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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

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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

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The model

S U I J R S

❍❍❍❍❍❍ ❍ ❥ ✟✟✟✟✟✟ ✟ ✯ ✻

• ✒

❅ ❅ ❅ ❅ ❘ ❄ ✲ ✛ ✲

I′ = (1 − τ)pκIS − (µ + γ1)I J′ = τpκIS − (µ + γ2)I R′ = γ1I + γ2J − (µ + ρ)R S = 1 − U − I − J − R

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distributed length of treatment

Contact age a, U(t) = ∞ u(t, a)da. u(t, a) =      τ(1 − p)κS(t − a)I(t − a)G(a), t > a u0(a − t) G(a) G(a − t), t < a 0 < p, τ < 1, G(a) = e−µaF(a). Ferguson, Mallett, Jackson, Roberts, Ward (2003)

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probability to be still treated, F(a)

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1

Gamma distribution age Prob

ν= 1 ν=2 ν=8 ν=100 ν=100 ν=1 ν=2 ν=8

(ASU) Tsing-Hua Univ, May 2011 6 / 31

treatment termination rate

U(t) = ∞ u(t, a)da.

F′(a) F(a) = −ζ(a).

(∂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) = u0(a). ζ constant U′ = τ(1 − p)κSI − (µ + ζ)U.

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Existence of solutions

three ODEs, one integral equation I′ = (1 − τ)pκIS − (µ + γ1)I J′ = τpκIS − (µ + γ2)J R′ = γ1I + γ2J − (µ + ρ)R S = 1 − U − I − J − R U(t) = t τ(1 − p)κS(r)I(r)G(t − r)dr + U1(t) U1(t) = ∞ u0(a)G(a + t) G(a) da. Volterra integral equations, contraction mapping theorem.

(ASU) Tsing-Hua Univ, May 2011 8 / 31

Disease extinction

R0 = (1 − τ)pκ µ + γ1 basic replacement ratio of the disease average number of secondary infections caused by one infectious individual that is introduced into an otherwise completely susceptible population

Theorem

If R0 ≤ 1, then I(t) → 0 and R(t) → 0 as t → ∞. I′(t) ≤ (1 − τ)pκI(1 − I) − (µ + γ1)I = (µ + γ1)I

• R0(1 − I) − 1
• ≤ 0.

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Disease persistence

Uniform (strong) disease persistence: There exists some ǫ > 0 such that lim inf

t→∞

I(t) ≥ ǫ for all solutions with I(0) > 0. Uniform weak disease persistence: There exists some ǫ > 0 such that lim sup

t→∞

I(t) ≥ ǫ for all solutions with I(0) > 0.

(ASU) Tsing-Hua Univ, May 2011 10 / 31

uniform weak disease persistence

Theorem

R0 > 1 = ⇒ uniform weak disease persistence. Proof by contradiction: suppose I∞ = lim supt→∞ I(t) < ǫ. J′ ≤ κI − (µ + γ2)J = ⇒ J∞ ≤ κ µ + γ2 I∞ R′ = γ1I + γ2J − (µ + ρ)R = ⇒ R∞ ≤ γ1I∞ + γ2J∞ µ + ρ . U(t) ≤ κ t I(t − a)G(a)da + ∞ u0(a)G(a + t) G(a) da. Fatou’s lemma: U∞ ≤ I∞ ∞ G(a)da < ǫ ˆ G(0).

(ASU) Tsing-Hua Univ, May 2011 11 / 31

the contradiction

lim inf

t→∞

I′(t) I(t) ≥ (1 − τ)pκ(1 − U∞ − I∞ − J∞ − R∞) − µ − γ1 ≥ (µ + γ1)

• R0(1 − Mǫ) − 1
• > 0,

if ǫ is sufficiently small. Then I(t) → ∞ exponentially fast.

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Uniform strong persistence

Uniform weak persistence semiflow on metric space compact attractor of points      = ⇒ uniform persistence (T 1993, 2003; Smith T 2011) state space X ⊆ L1(R+) × R3 X =              u, I, J, R     ; I, J, R ≥ 0, u ∈ L1

+(R+)

I + J + R + ∞ u(a)da ≤ 1 G(a) = 0 ⇒ u(a) = 0          metric space

(ASU) Tsing-Hua Univ, May 2011 13 / 31

semiflow, compact attractor

Φ : R+ × X → X Φ(t, x0) = (u(t, ·), I(t), J(t), R(t)), x0 = (u0, I0, J0, R0). semiflow: Φ(t + r, x0) = Φ(t, Φ(r, x0)) compact attractor: there exists a compact set A in X such that d(Φ(t, x0), A) → 0, t → ∞, x0 ∈ X. The convergence is even uniform in x0 ∈ X. A is stable.

(ASU) Tsing-Hua Univ, May 2011 14 / 31

transform to a scalar integro-diff. eqn

I′ = (1 − τ)pκIS − (µ + γ1)I J′ = τpκIS − (µ + γ2)J R′ = γ1I + γ2J − (µ + ρ)R U(t) = τ(1 − p) t κS(t − a)I(t − a)G(a)da + U1(t). κSI = 1 (1 − τ)p

• I′ + (µ + γ1)I
• .

U(t) = φ

• − I(0)G(t) + I(t) +

t I(t − a)dG(a) + (γ1 + µ) t I(t − a)G(a)da

• + U1(t).

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Transformation continued

I′ I = (1 − τ)pκ(1 − U − I − J − R) − (µ + γ1) I′(t) I(t) = (1 − τ)pκ

• 1 − I(t) −

t I(t − s)dβ(s)

• − v(t) − (µ + γ1)

R0 > 1: there exists a unique endemic equilibrium with infective component I∗ > 0. I′(t) I(t) = (1 − τ)pκ

• I∗ − I(t) −

t

• I∗ − I(t − s)
• dβ(s)
• − ˘

w(t)

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transformation completed

Set w = ln I(t) I∗ . (♠) w′(t) + t g(w(t − r))dα(r) = ˘ w(t). g(w) = ew − 1, wg(w) > 0, w = 0, ˘ w(t) → 0, t → ∞. α(t) = 0, t ≤ 0, left-continuous at t > 0,

• f bounded variation on R+

(ASU) Tsing-Hua Univ, May 2011 17 / 31

global stability

Theorem (Londen 1975)

Let . . . and (♦) ℜ ∞ eistdα(t) > 0 ∀s ≥ 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) I∗ is bounded on R+. w(t) t→∞ − → 0 = ⇒ I(t) t→∞ − → I∗.

Theorem (Smith T 2011)

Let . . . and (♦′) inf

s≥0 ℜ

∞ eistdα(t) > 0. Then every bounded solution w : R → R of (♣) w′(t) + ∞ g(w(t − r))dα(r) = 0, t ∈ R, 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 = {E0} ∪ C ∪ {E1}, the disease-free equilibrium E0 = (0, 0, 0, 0) is stable in X0 = {(u, I, J, R) ∈ X; I = 0} and attracts all solutions in X0, the endemic equilibrium E1 = (u∗, I∗, J∗, R∗) is stable and attracts all solutions in X1 = {(u, I, J, R) ∈ X; I > 0}. C consists of total orbits connecting E0 to E1. {E0} extinction attractor, {E1} persistence attractor.

(ASU) Tsing-Hua Univ, May 2011 20 / 31

Evaluation of (♦)

Theorem

If −(γ1 + µ) ℜ ∞ eistG(t)dt ≤ p 1 − p 1 − τ τ ∀s > 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, provided the integrals exist as improper integrals. 2nπ sin(t)f(t)dt =

n−1

• k=0

2π sin(t)f(t + 2kπ)dt =

n−1

• k=0

π sin(t)

• f(t + 2kπ) − f(t + (2k + 1)π)
• dt.

≥ 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, provided the integrals exist as improper integrals. 2nπ cos(t)f(t)dt = 2nπ sin(t)(−f ′(t))dt. −f ′ is decreasing.

(ASU) Tsing-Hua Univ, May 2011 23 / 31

short prophylactic treatment

Corollary

If (γ1 + µ) ∞ G(t)dt ≤ p 1 − p 1 − τ τ , the endemic equilibrium is globally asymptotically stable. ˜ DI = 1 γ1 + µ average duration of infectious period ˜ DT = ∞ G(t)dt average duration of treatment τ 1 − τ ˜ DT ≤ p 1 − p ˜ DI = ⇒ end. equil. GAS.

(ASU) Tsing-Hua Univ, May 2011 24 / 31

Instability of the endemic equilibrium

Theorem

Let ℜ ∞ eiytG(t)dt < 0 for some y > 0. Given the parameters ρ ≥ 0, µ ≥ 0, γ2 > 0, the parameters 0 < p < 1, 0 < τ < 1, γ1 > 0 and κ > 0 can be chosen such that the endemic equilibrium is unstable. p and κ can be changed such that a root of the characteristic equation crosses the imaginary axis causing the emergence of periodic solutions via a Hopf bifurcation. Adjust the parameters such that one obtains a purely imaginary root of the characteristic equation. Apply the implicit function theorem. Hopf: Fiedler 1986

(ASU) Tsing-Hua Univ, May 2011 25 / 31

Gamma-distributed treatment

length of the treatment (death neglected) is Gamma-distributed, i.e. F(a) = ∞

a f(t)dt,

f(t) = ανtν−1e−αt Γ(ν) , t ≥ 0. The expected length of the treatment, DT , is ν/α and the variance VT = ν/α2. Set α = ν/σ. Then DT = σ is independent of ν and VT = D2

T /ν → 0 as

ν → ∞. With this scaling, the Gamma-distribution interpolates between exponentially distributed treatment length and fixed treatment length as ν varies from 1 to ∞.

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Illustration for treatment duration 1

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1

Gamma distribution age Prob

ν= 1 ν=2 ν=8 ν=100 ν=100 ν=1 ν=2 ν=8

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Γ-distributed length of treatment.

Corollary

If VT /D2

T ≥ 1/2 , the endemic equilibrium is globally asymptotically

stable. If VT /D2

T < 1/2, after choosing µ > 0 sufficiently small the parameters

p, τ, γ1, and κ can be chosen so that the endemic equilibrium is unstable.

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Treatment of fixed length

100 200 300 400 500 600 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 time t 5I, 5J, R, U 5I 5J R U

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References

Thieme, Horst R., Abdessamad Tridane, Yang Kuang An epidemic model with post-contact prophylaxis of distributed length. I. Thresholds for disease persistence and extinction.

• J. Biol. Dyn. 2 (2008), 221-239.

An epidemic model with post-contact prophylaxis of distributed length. II. Stability and oscillations if treatment is fully effective.

• Math. Model. Nat. Phenom. 3 (2008), 267-293.

Smith, H.L., H.R. Thieme, Dynamical Systems and Population Persistence Graduate Studies in Mathematics 118 AMS Providence 2011

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Conclusion

Prophylactic post-contact treatment can lead to undamped oscillations, but only for rather extreme parameter values. If the treatment length is short or has a convex distribution, the disease dynamics converge to an equilibrium. This model is a gross caricature. However, it has lead us to some clear-cut analytic results. They can now serve as educated conjectures for the simulation of more realistic models.

(ASU) Tsing-Hua Univ, May 2011 31 / 31