1. The interplay of infectivity that decreases with virulence with limited cross-immunity (toy) models for respiratory disease evolution Hans (= J A J * ) Metz (formerly ADN ) IIASA VEOLIA- Ecole Poly- technique & Mathematical Institute, Leiden University

2. prelude: the Anderson-May framework Starting point: Equilibrium: Initial per capita growth rate of mutant with trait value y : ( Implicit assumption: full cross-immunity.) †††† R 0 : expected disease-lifetime number of infections produced by freshly infected individuals present in negligible numbers in an otherwise infection-free community x : trait S usceptibles I nfecteds R ecovereds

3. prelude: the Anderson-May framework Initial per capita growth rate of mutant with trait value y : This growth rate is positive, and hence y can invade, if, or equivalently Evolution maximises R 0.  Initial per capita growth rate of mutant with trait value y :

4. prelude: the Anderson-May framework Evolution maximises R 0. R0 R0 x x  To get interesting conclusions a negative trade-off is assumed: x x   x x   Evolution maximises R 0.

5. prelude: the Anderson-May framework Limitations : (i)Usually  will depend on x as well. (ii)The assumed life cycle of the disease is extremely simple. (iii)This is even worse for the host. (The optimising property is lost when host death rates are made density dependent !) (iv) The assumed negative trade-off is based on an oversimplified view of the body as one well-mixed compartment: faster growing agent populations do both more harm and produce a larger infective output. In reality the death toll often depends less on the growth capacity of the agent than on its location in the body. (v) Diseases that specialise on different body parts carry different antigens, and hence have limited cross-immunity.

6. The simplest model for the dynamics of diseases with limited cross-immunity c.f. Viggo Andreasen, Juan Lin, Simon A. Levin (1997) The dynamics of cocirculating influenza strains conferring partial cross-immunity J. Math. Biol. 35: 825 - 842

7. immune and disease states of individuals : # of disease strains present in the population at time t, : their trait values. : immune status of an individual, with the power set (i.e., set of all subsets) of. : probability of a p -host becoming ill from an encounter with an i -inoculum: Assumptions: (i) immunity only affects initial infection, (ii) immunity profile does not wane, (iii) infection order does not matter.

8. cross-immunity: ‘ all doors should be open’ A more specific assumption is: with the ‘crossimmunity profile’  a smooth function satisfying for, for.

9. population state : density of healthy individuals (or hosts) with immune status p, : density of individuals suffering from disease i with immune status. : force of i -infection. : primary infection rate constant, : disease related mortality rate, : recovery rate. & disease parameters

10. uninfected population dynamics : total birth rate, : per capita death rate

11. endemic dynamics neglect multiple infections : force of i -infection.

12. Interlude: the adaptive dynamics toolbox

13. community dynamical background  Populations are considered as measures over a space ot i(ndividual)-states (e.g. spanned by age and size).  Environments are delimited such that given their environment individuals are independent,  and hence their mean numbers have linear dynamics.  Resident populations are assumed to be so large that we can approximate their dynamics deterministically.  These resident populations influence the environment so that they do not grow out of bounds.  The resulting dynamical systems therefore have attractors, which are assumed to produce ergodic environments.

14. mutant population dynamics: fitness   Mutants enter the population singly.  Therefore, initially their impact on the environment can be neglected.   The initial growth of a mutant population can be approximated with a branching process.  Invasion fitness is the (generalised) Malthusian parameter (= averaged long term exponential growth rate of the mean) of this proces. (Existence guaranteed by the multiplicative ergodic theorem.)   Residents have fitness zero.

15.  as dominant transversal eigenvalue resident population size population sizes of other species mutant population size

16. resident population size population sizes of other species mutant population size or transversal natural Lyapunov exponent

17. trait substitutions C : = {X 1,..,X k } : trait values of the residents E nvironment: E attr (C) Y : trait value of mutant Fitness (rate of exponential growth in numbers) of mutant:  s (Y | C) : =  ( Y | E attr (C)) * Y has a positive probability to invade into a C community iff s (Y | C) > 0. * After invasion, X i can be ousted by Y only if s( X i | X 1,.., Y,.., X k ) ≤ 0. * For small mutational steps Y takes over, except near so-called “ess”es.

18.  Evolution proceeds through uphill movements in a fitness landscape that keeps changing so as to keep the fitness of the resident types at exactly zero. resident trait value(s) x evolutionary time fitness landscape picture of evolution 0 0 0 fitness landscape:  (Y,E(t)) mutant trait value y 0 0

19. resident trait value(s) x evolutionary time 0 0 0 fitness landscape:  (Y,E(t)) mutant trait value y 0 0 i.a. branching points and Evolutionarily Steady States evolutionarily singular strategies (ess-es), fitness landscape picture of evolution  Organisers of landscape change:

20. essential for most conclusions i.e., separated population dynamical and mutational time scales: the population dynamics relaxes before the next mutant comes 1. mutation limited evolution 2. clonal reproduction 3. good local mixing 4. largish system sizes 5. “good” c (ommunity) -attractors 6. interior c-attractors unique 7. fitness smooth in traits 8. small mutational steps essential conceptuallly essential underlying simplifying assumptions

21. t   ,  rescale time, only consider traits    rescale numbers to densities  = system size,  = mutations / birth the associated limit trait value x classical large number limit individual-based simulation adaptive dynamics limit For comparison:

22. y x + + - - - fitness contour plot x: resident y: potential mutant x x0x0 x1x1 x1x1 x2x2 Pairwise Invasibility Plot PIP trait substitution sequences x

23. Trait Evolution Plot TEP Pairwise Invasibility Plot PIP y x X X 1 x x2x2 trait substitution sequences + + - -

24. Pairwise Invasibility Plot PIP x Trait Evolution Plot TEP x2x2 trait substitution sequences y x + + - -

25. the evolutionarily singular strategies

26. evolutionary time t i m e t r a i t fitness minimum population geometry of adaptive branching

27. interrupted : branching prone (  trimorphically repelling) In a thought experiment where the other type does not mutate an isocline corresponds to a locus of monomorphic singular points matryoska principle

28. consistency conditions  There also exist various global consistency relations: The classification of the singular points was based on just a smoothness assumption and some ecologically reasonable consistency conditions. Use that on the boundaries of the coexistence set one type is extinct.

29. the bifurcation ESS  branching point

30. Many of the results discussed so far hold good just as well for higher dimensional trait spaces. Usually more traits evolve simultaneously, i.e., trait spaces have more than one dimension. more than one trait variable (Of course, some care is needed in how one generalises !)

31. The evolutionary model

32. nose alveoli xx 0 1 R0: R0: x x  0 1 evolutionary ingredients primary infection rate constant: disease induced mortality rate: recovery rate:

33. further ingredients cross-immunity: y-x y-x 

34. monomorphic populations endemic dynamics: 0 0 00 0 0 equilibrium:  for : with and S: I: R:S: I: R:

35. monomorphic ess-s, general theory X * is an evolutionarily singular strategy (ess) if A scalar ess x * is an Evolutionary Steady Strategy (ESS) if and a branching point if

36. monomorphic populations with etc. and Use and.

37. monomorphic populations Equation for the monomorphic ess:    Theorem: The maximima of R 0 attract, the minima repel.

38. monomorphic populations For scalar traits:  Branching point when

39. polymorphic populations endemic dynamics: For given these equations can be solved succesively to give  0 0 0 0 0 0 equilibrium:

40. polymorphic populations For k = 2 : H I H I H † † † †† etc.

41. polymorphic populations Consider the recurrence: Theorem: (i)The recurrence has the same equilibria as the full population dynamics. (ii) The equilibria of the full population dynamics are invasible by a new type Y if and only if this is the case for the recurrence. Observation: The recurrence is similar to an ecological competition model.

42. polymorphic populations Theorem: For k = 2 : For all all solutions of the recurrence converge to the same stable equilibrium. This equilibrium is internal if and only if both boundary equilibria are invadable. The latter equilibria are unique and cannot both be uninvadable.

43. some numerical results increasing c  R0R0 x  increasing c  R0R0 x 

44. increasing c  R0R0 x  some numerical results increasing c 

45. some numerical results increasing c  R0R0 x 

46. some numerical results increasing c 

47. some numerical results

49. aside: Lotka Volterra resource competition models resource distribution, expressed as the resulting carrying capacity, K scaled trait: x competition kernel, a, derived from the capacities to exploit and deplete (scaled) trait difference c

50. increasing c  aside: Lotka Volterra resource competition models

51. more trait variables, some speculations x 2  x1x1 x1x1 R 0  0 1 1 1 the further x 2 is from its optimal value the worse the disease performs: R 0 =1

52. The end Thank you for your attention ! Kevin Kleine Juan Keymer Vergara

54. Prelude: classical theory of virulence evolution

56.  For mutants the environment is set by the population dynamics of the resident types { X 1,.., X k } =: C.  The fitness  of a given type Y in a given stationary environment E can be defined as the exponential growth rate of a clone of individuals of that type in that environment. short adaptive dynamics refresher Invasion fitness: (asymptotic ) (hypothetical) Note that (1) as fitness is measured here on a logarithmic scale, zero is neutral, (2) residents have fitness zero. ~ dominant Lyapunov exponent (Furstenberg & Kesten, Oseledets) s ( Y | X ) : =  ( Y | E C ), average

57.  Fitnesses are not given quantities, but depend on (1) the traits of the individuals, X, Y, (2) the environment in which they live, E : (Y,E) |   (Y | E)  E is set by the resident community : E = E attr (C), C={X 1,...,X k ) fitnesses cause and change with evolution Evolutionary change is mainly determined by the fitnesses of potential mutants.

58. endemic dynamics,, neglect multiple infections etc. ††††

59. some numerical results increasing c  R0R0 x  increasing c  R0R0 x 

60. some numerical results increasing c  R0R0 x 

61. some numerical results increasing c  R0R0 x 

62. some numerical results increasing c 