Spread of Viral Plaque

Transcription

1 Spread of Viral Plaque Don Jones, Gergely Rost, Hal Smith, Horst Thieme A R I Z O N A S T A T E U N I V E R S I T Y Everything Disperses to Miami () Spread of Viral Plaque December / 2

2 Outline 1 Bacteriophage and Plaques 2 Prior work on Plaque spread 3 Model I: Latent Period of Fixed-Duration 4 Model II: distributed latent period and virus removal 5 Reduction of the Model System to a Single Equation 6 Traveling Wave of Virus Infection 7 Conclusions Everything Disperses to Miami () Spread of Viral Plaque December / 2

3 Bacteriophage and Plaques Phage Life Cycle: adsorption to lysis Latent Period: time from adsorption to burst 2 4 min. Burst size: 1-1 virus. Everything Disperses to Miami () Spread of Viral Plaque December / 2

4 Plaque Assay Bacteriophage and Plaques The plaque technique of virus assay has played an important role in the development of knowledge of the physiology and genetics of viruses. For bacteriophage the technique is quite simple and consists of adding a large number of susceptible bacteria and a few virus particles to a tube containing melted nutrient agar, which is then poured on a Petri plate that already contains a basal layer of nutrient agar. The virus adsorbs to the host bacteria, multiplies, and lyses the bacterial cell; the progeny viruses diffuse to neighboring bacterial cells and multiply further, yielding holes or plaques in the otherwise continuous sheet of bacterial growth." (A.L. Koch: JTB 1964) Everything Disperses to Miami () Spread of Viral Plaque December / 2

5 Prior work on Plaque spread Previous Work on Spreading Plaque Koch (JTB 1964) proposes that ( virus diffusion constant speed of plaque spread latent period ) 1/2 Yin & McCaskill (BioPhysics 1992) propose the model: V = virus, B = susceptible bacteria, I = infected bacteria. in R 2 with initial conditions: V = V t = d(v rr + 1 r V r) k + VB + (k 2 β + k )I B t = k + BV + k I, I t = k + BV (k + k 2 )I ( ) V, r r, B =, r > r ( ), r r, I = B, r > r Everything Disperses to Miami () Spread of Viral Plaque December / 2

6 Prior work on Plaque spread Previous Work on Spreading Plaque Koch (JTB 1964) proposes that ( virus diffusion constant speed of plaque spread latent period ) 1/2 Yin & McCaskill (BioPhysics 1992) propose the model: V = virus, B = susceptible bacteria, I = infected bacteria. in R 2 with initial conditions: V = V t = d(v rr + 1 r V r) k + VB + (k 2 β + k )I B t = k + BV + k I, I t = k + BV (k + k 2 )I ( ) V, r r, B =, r > r ( ), r r, I = B, r > r Everything Disperses to Miami () Spread of Viral Plaque December / 2

7 Model I: Latent Period of Fixed-Duration Our First Model An infected cell remains so for τ time units, then lyses, releasing β > 1 phage. First latent period: t τ V t = d V kvb B t = kbv, x D I t = kbv with initial data: V(, x) = V (x), B(, x) = B (x), I(, x) =. For t > τ: V t = d V kv(t, x)b(t, x) + βkb(t τ, x)v(t τ, x) B t = kb(t, x)v(t, x) I t = kb(t, x)v(t, x) kb(t τ, x)v(t τ, x) Everything Disperses to Miami () Spread of Viral Plaque December / 2

8 Model I: Latent Period of Fixed-Duration Our First Model An infected cell remains so for τ time units, then lyses, releasing β > 1 phage. First latent period: t τ V t = d V kvb B t = kbv, x D I t = kbv with initial data: V(, x) = V (x), B(, x) = B (x), I(, x) =. For t > τ: V t = d V kv(t, x)b(t, x) + βkb(t τ, x)v(t τ, x) B t = kb(t, x)v(t, x) I t = kb(t, x)v(t, x) kb(t τ, x)v(t τ, x) Everything Disperses to Miami () Spread of Viral Plaque December / 2

9 Model I: Latent Period of Fixed-Duration Spreading Phage Plaque: β = 1, kb τ = V(t, x)/b (β 1) is plotted. B(, x) B const. Everything Disperses to Miami () Spread of Viral Plaque December / 2

10 Model I: Latent Period of Fixed-Duration Bacteria are infected and lysed B(t, x)/b is plotted. Everything Disperses to Miami () Spread of Viral Plaque December / 2

11 Model I: Latent Period of Fixed-Duration Theoretical vs Simulated Spread Speed 7 6 Tube Disk C * C O 5 C K scaled c vs K = kb τ in black Everything Disperses to Miami () Spread of Viral Plaque December / 2

12 Model I: Latent Period of Fixed-Duration Ingredients of Model II: infection age and virus removal 1.5 Latent Period Distribution Probability not lysed a F(a) is the probability that an infected bacterium has not yet lysed a time units after infection. b(a) is number of virus progeny released when an infected cell lyses a time units after infection. infected bacteria given by I(t, x) = i(t, a, x)da where i(, a, x) = i (a, x) and i(t, a, x) is the infected cell density w.r.t. (a, x). Everything Disperses to Miami () Spread of Viral Plaque December / 2

13 Model I: Latent Period of Fixed-Duration Ingredients of Model II: infection age and virus removal 1.5 Latent Period Distribution Probability not lysed a F(a) is the probability that an infected bacterium has not yet lysed a time units after infection. b(a) is number of virus progeny released when an infected cell lyses a time units after infection. infected bacteria given by I(t, x) = i(t, a, x)da where i(, a, x) = i (a, x) and i(t, a, x) is the infected cell density w.r.t. (a, x). Everything Disperses to Miami () Spread of Viral Plaque December / 2

14 Model I: Latent Period of Fixed-Duration Ingredients of Model II: infection age and virus removal 1.5 Latent Period Distribution Probability not lysed a F(a) is the probability that an infected bacterium has not yet lysed a time units after infection. b(a) is number of virus progeny released when an infected cell lyses a time units after infection. infected bacteria given by I(t, x) = i(t, a, x)da where i(, a, x) = i (a, x) and i(t, a, x) is the infected cell density w.r.t. (a, x). Everything Disperses to Miami () Spread of Viral Plaque December / 2

15 Model II: distributed latent period and virus removal Model-II: variable latent period, virus removal rate V t = d V αv + J(t, x) + k B t = kbv, x R n, t > I = t t kb(t a, x)v(t a, x)f(a)da + b(a)b(t a, x)v(t a, x)( df)(a) where df is the Stieltjes measure associated with 1 F, α > is virus removal rate due to adsorption and decay. J(t, x) = t b(a) i (a t, x) F(a t) ( df)(a) t F(a) i (a t, x) F(a t) da is virus released when survivors of the initial infected cell cohort burst. Typically, i is so small that it can be ignored. Everything Disperses to Miami () Spread of Viral Plaque December / 2

16 Model II: distributed latent period and virus removal Simulations use a Gamma-Distributed Latent Period F(a) = a g m (s; r)ds g m (s, r) = r m s m 1 (m 1)! e rs Plot of g m (s,r) m=r=2 m=r=3 m=r=4 m=r=5 mean latent period m/r.1 variance m/r s Everything Disperses to Miami () Spread of Viral Plaque December / 2

17 Model II: distributed latent period and virus removal Simulations: virus at leading edge of plaque disk m=2 m=3 m=infinity 6 m=2 m=3 m=infinity Phage (Density) 4 Phage (Density) Length Length Virus at T = 5 Virus at T = 15 The mean latent period is 1. while its variance is.5 (solid lines),.33 (dashed lines), and (hashed lines). k = 1, α = 2, b = 3. Everything Disperses to Miami () Spread of Viral Plaque December / 2

18 Reduction of the Model System to a Single Equation Reduction of System to a Single Scalar PDE Let u(t, x) = t V(s, x)ds. Then, as B t = kbv, we have B(t, x) = B e ku(t,x). Integrating the V equation w.r.t. t, we find that: where and where t u t = d u αu + ˆV (t, x) + kb b(a)f(u(t a, x))( df)(a) t ˆV (t, x) = V (x) + J(s, x)ds f(u) = 1 e ku. k Note that f is bounded and f() =, f () = 1, and f(u) < u for all u >. Everything Disperses to Miami () Spread of Viral Plaque December / 2

19 Reduction of the Model System to a Single Equation Reduction to an Integral Equation Using the heat kernel Γ(t, x) and variation of constants formula t u(t, x) = u (t, x) + kb Φ(r, y)f(u(t r, x y))drdy R n where Φ(r, y) = and u (t, x) = t r e α(r a) Γ(r a, y)b(a)( df)(a) e αs Γ(s, y)ˆv (t s, x y)dyds. R n Everything Disperses to Miami () Spread of Viral Plaque December / 2

20 Reduction of the Model System to a Single Equation Virus Spreading Speed & Reproductive Number Spreading Speed: c = inf{c : λ >, G(c, λ) < 1} G(c, λ) = kb e λ(cs+y1) Φ(s, y)dyds R n c can be expressed as c = dc 1, where (c 1, λ 1 ) is unique solution of G 1 (c, λ) = 1, d dλ G 1(c, λ) = G 1 (c, λ) = kb λc + α λ 2 e λca b(a)d( F(a)). c > Basic Reproductive Number for Virus: R kb α b(a)d( F(a)) > 1. Everything Disperses to Miami () Spread of Viral Plaque December / 2

21 Reduction of the Model System to a Single Equation Virus Spreading Speed & Reproductive Number Spreading Speed: c = inf{c : λ >, G(c, λ) < 1} G(c, λ) = kb e λ(cs+y1) Φ(s, y)dyds R n c can be expressed as c = dc 1, where (c 1, λ 1 ) is unique solution of G 1 (c, λ) = 1, d dλ G 1(c, λ) = G 1 (c, λ) = kb λc + α λ 2 e λca b(a)d( F(a)). c > Basic Reproductive Number for Virus: R kb α b(a)d( F(a)) > 1. Everything Disperses to Miami () Spread of Viral Plaque December / 2

22 Reduction of the Model System to a Single Equation Virus Spreading Speed & Reproductive Number Spreading Speed: c = inf{c : λ >, G(c, λ) < 1} G(c, λ) = kb e λ(cs+y1) Φ(s, y)dyds R n c can be expressed as c = dc 1, where (c 1, λ 1 ) is unique solution of G 1 (c, λ) = 1, d dλ G 1(c, λ) = G 1 (c, λ) = kb λc + α λ 2 e λca b(a)d( F(a)). c > Basic Reproductive Number for Virus: R kb α b(a)d( F(a)) > 1. Everything Disperses to Miami () Spread of Viral Plaque December / 2

23 Reduction of the Model System to a Single Equation Spreading Theorem If R > 1 and ˆV is bounded & continuous, vanishes for {(t, x) : x η, t }, then lim u(t, x) =, c t, x ct > c Further, if u is the unique positive solution of Then, for every c (, c ), u = R ( 1 e ku k ). lim inf u(t, x) u t, x ct If R 1, then u(t, x), x, uniformly in t. Proof uses results of Thieme (JMB 1977,79) for integral equations: sub- & super solutions Everything Disperses to Miami () Spread of Viral Plaque December / 2

24 Reduction of the Model System to a Single Equation Spreading Theorem If R > 1 and ˆV is bounded & continuous, vanishes for {(t, x) : x η, t }, then lim u(t, x) =, c t, x ct > c Further, if u is the unique positive solution of Then, for every c (, c ), u = R ( 1 e ku k ). lim inf u(t, x) u t, x ct If R 1, then u(t, x), x, uniformly in t. Proof uses results of Thieme (JMB 1977,79) for integral equations: sub- & super solutions Everything Disperses to Miami () Spread of Viral Plaque December / 2

25 Traveling Wave of Virus Infection Traveling Wave Solutions Theorem: Assume that R > 1 and c > c. Then there exists solutions V(x + ct) and B(x + ct) of satisfying V t = dv xx αv + k B t = kbv b(a)b(t a, x)v(t a, x)d( F(a)) B( ) = B, B(+ ) = B e ku, V(± ) =. B and V are positive and B is strictly decreasing. Moreover, the total amount of virus in the wave is given by V(s)ds = cu = cr f(u ). R There is no non-trivial traveling wave with speed c < c. Everything Disperses to Miami () Spread of Viral Plaque December / 2

26 Conclusions Conclusions 1 Formulated a model of phage spread on immobile bacteria where: 1 length of latent period has prescribed distribution. 2 burst size depends on length of latency. 2 Identified basic reproductive number, R, for virus propagation on uniform bacterial lawn. 3 Showed that realistic initial data give rise to spreading solutions if R > 1. 4 No spreading if R < 1. 5 Identified spreading speed c of virus. 6 Proved existence of a traveling wave of virus infection for each speed c c ; no wave if c < c. Everything Disperses to Miami () Spread of Viral Plaque December / 2

27 Conclusions Thanks For Your Attention D. Jones, G. Rost, H.S.,H. Thieme, On Spread of Phage Infection of Bacteria in a Petri Dish, SIAM J. Applied Math., (212) Vol.72, No.2, D. Jones, H.S.,H. Thieme, Spread of Viral Infection of Immobilized Bacteria, to appear, Networks and Heterogeneous Media Analysis makes use of: H.R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biology 4, (1977), O. Diekmann, Limiting behavior in an epidemic model, Nonlin. Anal. 1, (1977), H.S. supported by NSF Grant DMS Everything Disperses to Miami () Spread of Viral Plaque December / 2