Multi-Scale Modelling of Infectious Diseases - A WICI Talk with Dr. Jane Heffernan by Waterloo Institute for Complexity and Innovation

WICI Waterloo 2016

Infectious disease modelling over many scales Jane Heffernan York Research Chair Centre for Disease Modelling Modelling Infection and Immunity Mathematics & Statistics York University

Multiple Scales

Modelling Infection and Immunity Lab  How does the immune system interact with a pathogen?  How do you develop immunity?  How is this related to transmission?  How does some population immunity affect epidemic size?

2 MI

 Multi-scale In-Host

Projects

Between Host

HIV

Pop’n Level

Experimental

HIV

HBV/HCV Measles

Measles

Measles Pertussis

Influenza

Multiple Exposure

Influenza

Herpes Amyloidosis

Amyloidosis Social Distancing

 Dynamical systems, stochastic models, computer simulations, bifurcations, stability

Background Mathematical Epidemiology Mathematical Immunology Basic Reproductive Ratio

Epidemiology S

 S - susceptible dS = λ − d S S − βSI  E – exposed dt dE  I – infectious (symptomatic = βSI − d E E − αE dt for part of the time) dI = αE − d I I − γI  R - recovered dt  λ - birth rate dR = γI − d R R  α, γ – disease progression dt  β – infection rate  Equilibria – uninfected, infected  di - death rate

Epidemiology  Basic reproductive ratio  The number of secondary infections produced by an initial infective in a totally susceptible population

 Threshold condition – transcritical bifurcation (usually)

Epidemiology  Basic reproductive ratio  The number of secondary infections produced by an initial infective in a totally susceptible population

dS = λ − d S S − βSI dt dE = βSI − d E E − αE dt dI = αE − d I I − γI dt dR = γI − d R R dt

 What can be measured?

Extra Complexity V

Epidemiology Model with drugs S

  Extensions dS = λ − d S S − (1 − ε ) βSI  Metapopulations dt dE = (1 − ε ) βSI − d E E − αE  Networks dt  Age dI = αE − d I I − γI  Space dt dR  And so on… = γI − d R dt

R Other compartments  Vaccinated  Partially immune  Asymptomatic infectious  Cancer  Chronic Infection  And so on…

General Disease and Immune Stages All states should depend on immune system status birth

infection

* death

recovery progression progression

death

waning immunity

vaccination death

recovery

R* death

waning immunity

infection vaccination

death

Immune System

Cells of Immune System

• Naive, activated, memory

• T – produced by thymus • CD4 – Helper T-cells – Activated by APCs – Activate the immune system

• CD8

– Killer T-cells

• B - produced in bone marrow • Mature in spleen • Plasma cells, antibodies

Pathogenesis Establishment of infection

Initiate adapative immune response

Adapative immune response

Immunological memory

Level of pathogen in plasma

Immune memory

Level of immunity

Chronic Infection

Acute Infection

Pathogen enters body

Pathogen enters plasma

Infectiousness begins

Symptoms appear

Infectiousness: little to none

Pathogen: clearance or persistence

Basic Model HIV

Protease Fusion

Reverse Transcriptase Viral RNA Viral RNA Transcribed to DNA

RNA + Viral Proteins Released

Viral DNA Incorporated Into Host Genome

New Proteins from Viral DNA

CD4 receptor

Budding of New Virion Protease Enables Capsid Assembly

CD4 T-cell

Basic Model dx = λ − d x x − βxv dt dy = βxv − d y y dt dv = ky − d v v dt

 x - uninfected cells  y - infected cells  v - free virus Nowak, M.A. and R.M. May, Virus Dynamics

 λ, k - production rate  β - efficacy of infection  dx, dy, dv - death rates/ clearance time

Basic Model- with virus loss dx = λ − d x x − βxv dt dy = βxv − d y y dt dv = ky − d v v − βxv dt

 x - uninfected cells  y - infected cells  v - free virus

 λ, k - production rate  β - efficacy of infection  dx, dy, dv - death rates/ clearance time

Basic Model- with virus loss dx = λ − d x x − βxv dt dy = βxv − d y y dt dv = ky − d v v − βxv dt

dx = λ − d x x − β xv dt dE ( X ) = λ − d x E ( X ) − β E ( XV ) dt E ( XV ) = E ( X ) E (V ) + COV ( X , V )

Moment closure

In-host Model  Basic reproductive ratio  The number of secondary infections produced by an initial infective in a totally susceptible population

βx0 k R0 = d v + βx0 d y

In-host Model  Basic reproductive ratio  The number of secondary infections produced by an initial infective in a totally susceptible population

βx0 k R0 = d v + βx0 d y

Basic Model- with drugs dx = λ − d x x − (1 − ε rt ) βxv dt dy = (1 − ε rt ) βxv − d y y dt dv = (1 − Qε p )ky − d v v − βxv dt dw = ε p Qky − d v w dt

 x - uninfected cells  y - infected cells  v, w - free virus

 λ, k - production rate  β - efficacy of infection  dx, dy, dv - death rates/ clearance time

Extensions to Basic Model  Drug therapy  Inhibit infection of a cell by virus  Inhibit production of new virions  Pharmacokinetics – doses of drugs, adherence

 Incorporate (some of) the immune system?    

Activation, development of memory CD8 cells, killer T-cells Antibodies and B-cells Innate immune system – cytokines

 Spatial effects – location of infection  And many more

CD8 T-cell immunity death ‘birth’

death Infection

budding

PBMCs – x, y

v+w proliferation

CD8’s – z conversion

za activation

death

zm activation death

death

Initiate adapative immune response

CD8 T-cell immunity Level of pathogen in plasma

Establishmen t of infection

Level of death immunity Infection ‘birth’

Immunological memory

Adapative immune response

death

budding Chronic Infection

PBMCs – x, y

Pathogen enters body

CD8’s – z

Immune memory

v+w

Acute Infection

Pathogen enters plasma

Infectiousness begins

proliferation

Symptoms appear

Infectiousness: little to none

Pathogen: clearance or persistence

conversion

za activation

death

zm activation death

death

CD4 T-cell immunity death â&#x20AC;&#x2DC;birthâ&#x20AC;&#x2122;

death conversion

activation

death

activation

proliferation

v+w

proliferation

Infection by infectious virus v

budding

ya activation

death

conversion

activation death

death

CD4 T-cell immunity death

death

death conversion

activation

Longlived!!

â&#x20AC;&#x2DC;birthâ&#x20AC;&#x2122;

activation

proliferation

v+w

Infection by infectious virus v

budding

activation

death

conversion

activation death

death

HIV and Latency death â&#x20AC;&#x2DC;birthâ&#x20AC;&#x2122;

death conversion

activation

death

activation

proliferation

v+w

proliferation

Infection by infectious virus v

budding

ya activation

death

conversion

activation death

death

Modelling Pathogen Dynamics and Immune System Memory In-Host Cao P, Yan AWC, et al. (2015) PLoS Comp. Biol., 11(8) e1004334. Laurie KL, Guarnaccia TA, et al. (2015) JID, published online jiv260. Du Y, Wu J, Heffernan JM (2014). Math. Pop. Stud., in press. Frascoli F, Wang Y, et al. (2014). CAMQ, in press. Wang Y, Brauer F, et al. (2014) Jour Math Analysis Appl, 414(2), 514-531 Wang Y, Zhou Y, et al. (2013) Jour Math Biol., 67(4), 901-934. Qesmi R, ElSaadany S, et al. (2011), SIAM J. Appl. Math. 71, 1509-1530. Qesmi R, Wu J, et al. (2010) Math Biosci. 4(2), 118-25. Heffernan JM, Keeling MJ (2008). Theor. Popul. Biol., 73, 134-147. Heffernan JM, Wahl L.M (2006). Jour. Theor. Biol., 243(2), 191-204.

HIV and Latency death â&#x20AC;&#x2DC;birthâ&#x20AC;&#x2122;

death conversion

activation

death

activation

proliferation

v+w

proliferation

Infection by infectious virus v

budding

ya activation

death

conversion

activation death

death

Viral Blips ď&#x201A;§ ODEs, include long lived memory cells

Frascoli F, Wang Y, et al. (2014). CAMQ, in press.

Viral Blips  ODEs, include long lived memory cells

Tat ‘on-off’ switch On-going MI2 work Bifurcations: Backward (1 or 2), Hopf (1 or 2), depend on s anf f(x,y,v)Tat ‘on-off’ switch

Frascoli F, Wang Y, et al. (2014). CAMQ, in press.

Acute - Influenza  Innate and B-cells – reinfection intervals  Ferrets  Viral shedding  Nasal wash – 24hrs

 Viral hierarchies Prevention of 2nd inf Co-inf Shortened 2nd inf Delayed 2nd inf No effect

Cao P, Yan AWC, et al. (2015) PLoS Comp. Biol., accepted. Laurie KL, Guarnaccia TA, et al. (2015) JID, published online jiv260.

Block/prevention A(H1N1)pdm09

log10 copy number / 100 µl nasal wash

    

Co-infection

A(H3N2) A(H1N1)pdm09

Delay A(H1N1)pdm09

B 8

Control

No effect A(H1N1)pdm09

limit of detection of infectious virus

Control

10 A(H1N1)pdm09

B 8

0 2 4 6 8 10 12 14 16 18 20 22

experimental day

Control A(H3N2)

Shortened

A(H1N1)pdm09

0 2 4 6 8 10 12 14 16 18 20 22

Acute - Influenza  Innate and B-cells – viral hierarchies

• • • •

productive co-infection (grey) early synchronised decrease (red) Desynchronised phase (green) removal of the challenge virus (blue)

Cao P, Yan AWC, et al. (2015) PLoS Comp. Biol., accepted. Laurie KL, Guarnaccia TA, et al. (2015) JID, published online jiv260.

Acute – Measles death ‘birth’

death Infection

Heffernan JM, Keeling MJ (2008). Theor. Popul. Biol., 73, 134-147.

budding

PBMCs – x, y

v+w proliferation

CD8’s – z conversion

za activation

death

zm activation death

death

Acute - Measles  CD8 T-cell – natural infection and vaccination Heffernan JM, Keeling MJ (2008). Theor. Popul. Biol., 73, 134-147.

Acute - Measles  CD8 T-cell – natural infection and vaccination Subclinical infection

Translation to epidemiology

Heffernan JM, Keeling MJ (2008). Theor. Popul. Biol., 73, 134-147.

Natural Inf + Booster Infs Subclinical Infection

Vaccine + Booster Infections Subclinical Infection

Modelling Immunity in a Population Qesmi R, Heffernan JM, Wu J (2015) Jour. Math. Biol. 70 (1-2), 343-366. Dafilis M, Frascoli F, et al. (2014) Theor Biol Med Model 11:43. Dafilis M, Frascoli F, et al. (2014) Jour Theor Biol 361, 124-132. Lou Y, Steben M, et al. (2014) Discrete and Continuous Dynamical Systems B 19(2), 447-466. Duvvuri VRSK, Heffernan JM, et al. (2012) BMC Inf Dis, 12:329 Lou Y, Qesmi R, et al. (2012) PLoS One, e46027. Ghosh S, Heffernan JM (2010). PLoS One, e14307. Heffernan JM, Keeling MJ (2009). Proc Roy Soc B, 276(1664), 2071-2080.

Waning Immunity - Measles ω p −k , ..., ωi

∑ β jY j σ  → γ ω , ..., ω αi j p p p −k +1 Si → Ei → Yi → R p    → S p −k  Parameterize epidemiological model using in-host output

β i , α i , γ i , ωi , σ

 Other Parameters  Host natural death rate  Host immunity  vaccination distribution Heffernan JM, Keeling MJ (2009). Proc Roy Soc B, 276(1664), 2071-2080.

Waning Immunity - Measles  No vaccine  Distribution of Immunity

Heffernan JM, Keeling MJ (2009). Proc Roy Soc B, 276(1664), 2071-2080.

 Symptomatic Infections

 Function of age  Variation from standard SEIR is slight and primarily occurs as a mild infection in older individuals

Vaccination + Waning  Distribution of immunity  30 and 80 years

 Avg Prevalence of Infection  Waning immunity can severely limit effects of vaccination – L vs. NL

not linear linear

42 Heffernan JM, Keeling MJ (2009). Proc Roy Soc B, 276(1664), 2071-2080.

Vaccination  High levels of vaccination (>70%) and moderate levels of waning immunity (>30 years) lead to large scale epidemic cycles  92 % vacc  30,40,50,60 years waning immunity

43 Heffernan JM, Keeling MJ (2009). Proc Roy Soc B, 276(1664), 2071-2080.

Hopf, Medium, Large

Waning Immunity     

What if immunity does not decay to zero? Consider waning to min of 6 mem T-cells /μL Can do similar analysis Not very different Level of infectious cases is higher, but more are asymptomatic  Magnitude and period is smaller for outbreaks  Suboptimal boosting 45

SIMPLE SIRWS Model  Waning immunity (1/κ=10 yr) and immune boosting (v) Dafilis M, Frascoli F, et al. (2014) Theor Biol Med Model 11:43. Dafilis M, Frascoli F, et al. (2014) Jour Theor Biol 361, 124-132. • Period diagrams (η,ν)-plane • 1/ξ=50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 years • saddle node lines and period-doubling cascades of different orbital periods overlap • system sensitive to small perturbations in parameters and prone 46 to multistable behaviour

SIMPLE SIRWS Model 1/1 1/2 1/3

2/4 2/5 3/6

4/6 4/10 6/12

Dafilis M, Frascoli F, et al. (2014) Theor Biol Med Model 11:43. Dafilis M, Frascoli F, et al. (2014) Jour Theor Biol 361, 124-132.

SIMPLE SIRWS Model ď&#x201A;§ Basins of attraction, vary duration of immunity

Period of Oscillations

Infecteds

2D slices through the full 4D initial-condition space. S(0)+I(0)+R(0)+W(0) = 1

Duration of Immunity

Dafilis M, Frascoli F, et al. (2014) Theor Biol Med Model 11:43. 48 Dafilis M, Frascoli F, et al. (2014) Jour Theor Biol 361, 124-132.

Understanding the Infectious Disease Landscape Collinson MS, Khan K, Heffernan JM (2015), PLoS One, 10(11) e0141423. Li X, Jankowski HJ, et al. (2014) BIOMAT 2015, proceedings. Li X, Jankowski HJ, et al. (2014) BIOMAT 2014, proceedings. Laskowski M, Dubey P, et al. (2014) BIOMAT 2014, proceedings. Collinson MS, Heffernan JM (2014) BMC Public Health 14(1), 376 Richardson K, Sander B, et al. (2014), AIMS Public Health, 1(4), 241-255. Duvvuri VRSK, Moghadas SM, et al. (2010). Influenza and other Respiratory Viruses 4(5), 249-258.

Influenza - clusters Li X, Jankowski HJ, et al. (2014) BIOMAT 2014, accepted

Influenza Clusters Li X, Jankowski HJ, et al. (2014) BIOMAT 2014, accepted

Effects of Mass Media V

vaccination

infection

Infected vs time

Waning

Social distancing recovery

Reports: confirmed symptomatic cases EI

Media Reports – 2009 H1N1

Collinson MS, Khan K, Heffernan JM (2015), PLoS One Collinson MS, Heffernan JM (2014) BMC Public Health 14(1), 376

Where Can We Use More Mathematical Modelling? Heffernan JM, Chit A. et al., in prep. Richardson K, Sander B, et al. (2014), AIMS Public Health, 1(4), 241-255.

Warning!  Important implications to public health initiatives to identify best population based strategies on the availability of vaccine and antivirals.  Individual vs. population health  Social behaviour  Vaccine uptake, drug therapy

Fast-slow analysis Game theory

 Mutation and evolution of resistance Too much to list  Seasonal forcing  Variability  In-host Stochastic models  Epidemic

Thank You!           

Lindi Wahl (Western) Matt Keeling (Warwick) Jianhong Wu (York) Yijun Lou (Hong Kong) Federico Frascoli (Swinburne) Yan Wang (U Petroleum China) Beni Sahai (Cadham Lab Hossein Zivari Piran (York) Jodie McVernon (Melbourne) James McCaw (Melbourne) Redouane Qesmi (Fez Morroco)

 Robert Smith? (Ottawa)  Shannon Collinson (MOHLTC)  Kamran Khan (St Michael’s Hospital, BlueDot)  MI2 Lab members (past/present)  Centre for Disease Modelling  Waterloo Institute for Complexity and Innovation (WICI)

Questions

Puzzle pieces Overlap Multiscale

Appendix

Chronic - HIV  Modelling/Understanding  Oscillations in viral load and T-cell count  Viral blips Frascoli F, Wang Y, et al. (2014). CAMQ, in press. Wang Y, Brauer F, et al. (2014) Jour Math Analysis Appl, 414(2), 514-531 Wang Y, Zhou Y, et al. (2013) Jour Math Biol., 67(4), 901-934. Heffernan JM, Wahl L.M (2006). Jour. Theor. Biol., 243(2), 191-204.

Variability in Viral Load ď&#x201A;§ Stochastic Model, measure SD in viral load T-cells NaĂŻve /mL

Lifetime Distributions

Time (Days)

Pharmacokinetics

Cell count .25 loge CD4 T-cells .22 loge CD4 T-cells Viral load .2-.4 log10 RNA copies .32 log10 RNA copies Heffernan JM, Wahl L.M (2006). Jour. Theor. Biol., 243(2), 191-204.

HBV/HCV Persistence  ODEs, two compartments of infection death

death

Infection

• Virus- one compartment • Liver perfusion •~400 to >4000 L of blood goes through liver every day

Liver – liver cells budding

Blood PBMCs

v+w

Infection death

yB death

Qesmi R, ElSaadany S, et al. (2011), SIAM J. Appl. Math. 71, 1509-1530. Qesmi R, Wu J, et al. (2010) Math Biosci. 4(2), 118-25.

HBV/HCV Persistence  k – production of virus particles  a – death rate of infected cells (k L − aL )( k B − aB ) < 0 (k L − aL )( k B − aB ) > 0

Qesmi R, ElSaadany S, et al. (2011), SIAM J. Appl. Math. 71, 1509-1530. Qesmi R, Wu J, et al. (2010) Math Biosci. 4(2), 118-25.

One Exposure

Multiple Exposures

Multiple Exposures  Interested in early stages of infection immediately after exposure to a pathogen  Assume individual is exposed to infectious dose ‘c’  Unit of pathogen load needed to produce an infection

 Therefore, assume infectious pathogen load will grow, overcoming non-specific immune response

Viral Load  When V(t,a) > A  New exposures do not increase pathogen load

 Assume

 Holling functional response type 2  b – number of effective contacts  K – adjustable parameter that measures how soon saturation can occur

Between Host Model ď&#x201A;§ SEIR model

In general  Highlights  Immune boosting induces cyclical behaviour in a model of infectious disease dynamics.  Seasonal forcing of transmission also induces cyclical behaviour in this system.  The birth rate, waning rate and forcing interact to generate complex dynamics.  Periodic cycles in the forced system are related to unforced limit cycle dynamics.  The “demographic transition” may lead to new dynamical regimes for certain diseases.