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Modeling Infectious Diseases from a Real World Perspective Wayne Getz Department of Environmental Science, Policy and Management What is disease? Disease is an abnormal condition that impairs bodily functions Infectious Disease is transmitted from one individual to another (airborne, waterborne, sexually transmitted, contact transmission) Vectored Disease requires an agent to be involved in the transfer Zoonotic Disease has a non human source Pathogens cause Disease microparasites: virus, bacteria, protozoans, fungi macroparasites: cestodes, nematodes, ticks, fleas Disease is an ecological process Basic Elements define species: single pop, vectored system, ecological system disease categories: infected vs infectious, latent vs active, normal vs superspreader demographic categories: gender, age, other interventions: vaccination, quarantine, drug regimens, circumcision, time: fast diseases (e.g. pneumonia, influenza) vs. slow diseases (e.g. TB, HIV, leprosy). Emerging Infectious Diseases: What?, Where? How? and Why? Cover: Vol 6(6), 2000 Emerging Infectious Disease (CDC Journal) Japanese color woodcut print advertising the effectiveness of cowpox vaccine (circa 1850 A.D.) WHAT? (Definition from MedicineNet.com) Emerging infectious disease: An infectious disease that has newly appeared in a population or that has been known for some time but is rapidly increasing in incidence or geographic range. Examples of emerging infectious diseases include: * Ebola virus (first outbreaks in 1976) * HIV/AIDS (virus first isolated in 1983) * Hepatitis C (first identified in 1989) * Influenza A(H5N1) (bird ‘flu first isolated from humans in 1997) * Legionella pneumophila (first outbreak in 1976) * E. coli O157:H7 (first detected in 1982) * Borrelia burgdorferi (first detected case of Lyme disease in 1982) * Mad Cow disease (variant Creutzfeldt-Jakob: first described 1996) More WHAT! CDC National Center for Infectious Disease information list for emerging and re-emerging infectious diseases drug-resistant infections, bovine spongiform encephalopathy (Mad cow disease) and variant Creutzfeldt-Jakob disease (vCJD), campylobacteriosis, Chagas disease, cholera, cryptococcosis, cryptosporidiosis (Crypto), cyclosporiasis, cysticercosis, dengue fever, diphtheria, Ebola hemorrhagic fever, Escherichia coli infection, group B streptococcal infection, hantavirus pulmonary syndrome, hepatitis C, hendra virus infection, histoplasmosis, HIV/AIDS, influenza, Lassa fever, legionnaires' disease (legionellosis) and Pontiac fever, leptospirosis, listeriosis, Lyme disease, malaria, Marburg hemorrhagic fever, measles, meningitis, monkeypox, MRSA (Methicillin Resistant Staphylococcus aureus), Nipah virus infection, norovirus (formerly Norwalk virus) infection, pertussis, plague, polio (poliomyelitis), rabies, Rift Valley fever, rotavirus infection, salmonellosis, SARS (Severe acute respiratory syndrome), shigellosis, smallpox, sleeping Sickness (Trypanosomiasis), tuberculosis, tularemia, valley fever (coccidioidomycosis), VISA/VRSA - Vancomycin-Intermediate/Resistant Staphylococcus aureus, West Nile virus infection, yellow fever More WHAT! CDC National Center for Infectious Disease information list for emerging and re-emerging infectious diseases drug-resistant infections, bovine spongiform encephalopathy (Mad cow disease) and variant Creutzfeldt-Jakob disease (vCJD), campylobacteriosis, Chagas disease, cholera, cryptococcosis, cryptosporidiosis (Crypto), cyclosporiasis, cysticercosis, dengue fever, diphtheria, Ebola hemorrhagic fever, Escherichia coli infection, group B streptococcal infection, hantavirus pulmonary syndrome, hepatitis C, hendra virus infection, histoplasmosis, HIV/AIDS, influenza, Lassa fever, legionnaires' disease (legionellosis) and Pontiac fever, leptospirosis, listeriosis, Lyme disease, malaria, Marburg hemorrhagic fever, measles, meningitis, monkeypox, MRSA (Methicillin Resistant Staphylococcus aureus), Nipah virus infection, norovirus (formerly Norwalk virus) infection, pertussis, plague, polio (poliomyelitis), rabies, Rift Valley fever, rotavirus infection, salmonellosis, SARS (Severe acute respiratory syndrome), shigellosis, smallpox, sleeping Sickness (Trypanosomiasis), tuberculosis, tularemia, valley fever (coccidioidomycosis), VISA/VRSA - Vancomycin-Intermediate/Resistant Staphylococcus aureus, West Nile virus infection, yellow fever =: first recognized ’93, rodent excretions, rare but deadly More WHAT! CDC National Center for Infectious Disease information list for emerging and re-emerging infectious diseases drug-resistant infections, bovine spongiform encephalopathy (Mad cow disease) and variant Creutzfeldt-Jakob disease (vCJD), campylobacteriosis, Chagas disease, cholera, cryptococcosis, cryptosporidiosis (Crypto), cyclosporiasis, cysticercosis, dengue fever, diphtheria, Ebola hemorrhagic fever, Escherichia coli infection, group B streptococcal infection, hantavirus pulmonary syndrome, hepatitis C, hendra virus infection, histoplasmosis, HIV/AIDS, influenza, Lassa fever, legionnaires' disease (legionellosis) and Pontiac fever, leptospirosis, listeriosis, Lyme disease, malaria, Marburg hemorrhagic fever, measles, meningitis, monkeypox, MRSA (Methicillin Resistant Staphylococcus aureus), Nipah virus infection, norovirus (formerly Norwalk virus) infection, pertussis, plague, polio (poliomyelitis), rabies, Rift Valley fever, rotavirus infection, salmonellosis, SARS (Severe acute respiratory syndrome), shigellosis, smallpox, sleeping Sickness (Trypanosomiasis), tuberculosis, tularemia, valley fever (coccidioidomycosis), VISA/VRSA - Vancomycin-Intermediate/Resistant Staphylococcus aureus, West Nile virus infection, yellow fever =: identified ’72, stomach flu on cruise ships, schools, hotels More WHAT! CDC National Center for Infectious Disease information list for emerging and re-emerging infectious diseases drug-resistant infections, bovine spongiform encephalopathy (Mad cow disease) and variant Creutzfeldt-Jakob disease (vCJD), campylobacteriosis, Chagas disease, cholera, cryptococcosis, cryptosporidiosis (Crypto), cyclosporiasis, cysticercosis, dengue fever, diphtheria, Ebola hemorrhagic fever, Escherichia coli infection, group B streptococcal infection, hantavirus pulmonary syndrome, hepatitis C, hendra virus infection, histoplasmosis, HIV/AIDS, influenza, Lassa fever, legionnaires' disease (legionellosis) and Pontiac fever, leptospirosis, listeriosis, Lyme disease, malaria, Marburg hemorrhagic fever, measles, meningitis, monkeypox, MRSA (Methicillin Resistant Staphylococcus aureus), Nipah virus infection, norovirus (formerly Norwalk virus) infection, pertussis, plague, polio (poliomyelitis), rabies, Rift Valley fever, rotavirus infection, salmonellosis, SARS (Severe acute respiratory syndrome), shigellosis, smallpox, sleeping Sickness (Trypanosomiasis), tuberculosis, tularemia, valley fever (coccidioidomycosis), VISA/VRSA - Vancomycin-Intermediate/Resistant Staphylococcus aureus, West Nile virus infection, yellow fever =: mosquito vector, 1st case N.Am. ’99 now ≈ 15000 cases 500 deaths WHERE? Global trends in emerging infectious diseases Jones et al. Nature 451, 990-993(21 February 2008) 335 events all pathogen types: 1940-2004 WHAT? by decade Jones et al. Nature 451, 990-993(21 February 2008) WHAT? by decade Jones et al. Nature 451, 990-993(21 February 2008) WHAT? by decade Jones et al. Nature 451, 990-993(21 February 2008) • HOW? Contacts with wildlife • Vulnerability to infection (elderly, HIV+) • Strains evolving to resist treatments • Contact networks particularly global travel • new diagnostic tools SARS Outbreak Current risk of an EID zoonotic pathogen from wildlife Jones et al. Nature 451, 990-993(21 February 2008) Disease Categories and Transmission in Kermack-Mckendrick Models W. O. Kermack and A. G. McKendrick: A Contribution to the Mathematical Theory of Epidemics, I, II (endemicity), and III (endemicity cont.) I. Proc. R. Soc. Lond. A, 1927, 115, 700-721 (doi: 10.1098/rspa.1927.0118) II. Proc. R. Soc. Lond. A, 1932, 138, 55-83 (doi: 10.1098/rspa.1932.0171) III. Proc. R. Soc. Lond. A, 1933, 141, 94-122 (doi: 10.1098/rspa.1933.0106) Hethcote, H. W. 2000. The mathematics of infectious disease. SIAM Rev. 42, 599–653. (doi:10.1137/S0036144500371907) Disease Categories and Transmission SIR Models S: susceptible, I: infected & infectious R: “recovered & immune” (V) or “removed” (D) N: Does N=S+I+V change with time? Units: numbers vs. densities. vs proportions. Transmission: mass action (densities of SxI) frequency dependent (proportion of SxI) Be Warned!: transmission = bSI holds for both frequency or mass action if N is constant or for variable N(t) if units are density (mass action) or proportions (frequency) Epidemics with “lumped” demography S: susceptible E: exposed (infected) I: infectious V: recovered immune D: dead N: S+E+I+V b0 bV: birth rate ! ! ! I !V µ ! D !V transmission rate refraction rate (latent period) reversion rate natural mortility disease induce mortality Outline of remaining material Preliminaries: Discrete versus continuous models in biology Discrete versus continuous models in epidemiology Discrete multi-compartment formulations based on probabilities Case studies: Bovine TB and Vaccination Group structure and containment of SARS TB and drug therapies, TB-HIV dynamics General theory of heterogeneous transmission Goals: Provide a flavor of how to incorporate complexity Illustrate how output used to understand complexities Lead you into some literature for you to explore further! Continuous versus discrete models in biology Simplest model: constant pop N = S + I; S → I, transmission β NS I: ! " ! " dI S I = βI = βI 1 − , I(0) = I0. dt N N Logistic model with solution: I0N Text I(t) = I0 + (N − I0)e−βt Discretized system ODE: ! " I(t) I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 − . N Discretized Solution: I(t)N I(t + ∆t) = I(t) + (N − I(t))e−β∆t Which is more appropriate? Continuous versus discrete models in biology Simplest model: constant pop N = S + I; S → I, transmission β NS I: ! " ! " dI S I = βI = βI 1 − , I(0) = I0. dt N N Logistic model with solution: I0N Text I(t) = I0 + (N − I0)e−βt Discretized system ODE: ! " I(t) I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 − . N Discretized Solution: I(t)N I(t + ∆t) = I(t) + (N − I(t))e−β∆t Which is more appropriate? Continuous versus discrete models in biology Simplest model: constant pop N = S + I; S → I, transmission β NS I: ! " ! " dI S I = βI = βI 1 − , I(0) = I0. dt N N Logistic model with solution: I0N Text I(t) = I0 + (N − I0)e−βt Discretized system ODE: ! " I(t) I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 − . N Discretized Solution: I(t)N I(t + ∆t) = I(t) + (N − I(t))e−β∆t Which is more appropriate? Continuous versus discrete models in biology Simplest model: constant pop N = S + I; S → I, transmission β NS I: ! " ! " dI S I = βI = βI 1 − , I(0) = I0. dt N N Logistic model with solution: I0N Text I(t) = I0 + (N − I0)e−βt Discretized system ODE: ! " I(t) I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 − . N Discretized Solution: I(t)N I(t + ∆t) = I(t) + (N − I(t))e−β∆t Which is more appropriate? Continuous versus discrete models in biology Simplest model: constant pop N = S + I; S → I, transmission β NS I: ! " ! " dI S I = βI = βI 1 − , I(0) = I0. dt N N Logistic model with solution: I0N Text I(t) = I0 + (N − I0)e−βt Discretized system ODE: ! " I(t) I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 − . N Discretized Solution: I(t)N I(t + ∆t) = I(t) + (N − I(t))e−β∆t Which is more appropriate? Which is the better discretization scheme? Continuous versus discrete models in biology Time (Δt=0.25) Time (Δt=0.05) Solid line: Iteration using solution Circles: Iteration using discretized equations Continuous Models with Demography dS = f recruitment (S, I, R) − f transmission (S, I, R)S − µS dt dI = f transmission (S, I, R)S − (α + µ ) I dt dR = α I − µR dt f recruitment : recruits and/or births µ: natural mortality rate Elaborations: α : infectious → removed/recovered 1. exposed class E 2. constant rate “exponential” transfers: → Weibull distribution OR → “box car” staging: gamma distribution breeders and long-lived species). Data reporting the proportion pµ of individuals step. Fortunately, a good approximation can be obtained by that die in a unit of time can be converted to a mortality rate parameter µ appearing linear modeling approach, as= follows. −µN by noting that the solution in a differential equation model of the form dN dt we write down time to this equation over First any time interval [k, kthe + 1]continuous is N (k + 1) = Nmodel (k)e−µof . interest This Proportion thatample, die or equations make transitions: e.g. mortalitytotal rate recruitment rate λ (4.1) dying with ais constant implies that the proportion of individuals Some basics on discrete epi models (6.1) class and a constant per-capita natural mortality rate µ. Then, N (k) − N (k + 1) N (k)(1 − e−µ ) pµτ= = (see equation=(3.1)), 1 − e−µthese equations be (I, N ) := pT C(N )I/N N (k) N (k) " ! 1 dS or, equivalently, µ = ln . 1−pµ = λ − µS − τ (I, N )S S(0) = S0 Continuous model SEI: It is often advantageous to formulate dt epidemiological and demographic models dE in discrete time. The primary advantage of differential equation (6.2) = τ (I, N )S − (δ +models µ)E disappears E(0) = E0 dt systems rather than trying to obtain once we resort to numerical simulation of MODELING THE INVASION AND SPREAD OF CONTAGIOUS dIimpossible analytical results, which are difficult if not to obtain forDISEASES most detailed 1 = δE − (α + µ)I I(0) = I0 . nonlinear models. Indeed, discrete time models can be implemented very naturally dt and easily indiscrete computer simulations, whilewenumerical solutions heEquivalent time interval [k, k +SEI: 1]. In this case obtain on the note transmission depends k: modelof differential equaNow assume over a small interval t ∈ [k, kAlso + 1]parameters that the propor tions requires algorithms that use discretizing approximations. in 6.5) over this interval due to a change in I(t) is sufficiently smallth discrete (1 −bepµmore )(1 −easily pτk ) related to0data that have been 0 collated S(k +time 1) models can overE(k discrete rates, N etc.). well approximated by τk(1=−τp(I(k), (k)) (e.g. over = (e.g. i + 1) intervals (1vital − pµand )pτktransmission 0 the time µ )(1 − pδ ) Discrete timein equations, cannot properly account for the interactions τ (I(t), Nhowever, (t)) percent, our I(k + 1) 0 may be a few (1 − pµ )pδ but (1 − estimates pµ )(1 − pαof) the of simultaneously pnonlinear processes, as individuals simultaneously subject to un N as defined byexpression (3.2), may have τ (I(t), (t)),such T in S(k) (1 Obviously −time pµ )λ the processes of infection death: in each interval weassumption either first account for several and times as large). this will influence +orvice versa. infection and thenstep natural mortality It does a difference how we E(k) 0 time × , make duration, with shorter steps required for faster-growin 28 schedule things [54]. Alternatively we can treat the two processes simultaneously, I(k) 0 replacing τ (I(t), N (t)) with the constant τ for t ∈ [k, k + 1], e k dI1 = δ(En − I1 ) Ex: Use analytical/ numerical methods dtto dIj Characterize the distribution of R(t) in the=SE n Ij−1 m R−model δ(I Ij ), j with = 2, .S(0) . . , m= tribution R(t) the with S(0) ==0 in terms of β, S0 , Ei (0) of = 0, i = in 1, ..., n, SE Ij (0) =R0, model j dt = 1, ..., m, R(0) n Im dR δ, µ, discrete and compare , ..., n,mIjand (0) n=for 0, the j =continuous 1, ..., m, and R(0) = 0 =informulations terms of β, δIm − µR (start with µ =and δ = 1 discrete and m = 1formulations and investigatedt inand the discrete model δ < 1) e continuous compare !model #δ < 1) dm = 1 and investigate in the discrete m " Continuous Discrete dS = −β Ij S ! m # ! m # dt " " j=1 dS ! m S(t # + 1) = S(t) − β = −β Ij S Ij (t) S(t) " dt dE1 j=1 j=1 Ij S − δE1 ! m # dt = β ! m # j=1 " " dE1 dEδE = β Ij S − i 1 E1E(t),+ i1)= 2,=. . . ,βn Ij S + (1 − δ)E1 = δ(Ei−1 − dt i j=1 dt j=1 dEi dt dI1 dt dIj dt dR dt dI1 = . .δ(E I1i)(t + 1) = δEi−1 (t) + (1 − δ)Ei (t) = δ(Ei−1 − Ei ), dt i = 2, . , nn − E i = 2, . . . , n dIj = δ(Ij−1 − Ij ), j = 2, . . . , m dt = δ(En − I1 ) I1 (t + 1) = δEn (t) + (1 − δ)I1 (t) dR = δIm − µR Ij (t + 1) = δIj−1 (t) + (1 − δ)Ij (t) dt = δ(Ij−1 − Ij ), j = 2, . . . , m = δIm − µR j = 2, . . . , m R(t!+" 1) =# δIm (t) − µR(t) m S(t + 1) = S(t) − β Ij (t) S(t) j=1 29 First Case Study: Bovine TB in African Buffalo Cross & Getz (2006) Ecological Modelling 196: 494-504. Important elements: Includes demography Herd structure: focus on one herd embedded in background prevalence assuming balanced movement into and out of herd SVEID structure (Susc, Vaccinated, Exposed, Infected, Dead) BTB model with demography & ecology ! 7#8!-,;!;/%(-%(!/,+45-)/$,!7 $89!(2/3#-)/$,!-,;!/22/3#-)/$,!7 %89!-,;!%4#</<-6!7.$48!)1(!2$,)16N! X! :7657!6%!9#(=0(/5<U4(3(/4(/)!:7(/! &!F+1/4!4(/%6)<!4(3(/4(/)!:7(/! &!FGO!V$ ! 2$;(6!(J4-)/$,%!+$,0$#2/,3!)$!)1(!-5$<(!-%%42.)/$,%!-#(W! Bovine TB model: X (susc), Y (infected), Z (infectious) & ! ! ! ! L! V (vac.), I (migr.) )76%!;$4(2!6%!#(21)6M(2<!5$/%)1/)D!%$!)71)!%(2(5)6/8!&!F!G!$#!+!6%!(%%(/)6122<!(=06M =D > * * -* ( + ( ++ # 11 5 $; 4 &( ' ( T! )7(!)#1/%;6%%6$/!5$(99656(/)! '(!H70%D!)7(!M120(!$9! &!6%!/$)!5#6)6512!)$!$0#!1/12<%6 ( + ( ++ $ 0= 4 0= (& ' & ' & ' & ' & ' 8 $ . +; 4 &( . =' 0 .$; 4 & 9 &( ''+ &= / $ $; 4 '+ += / = " ( 8 ( !7 ( 0 6 ( . / . $; 4 $; 4 $; 4 1 $:4 ( % ( ( & ' 9 ( +G! #(%02)%!$/2<!9$#!)7(!51%(! 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vs. S(1-3,3-5,5+) 1 9.49 0.00 0%%)/-'*)77/-"'+&3+"2),($"% 56719891:8:1;8;<=!2%>!5671:8:1;8;<= 7 ?>B? ?>:9 +> 11 ?>:7 11 9 "$N%!91B 5671:8:1;8;<=!2%>!5671;8;<= 7 7>A9 ?>7C +? 1 "$N%!B1: 11 ?>@B 11 9 S(1-8,8+) vs. S(c) 8.28 0.00 4&5'2&3&%2&%,& + @A "$N%!!:< 11 ?>@; 11 9 S(sex) vs. S(c) 1 10.86 0.00 B"%(C-D'2$.3&+./6!'2&3&%2&%,& $ 8:E98 YLL3)4#(!L3-(% ?>?7 ?>?/ ?>?B 7 56)D=!2%>!56E= 7 7>A? ?>7C $ F:=98 +3)4#(!L3-(% ?>/B ?>?C ?>?9 7 X-,!L3-(% Z(L3-(% $ 8GA98 $ 8A9; ?>B: ?>/@ ?>79 7 ?>?B ?>?/ ?>?7 7 +,1 $ 8GA98 $ 8A9; +,0 ?>B: ;9<(% ?>?B =(>9<(% ?>/@ ?>79 7 ?>?7 7 ?>?9B ?>?:@ ?>?B9 ?>/7 ?>?:9 7 7 B ? ?>??B9 ?>??;B : ? ?>??B9 ?>??;B : ? 11 7 %((!)(V) Model Parameters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prevalence for low, baseline and high transmission coef. values )#9?%>C%%C$?! 8$(DDC8C(?)!J! " +,. +,+/0 +,+0/ +,+1/ +,+ + ! =C7,!.,! ! -+ .+ F(9# /+ 0+ 1+ 9? ! !! Efficacy of Vaccination ! +,! ./4 ! ! ! ! ! ! ! ! ?(+#!4.!@#(<+;(:8( ! -,! "+;<(% =;;!+>(% ./3 ./4 ./3 ./2 ./2 ./1 ./1 ./0 ./0 ./. ./. ./. ./1 ./3 ./5 ./6 7+889:+)9$:!#+)( 0/. ! ! ! ! 50 years: ! "#$%%! Prevalence ' isopleths after calf only vaccination "#$%&'()*&+,)-".$.'#/,,$%/($"%' !'()*! 0.75 vaccination rate of longacting vaccine needed to reduce BTB below 1% ! Second Case Study: SARS Lloyd-Smith, Galvani, Getz (2003) Proc. Royal Soc. B 270: 1979-1989. Important elements: No demography but group structure for disease classes Group structure relates to intervention and control strategies Time iteration is daily: relates to reporting and data structure Global emergence of SARS, 2003 Hong Kong >195 cases Guangdong province, A China Canada F,G 29 cases F,G A A H,J H,J K Hotel M, Hong Kong B K Ireland 0 cases C,D,E I,L,M C,D,E B Vietnam 58 cases I,L,M United States Singapore 71 cases 1 case Adapted from Dr. J. Gerberding, Centers for Disease Control SARS transmission chain, Singapore 2003 Morbidity & Mortality Weekly Report (2003) Group-level heterogeneity for SARS Health care workers (HCWs) comprised 18-63% of cases in different locales • Main control measures were hospitalization and quarantine. • Strict infection control implemented in hospitals, and contacts with visitors were reduced. β ρβ Community ηβ HCWs ρβ SARS Patients κηβ Detailed structure of SARS: results from daily iterated stochastic simulations ij (c) symptomatic modifying transmission for different settings: ε, η, γ, substructure κ, and ρ (all on the interval 21 r 0.15 1– individuals 1– r (indaybaseline b 0.08–0.26 2 1transmission [0,) 1]). In particular, the reduced rate of exposed (E) transmission rate (day ) b I I I x1 x2 x3 x4 cluded because the extent of pre-symptomatic transmission of SARS was = 3) (R0 = 1.5–5) (R0Iunknown ((IR R h:when health workers; c: general managed r r 1 m: within 1 thecare model was created) is εβ. community; All transmission the patients hospital setting occursowing at a reduced rate ηβ to reflect contact precautions adopted by all hospital ionfactors rate, to: transmission d ) transmission substructure modifying rate, (owing to: and patients, such as the use of and gowns. Additionally, « masks, gloves ssionpersonnel 0–0.1 0.1 hbk « pre-symptomatic transmission I l ha factor γ, yieldm quarantine of exposed individuals reduces their contact rates by cautions h 0–1 0.5 hospital-wide contact rate precautions hidentified ing a total transmission of γεβ, while specific isolation measures for hb Ih + e Eh 0–1 ty mixing r 0.5 reduced HCW–community mixing r factor SARS patients (Im ) in the hospital reduces their transmission rby ba further Ic + e Ec k 0–1 1 isolation rb transmissionk rates beκ.case Finally, we considered the impact of measures to reduce 0–1 these assumptions, 0.5 quarantine g e Em tween HCWs and community members gby a factor ρ. Under lc bg the transmission hazards are: daily probability of: β(Ic + εEc ) + ρβ(Ih + εEh ) + γβεEm τ = c quarantining of community incubating individuals g individuals in the 0–1 ‘m’), representing 0q q Ninc the community case management (subscript qu or case isolation. We categorized disease classes a and(Ec) ηβ(Iin εEhcommunity + κIm )0–1(Ec , Eh , Em 0.3 isolation individuals hcsymptomat h +the individuals in of thesymptomatic community ), τh = ρτc +tible h(cSc , Sh ), incubating , (Ic) Nh owing to recovery or death (Rc , Im ) and removed where Ii , i = c, h, represent over all in sub-compartments in isolation of symptomatic HCWsmodel (sums Ih)hhupdates hhthe incu- t 1-day HCWs (IhE ) i and 0.9timesteps, representing 0.9 bating and symptomatic classes for est pool j, andfor which people’s activities can be th interval beEequivalent. Substructures associated with dai Nh = S h + h + Ih + Vh + Im ments are summarized in figure 1, with details and the caption. Stochastic transitions of individuals Hospital-wide contact precautions, such as the use at all possible. In cautions, such as N the use at all possible. In the case of SARS, non-sp + Vc + ρ(S Ehthe + Iprobabilities c = Sc + Ec + Ic classes, h +on h + Vh ). based shown in figur times of sterile gowns, filtration masks and gloves, modify ures may ha Equations: transmission hazard ation masks and gloves, modify ures may have helped or hindered earl and Epi Equations: Nc = Sc + Ec + Ic + Vc + ρ(Sh + Eh + Ih + Vh ). The detailed form of the SID equations that were formulated are: Community and HCW equations: q: Si (t + 1) = exp (−τi (t)) Si (t) Ei1 (t + 1) = [1 − exp (−τi (t))] Si (t) Eij (t + 1) = (1 − pj−1 )(1 − qij−1 )Eij−1 (t) j = 2, . . . , 10 !10 Ii1 (t + 1) = j=1 pj (1 − qij )Eij (t) i = c, h, Ii2 (t + 1) = (1 − hi1 )Ii1 (t) Ii3 (t + 1) = (1 − hi2 )Ii2 (t) + (1 − r)(1 − hi3 )Ii3 (t) Iij (t + 1) = r(1 − hi j−1 )Ii j−1 (t) + (1 − r)(1 − hij )Iij (t) j = 4, 5 i Vi (t + 1) = Vi (t) + rIi5 (t) + rIm5 (t) & i ' i Em,j (t + 1) = (1 − pc j−1 ) Em,j−1 (t) + qj−1 Ec j−1 (t) j = 2, . . . , 10 & ' ! 10 i i Im1 (t + 1) = j=1 pj Emj (t) + qij Eij (t) i i Im2 (t + 1) = hi1 Ii1 (t) + Im1 (t) ( ) i = c, h. i i i Im3 (t + 1) = hi2 Ii2 (t) + Im2 (t) + (1 − r) )hi3 Ii3 (t) + Im1 (t) ( i i Imj (t + 1) = r hi j−1 Ii j−1 (t) + Im( j−1 (t) ) i +(1 − r) hij Iij (t) + Imj (t) j = 4, 5 In the analysis presented here, the probabilities qij and hij vary between 0 quarantine h: hospitalization rates;forr:delays recovery/death and a fixed rates; value less than 1 and account in contact tracing or case identification. In addition, we did not analyze scenarios where health care workers are quarantined so that qhj = 0 for all j. Deterministic solutions to this SARS Parameter values used in simulations 1980 J. O. Lloyd-Smith and others SARS transmission in a community and hospital Table 1. Summary of transmission and case-management parameters, including the range of values used throughout the study and the three control strategies depicted in figure 3. parameter symbol range examined figure 3 (1) figure 3 (2) figure 3 (3) 0.15 (R0 = 3) 0.15 (R0 = 3) 0.15 (R0 = 3) baseline transmission rate (day2 1) b 0.08–0.26 (R0 = 1.5–5) factors modifying transmission rate, owing to: pre-symptomatic transmission hospital-wide contact precautions reduced HCW–community mixing case isolation quarantine « h r k g 0–0.1 0–1 0–1 0–1 0–1 q daily probability of: quarantining of incubating individuals in the community (Ec) isolation of symptomatic individuals in the community (Ic) isolation of symptomatic HCWs (Ih) 0.1 0.5 0.5 1 0.5 0.1 0.9 1 0.5 0.5 0.1 0.5 0.5 0.5 0.5 0–1 0 0.5 0.5 hc 0–1 0.3 0.9 0.9 hh 0.9 0.9 0.9 0.9 1984 J. O. Lloyd-Smith and others SARS transmission in a Individual runs: Cumulative cases for different R (effective reproduction numbers--i.e. R0 when some control is applied) (a) 100 R = 2.0 R = 1.6 R = 1.2 cumulative cases 80 60 40 20 0 0 50 100 150 days since initial case 200 Probability of epidemic containment for different effective (b) R’s 200 1.0 Pr(containment) probability of containment 1.0 0.8 0.6 0.9 0.8 0.7 0.6 0.8 0.4 1.0 R 1.2 0.2 0 0 2 4 6 reproductive number, R 8 10 R=1 contours (right side of curves guarentees control of epidemic) for the effects of isolation leves hc and transmission curtailment (1-κ) for epidemics with different R0 η: hospital precautions reduce transmission by 1/2 3 lines right to left: increasing delays in isolation of patients 20 0 50 0 probability probabil cumula 40 0.4 0.8 0.4 1.0 0.6 R 0.8 1.2 1.0 0.2 Combinations of policies that lead to containment: plots of R=1 contours 0 100 150 200 50 100 150 days since initial case days since initial case 0.2 20 0 1.2 R 4 6 8 10 0 reproductive 2 4 6 8 number, R reproductive number, R (three lines represent increasing delays in isolating patients) (d ) h=1 0.8 0.6 0.4 0.2 0 0.2 0 0.4 0.2 0.6 0.4 0.8 0.6 1.00.8 transmission intransmission isolation, kin isolation, k (e) (d R 0) = 5.0 1.0 h=1 0.8 0.6 0.4 0.2 0 1.0 (f) 1.0 daily probability of isolation, hc R 0 = 1.5 daily probability of isolation, hc daily probability of isolation, hc R 0 = 1.5 (c) 3.0 2.5 2.0 0 4.0 5.0 4.0 3.0 2.5 2.0 1.0 200 = 0.5 0.8 B B 0.6 0.4 0.2 0 0.2 0 0 4.0 3.0 2.5 2.0 3.0 2.5 R 0 = 5.0 4.0 h = 0.5 h A 1.0 0.8 transmissiontransmission in isolation,ink isolation, k (f) 0.40.2 A 0.6 0.4 0.8 0.6 proba cumulativ control ect case isolation. Here, h = 0.5, r = 1, 0.9, so the case isolation strategy is h h = 100 to commence after the first day of degree50 to which transmission is reduced es. From right to left, two solid lines of t 1-day 0and 5-day delays in contact 0 ntining begins. Solid50lines show100 the case 150 (days) mission can occur during thetime incubation ines show the case « = 0 (please note =) 2.44, 2.92 and 3.90, rather than 2.5, 3 (c) ty of 1.0 effective reproductivecontrol number R to precautions n-control parameters. In allquarantine and isolation , q = 0.5, h c = 0.3 and h h = 0.9; all measures combined h were0.8 set to 0.5, then varied one at a 0.2 Probability of containment in terms of implementation of control after epi onset 2000 0 50 100 0.6 1.0 probability of containment probability of containment in implementation Left: 3 strategies; Right: combineddelay measure for 3(days) R0 0.8 150 R0 = 2 R0 = 3 R0 = 4 0.6 egies and delays in on 0.4 0.4 the importance of various control meairs, we now consider the effects of intetegies0.2 on SARS outbreaks. We treat a 0.2 3, similar to outbreaks in Hong Kong e median and 50% confidence intervals 0 0 75th percentile values) of cumulative 0 50 100 150 0 50 100 hat such an epidemic is likely to spread delay in implementation (days) delay in implementation (days) population if uncontrolled (figure 3a, Figure 3. (Caption overleaf.) )ol strategies emphasizing contact pre1.0 lines) or quarantine and isoa, green 150 number 10 Importance of HCW mixing restrictions ρ in 5 preventing epidemics (control after 14 days): 0 20 and 4050060runs 80 100 988 J.histograms O. Lloyd-Smith and others transmission community -- 1SARSrun; piein 0acharts --hospital time (days) c=community pool, h=hospital pool Third, the hospital pool is considered to in (b) 25 c–to– c 13% number of infections 20 c– to –h 15% 15 h –to– h 57% h– to –c 15% 10 5 0 0 20 40 60 time (days) and SARS cases, but other patients are 25 explicitly. Infection of other patients has pla r = 0.1 c– to –c c –to– h r=1 cant part 4% in some 4% outbreaks, though it important in hospitals that eliminate nonh –to– c 20 cedures while SARS remains a significant ri 2% al. 2003; Maunder et al. 2003), and for regi h– to–hospitals h opened dedicated SARS or wards. 15 90% on hospital-community SARS outbreaks co ate patient dynamics, and could also evaluat staff reductions (Maunder et al. 2003) or of 10 tine of hospital staff following diagnosis of (as reported by Dwosh et al. 2003). 5 Some caution is required in identifying R0 with that obtained from incidence data for p breaks. As discussed already, reproduct 0 derived incorporate 80 100 0 20 from40data inevitably 60 80 100so control owing to routine healthcare practice (days) late R0 from itstime formal definition, however, as number of infections a) Third Case Study: TB in Humans Salomon, Lloyd-Smith, Getz, Resch, Sanchez, Porco, & Borgdorff, 2006. PLoS Medicine. 3(8), e273. Sánchez M. S., J. O. Lloyd-Smith, T. C. Porco,B. G. Williams, M. W. Borgdorff, J. Mansoer, J. A. Salomon, W. M. Getz, 2008. Impact of HIV on novel therapies for tuberculosis control. AIDS 22:963-972. Important elements: Includes important disease classes relating to latent vs. active, sputum smear positive vs. negative TB, DOTS vs Non-DOTS treatment, detectable vs. non-detectable Follows a competing rates formulation Time iteration is monthly: relates well to treatment regimen TB in and HIV background Core model of TB – elaborated SEIR framework Active TB Susceptible Latent Slo ss Fas ss Sus Re Recovered Not shown: all classes suffer natural mortality active cases suffer additional mortality TB treatment model Susc. Latent Slo Active TB ss ss− det ss ss+ det “Detectable” cases Sus Fas Re Recovered “Nondetectable” cases TB treatment model Susc. Latent Slo Active TB ss ss− det ss ss+ det Sus Fas Under treatment ss+ non-DOTS full rec Recovered partial rec. ss+ Partially recovered Defaulters Tx Completers TB treatment model Susc. Latent Slo Active TB ss ss− det ss ss+ det Sus Fas Under treatment ss+ non-DOTS full rec partial rec. ss+ partial rec. ss- TB treatment model Slo ss− DOTS ss ss− det ss− non-DOTS ss ss+ det ss+ DOTS Sus Fas ss+ non-DOTS full rec partial rec. ss+ TB/HIV treatment model TB HIV− TB HIV+ stage I TB HIV+ stage II TB HIV+ stage III TB HIV+ stage IV HIV incidence (external input) HIV progression HIV mortality partial rec. ss- TB treatment model Slo ss− DOTS ss ss− det ss− non-DOTS ss ss+ det ss+ DOTS Sus Fas ss+ non-DOTS full rec partial rec. ss+ TB REINFECTION HIV no HIV progression progression Slow Fas HIV+ stage II full rec Slow HIV+ stage III Fas TB-HIV CO-DYNAMICS IN KENYA: Monitoring Interacting Epidemics Sánchez M. S., J. O. Lloyd-Smith, B. G. Williams, T. C. Porco, S. J. Ryan, M. W. Borgdorff, J. Mansoer, C, Dye, W. M. Getz, 2009. Incongruent HIV and Tuberculosis Co-dynamics in Kenya: Interacting Epidemics Monitor Each Other. Epidemics 1:14-20. Tuberculosis notification rate, 2004 Notified TB cases (new and relapse) per 100 000 population 0 - 24 25 - 49 50 - 99 100 or more No report The boundaries and names shown and the designations used on this map do not imply the expression of any opinion whatsoever on the part of the World Health Organization concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. Dotted lines on maps represent approximate border lines for which there may not yet be full agreement. WHO 2005. All rights reserved HIV prevalence in adults, 2005 38.6 million people [33.4-46.0 million] living with HIV, 2005 Estimated HIV prevalence in new adult TB cases, 2004 HIV prevalence in TB cases, 15-49 years (%) 0-4 5 - 19 20 - 49 50 or more No estimate The boundaries and names shown and the designations used on this map do not imply the expression of any opinion whatsoever on the part of the World Health Organization concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. Dotted lines on maps represent approximate border lines for which there may not yet be full agreement. WHO 2005. All rights reserved 350 TB case notification rate 300 Misfit of TB Data in Kenya in HIV-TB model Data Calibration to data up to 2004 Calibration to data up to 1997 Uncertainty bounds to pre-1997 250 200 150 100 50 0 1980 1985 1990 Year 1995 2000 2005 Case Study: Circumcision & HIV Williams, B.G., Lloyd-Smith, J.O., Gouws, E., Hankins, C., Getz, W.M., Dye, C.,1, Hargrove, J., de Zoysa, I., Auvert, B, 2006. The potential impact of male circumcision on HIV incidence, HIV prevalence and AIDS deaths in Africa. PLoS Medicine 3(7):e262. Important elements: two sex model circumcised versus uncircumcised male categories Weibull Circumcision reduces female to male transmission of HIV by 70% Circumcision reduces female to male transmission of HIV by 70% green: S. Af.; red. E. Af.; orange: cent. Af.; blue, W. Af. Circumcision% reduces female to male transmission of HIV by 70% % prevalence circumcised Currently MC equivalent to a vaccine with 37% efficacy %prevalence reduction impact numbers 1000s p.y. Stochastic models in homogeneous populations Discrete Markov Chain Binomial Models Reed-Frost (class room lectures late 1920s at Johns Hopkins) E.g. Daley and Gani’s book: Epidemic Modelling, 1999 Graph theory interpretations of Reed-Frost models unidirected graph on N nodes, probability p of connections Giant component iff R0=pN>1 ⇒ z = 1 - exp(R0z) where z is expected value for (1-S∞) Stochastic models in homogeneous populations Continuous time stochastic jump process models SIR + demography E.g Ingemar Nasell, Math. Biosci. 179:1-19, 2002. Stochastic simulation of discrete time equivalents of SIR models with demography (including age structure) (e.g. HIV models, TB models, SARS models, bovine TB models) Problem with homogeneity! 1. Variation in host behavior: contact rates 2. Variation in host susceptibility: probability of infection 3. Variation in intensity of host infectivity: probability of infection 4. Variation in period of infectiousness: number of contacts and probability if infection 5. Several host strains with varying transmissibility and virulence. 6. Lots of others! Superspreaders: the effect of heterogeneity on disease emergence Lloyd-Smith, J. O., S, J. Schreiber, P. E. Kopp, and W. M. Getz, 2006. Superpreading and the impact of individual variation on disease emergence. Nature 438:335-359. Heterogeneity and epidemiology We have discussed disease models that assume homogeneous What about populations with heterogeneity? Common approach: break population into many sub-groups, each of which is homogeneous. What about continuous variability among individuals within well-mixed groups? Homogeneous models of disease: Individual Level Galton-Watson branching process theory: A probability generating function approach ∞ 1. Probability that I infects k individuals is qk : q = {qk }k = 0 ∞ 2. Probability generating function gq (z) = ∑ qk z k , 0 ≤ z ≤ 1 k =1 3. zn is probability I(t) = 0 at generation n : zn = gq (zn −1 ), z1 = q0 4. gq (0) = q0 , gq (1) = 1, gq′ (1)=R0 5. Each individual expects to infect ν : Poisson process: gq (z) = eν (z −1) Invasion condition (infinite pop size assumption, fixed generation time): Determistic: R0>1 Stochastic (homogeneous): R0>1 ⇒ prob{invasion}=1-1/R0 Heterogeneous models of disease: Individual Level 5. Each individual expects to infect ν (homogenous ⇒ Poisson process) 6. If ν is itself distributed (e.g. gamma) then process is not Poisson (e.g negative binomial) Parent distribution: Individual reproductive number ν Offspring distribution: Distribution of cases caused by particular individuals Standard Model I Completely homogeneous population, all ν = R0 Constant Parent distribution ν Poisson Offspring distribution Z fν (x) = δ ( x − R0 ) ∞ g(z) = ∫ e − x (1− z ) 0 = e − R0 (1− z ) fν (x)dx Standard Model II (SIR) Homogeneous transmission, constant recovery Exponential Parent distribution ν Geometric 1 − x R0 fν (x) = e R0 ∞ Offspring distribution g(z) = ∫ e − x (1− z ) fν (x)dx Z = 1 + R0 (1 − z ) 0 New Model Heterogeneous force of infection (superspreaders in right-hand tail) Gamma Parent distribution 1 kx fν (x) = Γ ( k ) R0 k −1 k − kx R0 R e 0 ν ∞ Negative Binomial Offspring distribution Z g(z) = ∫ e − x (1− z ) fν (x)dx 0 R0 = 1 + (1 − z ) k −k offspring Z ν ~ gamma Z ~ negative binomial Geometric k=1 Relatively few Poisson k→∞ parent ν 0.1 1 10 100 k greater Dispersion individualparameter, heterogeneity (parameter k) ∞ Empirical distributions The unprecedented global effort to contain SARS generated extensive datasets through intensive contact tracing: unique opportunity to study individual variation in a disease of casual contact. Beijing: Shen et al. EID (2004) Singapore: Leo et al. MMWR (2003) Superspreading events: Definition? Useful concept? Currently not useful! Should measure variation Beijing SARS hospital outbreak, 2003 Number of secondary cases: note superspreader events in tail What fits best? 1. ν ~ constant ⇒ Z ~ Poisson 2. ν ~ exponential ⇒ Z ~ geometric 3. ν ~ gamma ⇒ Z ~ negative binomial Singapore SARS outbreak, 2003 Singapore SARS outbreak, 2003 Singapore SARS outbreak, 2003 Singapore SARS outbreak, 2003 ν parent distribution Z offspring distribution ΔAICc Akaike weight ν ~ constant Poisson 250.4 < 0.0001 ν ~ exponential Geometric 41.2 < 0.0001 ν ~ gamma Negative binomial 0 >0.9999 Model selection strongly favours NB distribution with MLE parameters R0=1.63, k=0.16. ν Singapore SARS outbreak, 2003 Parent distribution ν is highly overdispersed: variance-to-mean ratio = 16.4 Singapore SARS outbreak, 2003 c.f. “20/80 rule”: 20% of cases cause 80% of transmission Evidence heterogeneity in other diseases SARS, smallpox, monkeypox, pneumonic plague, avian influenza, rubella All show strong evidence for individual variation P = Poisson model for Z generally rejected G = geometric model NB = negative binomial model Revisiting the 20/80 rule Superspreaders Superspreading Events (SSEs) How many cases make an SSE? SARS, 2003: • • • • Z ≥ 8, Shen et al. Emerg. Infect. Dis. (2003) Z > 10 Wallinga & Teunis, Am. J. Epidem. (2004) Z ≥ 10 Leo et al. MMWR (2003) “many more than the average number”, Riley et al. Science(2003) But what about measles (R0~18) or monkeypox (R0~0.8)? How to account for the influence of stochasticity? We need a general, scaleable definition of a SSE, based on probabilistic considerations. Proposed definition for superspreading events 1. 2. 3. Set context for transmission by estimating effective R0. Generate Poisson (R0) representing expected range in Z due to stochastic effects in absence of individual variation Define an SSE as any case who infects more than Z(99) others, where Z(99) is the 99th percentile of Poisson (R0). Superspreading events (SSEs) ■ R0 + 99th percentile of Poisson (R0) ♦ reported SSEs SSEs with >1 index case Superspreading Load Calculate R0 from data and ZPois-99 using Poisson model (number of infections demarcating 99 percentile) Fit negative binomial NegB(R0,k) to data Construct cummulative distribution ΦNB(ZPois-99 ) Calculate proportion in tail beyond ZPois-99 ΨNB(ZPois-99 ) =1-ΦNB(ZPois-99 ) Superspreader load (SSL) is 1-ΨNB(ZPois-99 ) /0.01 Predicting frequency of SSEs in Negative Binomial epidemics NegB(R0,k) NegB(10.3,1): SSL≈18 Implications for disease invasion Data from 10 diseases of casual contact show that individual variability in ν is a universal phenomenon. How does this variability affect: • Probability of stochastic extinction? (infinite population) • Timing of extinction? • Size of minor outbreak? (i.e prior to extinction) • Rate of growth if major outbreak occurs? We explored these questions using branching process models for ν ~ gamma Various Gamma distributions with R0=1.5 Special cases: k = 1 k=infty exponential ν: Geometric offspring dist. constant ν: Poisson offspring dist. smaller k greater variance in ν: Neg Biomial offspring dist. more aggregated Probability of disease extinction Greater variation in ν favors stochastic extinction, due to higher Pr(Z=0). Time to stochastic extinction High variability in ν (small k) ⇒ extinction happens fast or not at all. Implications for detection of emerging pathogens Expected size of minor outbreak (i.e. epidemic in infinite pop goes extinct) R0 < 1 E(total # cases) = 1/(1-R0) i.e. independent of k R0 > 1 E(total # cases) depends very weakly on k Rate of growth of major epidemic Greater variability ⇒ major outbreaks are rare but explosive! Conclusion • Data imply considerable heterogeneity in epidemics • Heterogeneity needed to explain rare explosive outbreaks, as in SARS • To estimate level of heterogeneity we need BOTH R0 and p0 (proportion of cases NOT transmitting) or SSL statistic • Control measures should target individuals in tails of parent distribution and hence reduce probability of explosive outbreaks How to do this an important area of research? Thanks! The End This research is supported by the US NIH, NSF, and James S. McDonnell Foundation