Apparent paradoxes in disease models with horizontal and vertical transmission

Transcription

1 Apparent paradoxes in disease models with horizontal and vertical transmission Thanate Dhirasakdanon, Stanley H.Faeth, Karl P. Hadeler*, Horst R. Thieme School of Life Sciences School of Mathematical and Statistical Sciences Arizona State University now Department of Mathematics and Statistics, University of Helsinki now Department of Biology, University of North Carolina at Greensboro partially supported by NSF grant DEB , *NSF grant DMS , NSA grant H and the ASU-Sloan National Pipeline Program in Mathematical Sciences, NSF grant and NSF grant (ASU) Tsing-Hua Univ., May / 39

2 The mystery The victim: Arizona grass Festuca arizonica, Arizona fescue The perpetrator: endophytic fungus Neotyphodium The detective: Stan Faeth, SoLS, ASU Transmission seems completely vertical. Vertical transmission is not perfect. Infection seems harmful though not much. How does the fungus persist? (ASU) Tsing-Hua Univ., May / 39

3 The paradox Hypothesis: Horizontal transmission makes up for imperfect vertical transmission, but is hard to detect at endemic equilibrium. Paradox: Horizontal transmission is the harder to detect the higher the horizontal transmission coefficient. (ASU) Tsing-Hua Univ., May / 39

4 Parasite-free host dynamics N density of host population N N = β(n) µ(n). The per capita birth rate β(n) is decreasing in N. The per capita mortality rate µ(n) is increasing in N. reproduction ratio β(n) µ(n) ց 0 strictly as N ր. β(0) µ(0) > 1: There exists a unique host carrying capacity K with β(k) µ(k) = 1. (ASU) Tsing-Hua Univ., May / 39

5 Some crucial parameters Key parameter: σ horizontal transmission coefficient prob. of infection if one spore lands on a plant C(N) rate at which spores produced by one average infective plant land on another plant if plant density is N. The per capita contact rate C(N) is increasing in N. C(N) does not depend on N: standard (frequency-dependent) incidence, C(N) is proportional to N: mass action (density-dependent) incidence. (ASU) Tsing-Hua Univ., May / 39

6 functional forms for contact Data for human diseases from England and the U.S. suggest that C(N)/N is a decreasing function of N. C(N) = ζn ν with 0 ν < 1 (Anderson 1982) C(N) = ζn 1 + νn (Dietz 1982) C(N) = ζ(1 e νn ) (Roughgarden 1979) C(N) = κ 1 N 1 + κ 2 N κ 2 N complex-formation approach (Heesterbeek Metz 1993) For insect diseases, C(N) = ζ ln(1 + νn) (Briggs Godfray 1995) (ASU) Tsing-Hua Univ., May / 39

7 Cross-protection All rate functions are strictly positive and cont. diff. cross-protection: a second infection of a host does not add to the first infection (even if the mode of infection is different) (ASU) Tsing-Hua Univ., May / 39

8 A model with one parasite strain S I h I v I = I h + I v N = S + I susceptible hosts horizontally infected hosts vertically infected hosts infected hosts host population S = ( S + q(1 p)i ) β(n) µ(n)s σc(n)si, N I h =σc(n)si N µ(n)i h αi h, I v =qpiβ(n) µ(n)i v αi v. (ASU) Tsing-Hua Univ., May / 39

9 Model reformulated N =N ( β(n) µ(n) ) I ( (1 q)β(n) + α ), I I =σc(n)(n I) N complete mathematical analysis by Zhou, Hethcote (1994) + qpβ(n) µ(n) α. Reformulation in terms of N and f = I/N. All solutions converge towards an equilibrium. Lipsitch, Nowak, Ebert, May (1995) computational, C(N) = N (mass action transmission) (ASU) Tsing-Hua Univ., May / 39

10 Thresholds for disease persistence Threshold σ 1 > 0: Case 1. 0 σ σ 1 : disease dies out H(N) = qβ(n) µ + α host reprod. ratio at host density N if all hosts are infected. Case 2. σ > σ 1, C(0) = 0 or H(0) 1: persistence equil. which attracts all solutions with I(0) > 0. Case 3. C(0) > 0 and H(0) < 1. Threshold σ 2 > σ 1 : σ 1 < σ < σ 2 : PE which attracts all sols with I(0) > 0. σ σ 2 : parasite drives host into extinction. (ASU) Tsing-Hua Univ., May / 39

11 The paradox in formulas Assume: some harm, q < 1 or α > 0. VT not perfect, 0 < p < 1. At the endemic equilibrium N is a strictly decreasing function of σ. f = I /N is a strictly increasing function of σ. Ih Iv ց 1 p p, σ ր. As disease prevalence increases, there is less room for horizontal transmission. (ASU) Tsing-Hua Univ., May / 39

12 Transient dynamics r = I h I v fraction of horizontally to vertically infected indiv. ( ) r = (1 + r) σc(n)(1 f) pqrβ(n). Assume that (N(0), f(0)) (K, 0) disease-free equil. r moves to a transient quasi-steady state close to r = σc(k) pqβ(k) (increases in σ). Eventually r converges to the steady state r < r, r = σc(n )(1 f ) pqβ(n ) (decreases in σ). (ASU) Tsing-Hua Univ., May / 39

13 From one to two strains The first model explains why HT is hard to detect in the field, but not why it is hard to observe in the lab. Two strains one completely vertically transmitted: another horizontally (& vertically) transmitted: VT strain HT strain Cross-protection: Infection with one strain protects against infection with the other strain. (ASU) Tsing-Hua Univ., May / 39

14 A model with two parasite strains I 1 I 2 infected by HT strain infected by VT strain N =S + I 1 + I 2, ( 2 ) S = S + q k (1 p k )I k β(n) µ(n)s C(N)S N σi 1, k=1 I 1 I 1 = C(N)S N σ + q 1p 1 β(n) µ(n) α 1, I 2 I 2 =q 2 p 2 β(n) µ(n) α 2. (ASU) Tsing-Hua Univ., May / 39

15 Virulence vertical transmission not perfect: p 2 < 1. VT strain dies out in absence of HT strain. If there is an endemic coexistence equilibrium (ECE), then its host population size N is determined by and does not depend on σ. 1 = R 2 (N ) := q 2p 2 β(n ) µ(n ) + α 2 Assume: the VT strain is less virulent than the HT strain: α 1 α 2 + (q 2 p 2 q 1 )β(n ) > 0. (ASU) Tsing-Hua Univ., May / 39

16 The paradox for two strains A unique coexistence equilibrium exists iff σ σ. relation to other thresholds: σ 1 < σ Case 3: σ < σ 2 I1 I2 is a strictly decreasing function of σ. No competitive exclusion between HT strain and VT strain. Already computationally observed for C(N) = N by Lipsitch, Siller, Nowak (1996) adds to the examples by Ackleh, Allen (2003, 2005), Andreasen, Pugliese (1995) (ASU) Tsing-Hua Univ., May / 39

17 Persistence If the ECE exists, we have uniform strong dynamical coexistence: There exists some ǫ > 0 such that lim inf t I j(t) ǫ, j = 1,2, for every solution with I j (0) > 0, j = 1,2. The VT strain persists in the presence of the HT strain by protecting the host against the more virulent HT strain. Conjecture: The coex equil. is globally as. stable if σ is suff. large. (ASU) Tsing-Hua Univ., May / 39

18 Protection in detail Recall: VT strain goes extinct in absence of the HT strain. Case 3, σ > σ 2 The HT strain drives the host into extinction in the absence of the VT strain, while the host persists when both strains are present. Case 3, σ < σ < σ 2 or Case 2, σ < σ The host persists whether or not the VT strain is present, but its equilibrium density is lower and decreases in σ when the VT strain is absent. (ASU) Tsing-Hua Univ., May / 39

19 Towards the persistence proof: fractions of infectives N N =(β(n) µ(n) 2 f 1 = I 1 N, f 2 = I 2 N, k=1 f k ( (1 qk )β(n) + α k ), f 1 =σc(n)(1 f 1 f 2 ) (1 q 1 p 1 )β(n) α 1 f 1 2 ( ) + f k (1 qk )β(n) + α k, k=1 f 2 f 2 = (1 q 2 p 2 )β(n) α ( ) f k (1 qk )β(n) + α k. k=1 (ASU) Tsing-Hua Univ., May / 39

20 remarks on persistence equilibria: ECE (N,f 1,f 2 ), boundary equilibria: (0, 0, 0), (K, 0, 0), possible boundary equilibrium: (N,f 1,0) C(0) > 0: more possible boundary equilibria (0,f # 1,0), (0,f 1,f 2 ) Special case: β and C independent of N. Then the ECE is globally asymptotically stable for {N > 0,f 1 > 0,f 2 > 0}. (ASU) Tsing-Hua Univ., May / 39

21 The acyclicity approach to persistence Butler, Freedman, Waltman 1986 (ODE) Hofbauer, So 1989 (difference equations) Hale, Waltman 1989 (semiflows in infinite dimensions) Hal. L. Smith, Horst R. Thieme Dynamical Systems and Population Persistence Graduate Studies in Mathematics American Mathematical Society 2011 (ASU) Tsing-Hua Univ., May / 39

22 prepare for persistence theory Consider a differential equation x = F(x), where F : X R n is locally Lipschitz continuous, X closed subset of R n. Let x R n + be an equilibrium, F(x ) = 0, and x Y R n +. Obviously {x } is an invariant set. {x } is called an isolated invariant set in Y if there exists some δ > 0 such that Y B δ (x ) contains no invariant set except {x }. Example: A saddle is an isolated invariant set. (ASU) Tsing-Hua Univ., May / 39

23 acyclicity An equilibrium x Y is called chained in Y to an equilibrium y in Y, x Y y, if there exists a solution x = F(x), x : R Y, such that x(t) x as t and x(t) y as t and there exists some t R such that x(t) x and x(t) y. A set M of equilibria in Y is called cyclic in Y if there exists some x M with x Y x or if there exist x 1,...,x k in M such that x 1 Y x 2 Y Y x k Y x 1, M acyclic in Y if M is not cyclic in Y. (ASU) Tsing-Hua Univ., May / 39

24 Persistence and repellers persistence function ρ : X R + continuous, not identically zero. X 0 = {x X; x = x(0), ρ(x(t)) = 0 t 0} Uniform ρ-persistence: There exists some ǫ > 0 such that x(0) X,ρ(x(0)) > 0 = lim inf t ρ(x(t)) ǫ. Y X uniformly weakly ρ-repelling if there exists some ǫ > 0 such that x(0) X,ρ(x(0)) > 0 = lim sup d(x(t),y ) ǫ. t (ASU) Tsing-Hua Univ., May / 39

25 Persistence Theorem THEOREM. Let X R n closed, F : X R n be locally Lipschitz continuous. Assume that X is forward invariant and x = F(x) is dissipative on X. Let ρ : X R +, X 0 X. Assume there are no solutions x and no t > s > 0 such that ρ(x(0)) > 0, ρ(x(s)) = 0 < ρ(x(t)). Let M be a finite set of equilibria in X 0 such that every solution in X 0 converges to one of the equilibria in M, every equilibrium in M is an isolated invariant set in X 0, M is acyclic in X 0. Then uniformly ρ-persistence if and only if every equilibrium in M is uniformly weakly ρ-repelling. (ASU) Tsing-Hua Univ., May / 39

26 Application for C(0) = 0 X =R 3 + = [0, ) 3, ρ(n,f 1,f 2 ) = min{n,f 1,f 2 }. X 0 = {(N,f 1,f 2 ) R 3 + ; N = 0 or f 1 = 0 or f 2 = 0}. M = {(0,0,0),(K,0,0), (N,f 1,0)} N(0) = 0 convergence to (0,0,0). N(0) > 0, f 1 (0) = 0 convergence to (K,0,0). N(0) > 0,f 1 (0) > 0,f 2 (0) = 0 convergence to (N,f 1,0). (ASU) Tsing-Hua Univ., May / 39

27 Stability of the coexistence equilibrium The ECE (provided it exists) is loc. as. stable if one of the following holds: σ is very large or just a little larger than the threshold σ β does not depend on N mass action incidence the HT strain does not reduce the fertility of the host more than the VT strain (but causes more disease deaths): q j > 0 and p j < 1 for j = 1,2 and q 2 p 2 q 1, α 1 α 2 + (q 2 p 2 q 1 )β(n ) > 0. (ASU) Tsing-Hua Univ., May / 39

28 Unstable endemic coexistence equilibrium ECE unstable in an oscillatory way (the associated Jacobian matrix has one negative eigenvalue and two complex conjugate eigenvalues with positive real part), if all of the following hold: Standard (alias frequency-dependent) incidence, and the per capita birth rate β depends on the population density N (but not too strongly), and the per capita mortality rate µ depends on the population density N only very weakly, and the VT strain is almost perfectly vertically transmitted and causes very little fertility reduction, and the HT strain is almost perfectly vertically transmitted or causes almost complete sterilization. (ASU) Tsing-Hua Univ., May / 39

29 Stability: coefficient of horizontal transmission Figure: Frequency-dependent incidence, left (σ, α 1 ), right (σ, α 2 ). (ASU) Tsing-Hua Univ., May / 39

30 Stability: vertical transmission I Figure: Curves of stability change for frequency-dependent incidence, left (p 1, q 1 ), right (p 2, q 1 ). (ASU) Tsing-Hua Univ., May / 39

31 Stability: vertical transmission II Figure: frequency-dependent incidence, left (p 1, q 2 ), right (p 2, q 2 ). (ASU) Tsing-Hua Univ., May / 39

32 Stability: density dependent incidence Figure: The graphs of N (blue), I 1 (green), and I 2 (red). (ASU) Tsing-Hua Univ., May / 39

33 Instability: frequency dependent incidence Figure: The graphs of N (blue), I 1 (green), and I 2 (red). (ASU) Tsing-Hua Univ., May / 39

34 Density versus frequency dependent incidence Figure: Phase plane projection of f1 and f2. (ASU) Tsing-Hua Univ., May / 39

35 Remarks undamped oscillations in a model of SI type three ODEs only two strains only frequency dependent rather than density dependent incidence The oscillations are driven by fertility reduction through the disease. (ASU) Tsing-Hua Univ., May / 39

36 Oscillations in multiple strain models SIR with three strains and nine ODEs and mass action incidence and partial cross-immunity (Lin Andreasen Levin 1999) SIR with two strains and age structure (PDE) and mass-action incidence and partial cross-immunity, (Andreasen 1989, Castillo-Chavez Hethcote Andreasen Levin Liu 1989) SI with two strains, three ODEs, and mass action incidence and ongoing mutation from the first to the second strain (Li Zhou Ma Hyman 2004) (ASU) Tsing-Hua Univ., May / 39

37 Oscillations: density versus frequency dependence For density but not for frequency dependent incidence SEI (Pugliese 1991, Gao Mena-Lorca Hethcote 1996, Greenhalgh 1997) SIS with two hosts and one parasite (Hethcote Wang Li 2005) Two competing host species, only afflicted by a parasite, SI type (van den Driessche Zeeman 2004, Venturino 2001) For incidence SI a+s+i, a > 0, but not for frequency dependent incidence SI with pair formation (Diekmann Kretzschmar 1991), For frequency but not density dependent incidence SI model for HIV with infection-age dependent infectivity: (Thieme Castillo-Chavez 1993, Milner Pugliese 1999) (ASU) Tsing-Hua Univ., May / 39

38 References Stanley H. Faeth, Karl P. Hadeler, HRT, An apparent paradox of horizontal and vertical disease transmission, J. Biol. Dynamics 1 (2007), Thanate Dhirasakdanon, HRT, Persistence of vertically transmitted parasite strains which protect against more virulent horizontally transmitted strains, Modeling and Dynamics of Infectious Diseases (Z. Ma, Y. Zhou, J. Wu, eds.), , World Scientific, Singapore 2009 Thanate Dhirasakdanon, HRT, Stability of the endemic coexistence equilibrium for one host and two parasites, Mathematical Modelling of Natural Phenomena 5 (2010), (ASU) Tsing-Hua Univ., May / 39

39 Conclusion Two Strain Model The VT strain dies out in absence of the HT strain. The VT strain persists in the presence of the HT strain by protecting against it. Paradox: At endemic coexistence equilibrium, the ratio decreases as the HT coefficient increases. HT/VT If VT is almost perfect, this ratio is very small if the HT coefficient is very large. Undamped oscillations for frequency-dependent incidence. (ASU) Tsing-Hua Univ., May / 39