) da + U 1 ( t ) . 0 1 � I ′ + ( µ + γ 1 ) I � κSI = . (1 − τ ) p � t � U ( t ) = φ − I (0) G ( t ) + I ( t ) + I ( t − a ) d G ( a ) 0 � t � + ( γ 1 + µ ) I ( t − a ) G ( a ) da + U 1 ( t ) . 0 (ASU) Tsing-Hua Univ, May 2011 15 / 31

Transformation continued I ′ I = (1 − τ ) pκ (1 − U − I − J − R ) − ( µ + γ 1 ) � t I ′ ( t ) � � I ( t ) = (1 − τ ) pκ 1 − I ( t ) − I ( t − s ) dβ ( s ) − v ( t ) − ( µ + γ 1 ) 0 R 0 > 1 : there exists a unique endemic equilibrium with infective component I ∗ > 0 . � t I ′ ( t ) � I ∗ − I ( t ) − I ∗ − I ( t − s ) � � � I ( t ) = (1 − τ ) pκ dβ ( s ) − ˘ w ( t ) 0 (ASU) Tsing-Hua Univ, May 2011 16 / 31

transformation completed w = ln I ( t ) Set I ∗ . � t w ′ ( t ) + ( ♠ ) g ( w ( t − r )) dα ( r ) = ˘ w ( t ) . 0 g ( w ) = e w − 1 , wg ( w ) > 0 , w � = 0 , w ( t ) → 0 , ˘ t → ∞ . α ( t ) = 0 , t ≤ 0 , left-continuous at t > 0 , of bounded variation on R + (ASU) Tsing-Hua Univ, May 2011 17 / 31

global stability Theorem (Londen 1975) Let . . . and � ∞ e ist dα ( t ) > 0 ( ♦ ) ℜ ∀ s ≥ 0 . 0 Then w ( t ) → 0 as t → ∞ for all bounded solutions w of ( ♠ ). The proof of this theorem uses frequency domain techniques, i.e. properties of Fourier transforms. Stech&Williams (1981) were the first to use this result for epidemic models. (ASU) Tsing-Hua Univ, May 2011 18 / 31

Does it apply? I ( t ) ≤ 1 , lim inf t →∞ I ( t ) ≥ ǫ > 0 if I (0) > 0 . w ( t ) = ln I ( t ) is bounded on R + . I ∗ w ( t ) t →∞ I ( t ) t →∞ → I ∗ . − → 0 = ⇒ − Theorem (Smith T 2011) Let . . . and � ∞ e ist dα ( t ) > 0 . ( ♦ ′ ) s ≥ 0 ℜ inf 0 Then every bounded solution w : R → R of � ∞ w ′ ( t ) + ( ♣ ) g ( w ( t − r )) dα ( r ) = 0 , t ∈ R , 0 satisfies w ( t ) = 0 for all t ∈ R . (ASU) Tsing-Hua Univ, May 2011 19 / 31

Partition of the attractor and stability Back to the semiflow Φ (Smith T 2011) The compact attractor of X is the disjoint union A = { E 0 } ∪ C ∪ { E 1 } , the disease-free equilibrium E 0 = (0 , 0 , 0 , 0) is stable in X 0 = { ( u, I, J, R ) ∈ X ; I = 0 } and attracts all solutions in X 0 , the endemic equilibrium E 1 = ( u ∗ , I ∗ , J ∗ , R ∗ ) is stable and attracts all solutions in X 1 = { ( u, I, J, R ) ∈ X ; I > 0 } . C consists of total orbits connecting E 0 to E 1 . { E 0 } extinction attractor, { E 1 } persistence attractor. (ASU) Tsing-Hua Univ, May 2011 20 / 31

Evaluation of ( ♦ ) Theorem If � ∞ p 1 − τ e ist G ( t ) dt ≤ − ( γ 1 + µ ) ℜ ∀ s > 0 , 1 − p τ 0 the endemic equilibrium is globally asymptotically stable. Corollary If G is convex, the endemic equilibrium is globally asymptotically stable. Corollary If the length of treatment is exponentially distributed, the endemic equilibrium is globally asymptotically stable. (ASU) Tsing-Hua Univ, May 2011 21 / 31

sine Lemma If f : R + → R is decreasing, then � ∞ sin( st ) f ( t ) dt ≥ 0 , s ≥ 0 , 0 provided the integrals exist as improper integrals. � 2 nπ � 2 π n − 1 � sin( t ) f ( t ) dt = sin( t ) f ( t + 2 kπ ) dt 0 0 k =0 n − 1 � π � � � = sin( t ) f ( t + 2 kπ ) − f ( t + (2 k + 1) π ) dt. 0 k =0 ≥ 0 . (ASU) Tsing-Hua Univ, May 2011 22 / 31

cosine Lemma Let f : R + → R be convex and continuous. Then � ∞ cos( st ) f ( t ) dt ≥ 0 , s > 0 , 0 provided the integrals exist as improper integrals. � 2 nπ � 2 nπ sin( t )( − f ′ ( t )) dt. cos( t ) f ( t ) dt = 0 0 − f ′ is decreasing. (ASU) Tsing-Hua Univ, May 2011 23 / 31