1. Disease modeling Maya Groner University of Prince Edward Island with slides from Mark Lewis University of Alberta

2. Why do we Model? Provide new understanding of the ecological and evolutionary processes Assess the weight of evidence for certain hypotheses Project/predict outcomes under different management scenarios Evaluate new management/control methods Design experiments/studies or allocate funding that will further the science Summarize/synthesize large amount of data Identify areas of ignorance

3. Formulation Verification Calibration Analysis and Evaluation Modelling Process Formulation: Objectives: What is system? What are questions? What is the scale and timeframe? How will output be analyzed, summarized and used? Hypotheses: verbal statements that connect objectives and current knowledge. Mathematical formulation: hypotheses are translated in specific quantitative relationships. These form a model that, given inputs, yield an output (prediction or answer to question). Verification: Verify that the method used to solve the model is correct. Calibration: Measure inputs for the model. Analysis and Evaluation: Computation, mathematical analysis etc. is used to answer questions. The answer is validated or corroborated against independent data sets.

4. Formulation Verification Calibration Analysis and Evaluation Haefner (2005) Modelling Process

5. Formulation Verification Calibration Analysis and Evaluation Haefner (2005) Modelling Process

6. Formulation Verification Calibration Analysis and Evaluation Haefner (2005) Modelling Process

7. Formulation Verification Calibration Analysis and Evaluation Haefner (2005) Modelling Process

8. Classical Hypothesis Testing Key elements are a confrontation between a single hypothesis and data. A test data set or “critical experiment” is used to evaluate hypothesis Relies on “doctrine of falsification” (Popper, hypotheses cannot be proved, only disproved). The hypothesis is given as a “null” statement that could possibly be falsified using a test of observed data. Two kinds of errors can arise: Erroneously rejecting the null hypothesis when it is true (Type I) Failure to reject the null hypothesis when it is false (Type II)

9. Classical Hypothesis Testing Advantage: provides a rigorous framework for assessing the weight of scientific evidence for a particular hypothesis Limitations: If we derive a model we cannot reject, then we may believe it is the correct model. It may be that we just need more data to be able to reject. Sequential passes through the modelling process should use really new data sets for testing

10. How do we Fit Parameters to Models? Real data is best, but not always possible Sometimes we use data from other models (e.g., coupled models) If we don’t know, try several values and test the sensitivity of the model to these values

11. Multiple Working Hypotheses Evaluates several competing hypotheses and models simultaneously (in parallel). Models are compared to see which is the most likely. A common auxiliary is simplicity, which is the basis for the Principle of Parsimony or Occam’s Razor. Advantage: multiple hypotheses can be evaluated simultaneously with a single data set. Limitation: if we start off with a poor set of hypotheses, the one we choose as “best” may still be poor.

12. Multiple Working Hypotheses Eg. What are the factors governing movement behaviour of ants foraging for seeds (Haefner and Crist)? – H1: ants are moving randomly, H2: ants rely on memory of places previously foraged, H3: ants rely on pheromone trails, H4: ants rely on memory and pheromones, H5: ants are omniscient and thus forage optimally. – Models: simulations of ants moving according to rules that describe the hypotheses. – Analysis: compare resulting ant movement paths to those observed. – Evaluation: assess which which model “best” describes the movement paths.

18. Types of Models Phenomenological (posits a relationship without delving into the underlying processes driving the relationship) versus mechanistic (describes relationship in terms of the underlying processes). Dynamical (changes with time) versus static. Spatial (changes with location) versus nonspatial Stochastic (uncertainty included) versus deterministic. Some common sorts of models aso include: compartment, stage- structured, transport, particle, finite state.

19. Yersinia pestis bacillus (Science Picture Library) A model for the plague Removed Infectious Susceptible

20. Yersinia pestis bacillus (Science Picture Library) Susceptible Infectious Removed A classic model (SIR)

21. Yersinia pestis bacillus (Science Picture Library) Susceptible Infectious Removed Kermack and McKendrick (1927) A model for the plague

22. Yersinia pestis bacillus (Science Picture Library) Susceptible Infectious Removed Bombay plague of 1905-6 Time (Weeks) Deaths (dR/dt) A model for the plague

23. A key prediction of epidemiological modeling is the disease basic reproduction number, R 0. R 0 > 1 R 0 < 1 Time Number of infectious R 0 =2 In the model for the plague R 0 is defined as the number of secondary infections arising from the introduction of a single infective into an otherwise susceptible population.

25. Use of R0 to understand global warming and disease Harvell et al. 2002

26. Many variations on the SIR model Susceptible-Exposed-Infectious-Recovered (SEIR) Susceptible-Infected (SI) S-I-I-I-I-I-I-R (Waning Infection) Multi-host SIR model (e.g, coupled SIR-SIR model) Etc.

27. Sea lice as a ‘model’ system for modeling

28. What are we modeling?

29. Ricker Model for Stock-recruitment Nonlinear model Carrying capacity Compensation Overcompensation Common in Fisheries Common in Ecology n i (t)=n i (t-2)exp[r - bn i (t-2)] Population growth rateDensity dependent mortality

30. Model for Pink Salmon: Stochastic Ricker Parameter estimate: r=0.62 95% Confidence Interval: (0.55, 0.69) Conclusion: r>0 for this population Stochastic Ricker equation: n i (t)=n i (t-2)exp[r-bn i (t-2)+Z(0,  2 )] r is growth rate r>0 means population will grow r<0 means population will decline Is a given pink salmon population (unexposed to sea lice but exposed to fishing) growing or declining? Log transformed Ricker: log[n i (t)/n i (t-2)] = r – bn i (t-2) +Z(0,  2 ) Can fit to data using least squares

31. Adult Egg Gravid II Gravid I PreadultChalimu s Betwee n- Clutch Larvae (free- swimming) P1P1 P2P2 P3P3 P5P5 P4P4 P6P6 P7P7 G5G5 G7G7 G4G4 G2G2 G3G3 G1G1 F7F7 F5F5 G6G6 Quantifying the role of temperature and mate limitation in sea lice epidemiology Population matrix model

32. Quantifying the role of temperature and mate limitation in sea lice epidemiology Temperature causes dramatic increases in population growth as a result of increases in net reproductive rate and decreases in generation time Groner et al. 2014 PLoS One

33. Quantifying the role of temperature and mate limitation in sea lice epidemiology

34. Identification of sensitive stages 1)What stages should be targeted for control? 2)When should controls be administered?

35. Initial questions Can we use modeling techniques to investigate the optimal ratio of wrasse: salmon? To what extent does using wrasse reduce the need for chemical treatments?

36. Modeling approaches Factors to incorporate Differential Equation Delay-differential Equation Individual-Based Model Sea Lice Growth and Survival Effects of Wrasse (Dynamic process) Bath Treatments (Instantaneous event) Behaviour and individual variation Stochastic processes Solve for equilibriums

37. Individual-based model of sea lice in Anylogic

39. Temperature ⁰ C after meta-analysis data from Stien et al. 2005 Temperature-dependent development

40. Feed at a constant rate Wrasse: salmon ratios 0, 1:200, 1:100, 1:50, 1:25, 1:10 Treatments Fish surveyed weekly. If Mobiles > treatment threshhold, we ‘treat’ Wrasse Predation Additional causes of mortality

41. What is driving infestations? External Source

42. What is driving infestations? Reinfection

43. What is driving infestations?

44. Run the model http://www.runthemodel.com/models/k- oketzEcHJltLMX4KZ5Zs/

45. Time (days) No control of sea lice Wrasse: Salmon is 1:50 Treated if mobiles > 4 Treated if mobiles > 4 Wrasse: Salmon is 1:50 Low reinfection, High external infection 17 Treatments 10 Treatments

46. Using wrasse can reduce the number of chemical treatments for all infection scenarios

47. Individual-based model of evolution of pesticide resistance in sea lice How do treatment regimens affect the rate the resistance evolves? How does the genetic mechanism of resistance effect the rate of evolution? Can integrated pest management slow the rate that resistance evolves? How do chemical refugia (e.g., wild salmon) influence the rate that pesticide resistance evolves in sea lice?

48. Model Structure Upper level variables

49. Preliminary data Temperature influences the rate that resistance evolves and the number of treatments used *these results vary with each iteration, so this will need to be scaled up to look at average effects.

50. Transmission processes Most disease models assume that transmission is density- or frequency- dependent (e.g., from a host perspective), however, transmission in aquatic systems is dynamic and environmentally dependent

51. Transmission processes Hydrodynamic approaches, 3- D particle tracking model Erin Rees

52. Transmission processes Calculate force of infection among farm network and analyze with Who Acquires Infection from Whom (WAIFW) matrix Paull et al. 2012Dobson & Fofopolous 2002

53. Various levels of complexity Deterministic SIR model Population matrix model w/ temperature- dependency and demographic stochasticity Agent Based Model w/ demographic stochasticity Hydrodynamic transmission model coupled with ODE model of farm transmission processes

54. Additional Considerations Sensitivity analysis: How sensitive are model predictions to small changes in model inputs (parameters or initial/boundary conditions)? Model validation: does the model accurately predict outcomes from a new experiment/field study? Bayesian versus frequentist approach: How do we deal with prior information? What philosophical approach do we use to assess the weight of scientific evidence? Hierarchical modeling: Is there a hierarchy of structure in the model? Are parameters fixed or do they they actually have a probability distribution?

55. Additional Considerations Consider your audience! Who tunes out during a modeling talk? Can you add visualizations? When to simplify or not? Consider your strengths If you aren’t a modeler, can you find one? How can you collect your data to be model-friendly?

57. New modeling frontiers with ‘big data’ Googling ‘influenza’

58. New modeling frontiers with ‘big data’ Syndromic surveillance real-time monitoring early detection use of technology (apps, cellphones)

59. Law of Unintended Consequences: An intervention in a complex system invariably creates unanticipated and often undesirable outcomes In Borneo in the 1950s many Dayak villagers had malaria and the World Health Organization had a solution that was simple and direct. Spraying DDT seemed to work: mosquitoes died and malaria declined. But then an expanding web of side effects…started to appear. The roofs of peoples houses began to collapse because DDT had killed tiny parasitic wasps that had previously controlled thatch-eating caterpillars. The colonial government issued sheet-metal replacement roofs, but people could not sleep when tropical rains turned the tin roofs into drums The DDT-poisoned bugs were eaten by geckoes which were eaten by cats. The DDT invisibly built up in the food chain and began to kill cats. Without the cats the rats multiplied. The World Health Organization, threatened by potential outbreaks of typhus and sylvatic plague, which it had itself created, was obliged to parachute fourteen thousand live cats into Borneo Thus, occurred Operation Cat Drop, one of the odder missions of the British Royal Air Force. Hawken (1999)