Can we Model Dengue Fever Epidemics in Singapore?

Transcription

1 anzapw15.tex iv.15 Printed 9 April 2015 Can we Model Dengue Fever Epidemics in Singapore? D. J. DALEY Department of Mathematics and Statistics The University of Melbourne Yes/No Barely? 1

2 Curiosity driven research: sparked by following figure (Stefan Ma, UniMelb graduate, Statistician, Singapore MoH) (co-organiser, Epidemic Modelling Workshop, NUS, 2005) Observed cases per 100 K year Annual DF notifications per 100K Singapore population

3 Background Singapore suffered explosion of Dengue Fever (DF) cases in early 60s: majority of all adults had dengue antibodies. DF became dominant cause of childhood illness & mortalitys. DF: Viral disease, spread by Aedes aegypti mosquito (daybiting, peri-domestic, likes clean water; same mosquito as yellow fever [building of Panama Canal later C.19 ], overcome by development of vaccine). c SIN govt. started larval source reduction program : Eliminate breeding places for the mosquito.? Successful: Proportion of housing with any larval mosquitoes nearby fell from c. 50% to < 2%; DF notifications fell away for 15 years as in figure data: 6948, 4552, 5330, 5309, 4602, c

4 DF virus: one of four serotypes DENV-1,..., DENV-4 A DF vaccine must protect against all serotypes: not yet found. Infection by DENV-n gives lifelong immunity to that serotype, limited immunity (weeks or months or 1 2 years) to others. Subsequent infection by other serotype can cause severe dengue (formerly called D Haemorrhagic Fever or D Shock Syndrome). [Antibody-dependent enhancement theory.] Data: Annual data do not distinguish between serotypes (standard serotype tests [ do you have dengue antibodies?] react to any of DENV-1 4). Specific serotyping possible but more expensive: requires infection to be recent done in lab.-confirm n of notified cases. Data give notified cases (lab. confirmed). Only 5 10% cases are clinical: remainder asymptomatic(seroprevalence survey). 4

5 DF endemic and more widespread in Indonesia and Malaysia (and Philipines and Thailand and... ) Worldwide: est d million cases annually; only malaria is more widespread. Modelling: Most modelling treat all serotypes as one. Most epidemic outbreaks of common vector-borne diseases are a consequence of sufficiently dense urbanization without adequate public health standards. Transmission of DF: DF infected human bitten by female mosquito (blood meal for food for eggs/larvae) Virus takes 5 7 days to reach and multiply in salivary glands as anti-coagulant sting fluid of mosquito. Subsequent stings on humans then infect them with virus. Belief: No vertical (inter-generational) DF transmission in hosts 5

6 DF/DHF in Singapore Table 1: Age distribution (%) of notified indigenous cases (i.e. acquired locally by Singaporeans, permanent and temporary residents) Age range Table 2: Incidence rates per 100K Age range Source: Singapore MoH Annual Reports, Vector-borne Communicable Diseases. 6

7 What can we learn from these data? In Table 1, the relative frequencies for a given age-group are more nearly constant than the frequencies for different agegroups. Table 2 expresses the same data via infection frequencies within a year on a per capita basis for each age-group. These show considerable variation between different years. Taken together, these tables support the product relation f(x,t) = infection rate in year t for age-group x f(x)λ(t) for functions f(x) and λ(t). Let the age-group x for the function f( ) be of length one year. Then f(x) is (approx.) estimated by the tabular entries of Table 1 divided by 5 (or perhaps 4.5) for the first column, and divided by 10 for the next five columns. 7

8 Mathematical modelling? Compartmental: Susceptibles, Infectives, Removed (recovered etc) Standard model for measles, chickenpox, mumps, pertussis: (X,Y,Z)(t) = counts of S, I, R at time t Approximatechangesin (X,Y,Z) via d.e. withx+y +Z = N: Ẋ = αxy/n, Ẏ = αxy/n βy, (1) Ż = βy. Solution for Y(t) v. t is approximately Epidemic curve. (Notified cases v. time hence, need new cases). System(1) NOT quite appropriate for vector-borne disease approximately true for an outbreak confined to a few months (e.g. DJD & Swift in Asmussen Festschrift = JAP Vol.48A). 8

9 Vector-borne: Infection occurs in two populations, with count functions (X 1,Y 1,Z 1 )(t) for humans, (X 2,Y 2 )(t) for mozzies. If a single outbreak, then because generation time 2 weeks (i.e. time between successive infections of (different) humans in a human-to-mosquito-to-human chain), human population size N 1 changes sufficiently slowly to be regarded as constant, hence the relations: Ẋ 1 = γ 12 X 1 (Y 2 /N 2 ), Ẏ 1 = γ 12 X 1 (Y 2 /N 2 ) βy 1, (2) Ż 1 = βy 1, Ẋ 2 = λ 2 N 2 γ 21 X 2 (Y 1 /N 1 ) µ 2 X 2, Ẏ 2 = γ 21 X 2 (Y 1 /N 1 ) µ 2 Y 2. [Parameters: λ h,µ h birth-, death-rates for popul ns h = 1,2. γ gh = effective contact-and-(infection)-transmission rate per g from g-to-h.] 9

10 To study endemic conditions over a longer time-scale, replace human population eqns by Ẋ 1 = λ 1 N 1 γ 12 X 1 (Y 2 /N 2 ) µ 1 X 1, Ẏ 1 = γ 12 X 1 (Y 2 /N 2 ) βy 1 µ 1 Y 1, (3) Ż 1 = βy 1 µ 1 Z 1, Non-trivialendemicconditionsrequire non-zerosolutionsy h to these eqns with all derivatives set = 0, AND verification that such a solution is a stable solution (d.e. theory). This approach can be ramped up to cover (a) age stratification; (b) social stratification of human population; (c) amplification to several serotypes (this needs variable coeffs. γ 12,γ 21 for var ble viral establishment ). [work in progress: can this explain data? prob. not, because swings in infection levels due predominantly to swings in dominant serotype (comment on tables)]. 10

11 Where is stochastic modelling? Consider human sample path: (birth).. (infection).. (recovery).. (death) J i (t) = S or I or R provided t lies in (birth, death) timeinterval for individual i. Probability of contact with insect in any time interval is relatively small, AND prob. that this insect is infectious is also small, i.e. rare events. AND contact with any particular insect is small. This is analogous to Poisson process conditions, hence develop eqns for p i,s (t), p i,i (t), p i,r (t). Add these eqns over i and deduce relations similar to (2) or (3) in which (X 1, Y 1, Z 1 )(t) are the first moments of the counting functions. [cf. Shupra Shah s talk later this morning.] Eqns (1) are known to be approximate first moment eqns obtained by neglecting cov(x(t), Y(t)). 11

12 (1) Eureka prize 4 Sept: Scott O Neill (Monash Univ) et al. (several places, incl. Qld DoH & James Cook Univ. Cairns: Scott Ritchie) Eliminate Dengue Research Program: Bacterium Wolbachia acts as vaccine in mosquito population stemming development of dengue virus within Aedes aegypti mosquitoes (= vector for DF and yellow fever). [Hence: transmission = life-path of virus is broken.] (2) The Age, Mon 9 Sept, p.4: Mozzie alert amid big virus increase Ross River virus (3500 cases to date; 4684 in 2012) Barmah Forest virus (3465 cases to date; 1722 in 2012) Dengue fever (1238 cases to date) (Annual dengue cases in last decade: max. about 1200) RR and BF viruses: endemic to Australia DF virus: All cases start with imported initial case In recent times: DF eliminated (spraying etc.) every year BUT: A. aegypti can live in mid-nsw (Coffs Harbour). 12

13 Bon jour 13

14 DF in N Queensland c. 1897: Hare (hosp. superintendant, Qld; Aust. Med. Gazette (1898) gave first good clinical description of the disease. After 1905 Brisbane epidemic, Bancroft (Brisbane GP) correctly identified A. aegypti as mosquito vector; Cleland (Sydney) confirmed B. s work via Coffs Harbour outbreak (1916). Dengue Fever infection is typically asymptomatic: else: breakbone fever (severe pain in bones, joints, muscles), sudden onset of high fever, nausea and vomiting, headache (esp. behind eyes), pink rash. Recovery after a week, but fatigue, depression may linger. Symptomatic rate 1 in 10 or 20 Notifications approx. 5 to 10% of all infections. [Serological surveys for DF antibodies in e.g. Singapore ( s), Charters Towers (Qld) 1993 epidemic, survey in 1995] 14

15 Parasite host systems: The parasite s view: cause minimal disturbance to host (DENV-(1 4) has no obvious effect on mosquito host typically asymptomatic in (most) human hosts) DENV emerged c years ago, diverged to four viruses that compete :?? Biodiversity (!) antibody-dependent enhancement theory (Halstead) Virus life-cycle : lives in human blood-stream After mosquito ingests virus with blood-meal, virus takes 5 7 days to (a) replicate and (b) reach salivary glands from where (c) mosquito injects anti-coagulant + virus into human blood Pr{a biting mosquito is infectious} = O(0.01) (because mozzie must survive a week after ingestion). Most epidemic outbreaks of vector-borne diseases are a consequence of sufficiently dense urbanization without adequate public health standards. Belief: no vertical (inter-generation) DF transmission Implies: virus lives via transmission between two hosts [If not: PROBLEM (!) ] 15

16 Mathematics (and statistics, like most of science) uses Toy models Models focus on key assumptions Suitably refined, for maths/physics/engineering, models enable lunar travel Mathematical models (with assumptions (!) ) describe (1) computation, (2) structure and structural relationships (logic) Maths in biological setting: both (1) and (2) [Economics setting: wilder fit than biological] But, quantitative relations more approximate than physics; logic remains. Paradigm math l model: branching process (e.g. Richard Duncan, founding population size). generation-wise evolution: viral infect n generation-time = weeks(animals: months years) offspring no./generation: either 0, else potentially high (e.g. measles > 10 ; DF?? ) typically R 0 BUT R 0 = Basic Reproductive Ratio, reflects social milieu vector-borne disease model approximated by standard model 16