Optimal Interventions in Infectious Disease Epidemics: A Simulation Methodology Jiangzhuo Chen Network Dynamics & Simulation Science Laboratory INFORMS at Virginia Tech November 30 th, 2011

Network Dynamics & Simulation Science Laboratory Talk Outline Background: –Propagation of infectious disease on social contact networks –Intervention strategies Vaccine Assignment Problem –Mathematical formulation –Simulations

Network Dynamics & Simulation Science Laboratory Background: SEIR Disease Model Influenza like illness Each person in one of four states: S, E, I,R Disease can only transmit from infectious person to susceptible person r(u,v): prob. that u transmits disease to v per unit time

Network Dynamics & Simulation Science Laboratory Background: Social Contact Network Daily activities move people between locations People staying at the same location at the same time may have contacts with each other (physical proximity) Contact network G(V,E) –V: people –E: (u,v) if u and v have contacts –w(u,v): edge weight for contact duration Disease may spread from node to node along the edge (contact)

Network Dynamics & Simulation Science Laboratory Disease Spread in Contact Network Within-host disease model: SEIR –State transitions are probabilistic and timed. Between-host disease model: transmission occurs along edges of a social contact network –People + Locations => Contacts. –Transmissions are probabilistic.

Network Dynamics & Simulation Science Laboratory Disease Spread in Contact Network Transmission depends on –Duration of contact –Type of contact –Characteristics of the infectious person –Characteristics of the susceptible person

Network Dynamics & Simulation Science Laboratory Synthetic Social Contact Network Synthetic population based on census data –Individual demographics: age, gender… –Household characteristics: size, income…

Network Dynamics & Simulation Science Laboratory Synthetic Social Contact Network Locations: Dun&Bradstreet data Synthetic activities based on activity surveys. –Matched to individuals by demographics –Matched to locations by activity type Synthetic social contact network –People follow activity schedules –Activities take them to locations –At locations they interact with each other

Network Dynamics & Simulation Science Laboratory Synthetic Social Contact Network

Network Dynamics & Simulation Science Laboratory Synthetic Social Contact Network We have generated networks for major urban regions of US: Miami, Seattle, Chicago, NYC, etc. We have generated network for regions outside US: Beijing, Delhi. These networks are of large-scale and very complex –E.g. NYC synthetic contact network has 18 million people and about 1 billion contacts

Network Dynamics & Simulation Science Laboratory Background: Interventions Pharmaceutical interventions: vaccination or antiviral changes an individual’s role in the transmission chain –Lower susceptibility or infectiousness Non-pharmaceutical interventions: social distancing measures change people activities and hence the social network –Sick leave, school closure, isolation, etc.

Network Dynamics & Simulation Science Laboratory Complications in Interventions Supply: vaccines may not be ready; antiviral stockpile; production capacity; available leave days Compliance: not all individuals will be able or willing to comply with an intervention policy Cost: drug cost; productivity loss Delay: vaccine takes a few days to become effective

Network Dynamics & Simulation Science Laboratory Optimal Interventions Effectiveness of an intervention depends on when and to whom it is applied When is it applied? –Too early: unnecessary cost; too late: outbreak out of control Who are targeted? –Supply constraints may require prioritization of groups for different interventions Objective varies Mitigating epidemic: minimize number of cases; reduce mortality; delay outbreak Cost-benefit analysis

Network Dynamics & Simulation Science Laboratory Talk Outline Background: –Propagation of infectious disease on social contact networks –Intervention strategies Vaccine Assignment Problem –Mathematical formulation –Simulations

Network Dynamics & Simulation Science Laboratory Vaccine Assignment: A Mathematical Formulation VA(G, r, τ E, τ I, x, k) –G(V,E) contact network –SEIR disease model (r,τ E,τ I ) r(u,v): prob. that u infects v per unit time τ E : incubation duration (time in state E) τ I : infectious duration (time in state I) –x in [0,1] n : prob. that each node is infected initially –k: vaccine supply Choose subset of nodes S with |S| at most k, so that expected number of infected nodes is minimized –Nodes in S are removed (assuming 100% vaccine efficacy) Stochastic combinatorial optimization

Network Dynamics & Simulation Science Laboratory Vaccine Assignment Problem is Hard Theorem VA(G, r, τ E, τ I, x, k) is NP-complete if r(u,v)=1 and there is a node s such that x(s)=1 and x(v)=0 for any other node v. Difficult to solve analytically for –realistic settings –large scale, unstructured network –Complicated intervention strategies

Network Dynamics & Simulation Science Laboratory Simulation Methodology Synthetic contact network Fast simulation tool: EpiFast (MPI code) –A few seconds for simulating a flu season in a multi-million population (e.g. Seattle) –Can handle sophisticated intervention strategies Find optimal from a set of feasible intervention strategies by comparing simulation results Factorial experiment design + replicates = many runs! Realistic suggestions for public health policy makers

Network Dynamics & Simulation Science Laboratory Simulation Design: Populations Two US cities City MiamiSeattle population average age average household size average household income average degree 4954

Network Dynamics & Simulation Science Laboratory Simulation Design: H1N1 Flu Catastrophic flu: very high infectivity Average incubation duration = 1.2 days Average infectious duration = 4.1 days 20 random seeds at beginning of epidemic 25 replicates for every configuration

Network Dynamics & Simulation Science Laboratory Simulation Design: Vaccines Limited supply: number of doses equal 10% of city population size Vaccines are applied one month after epidemic starts Vaccine reduces transmission probability by 80%

Network Dynamics & Simulation Science Laboratory Simulation Design: Vaccine Assignment To minimize attack rate, it is intuitive to give vaccines to most vulnerable people. Vulnerability of each person is his probability of getting infected. We use EpiFast simulations to compute the vulnerability measure: 1000 replicates. How good is this strategy?

Network Dynamics & Simulation Science Laboratory Assign Vaccines to Most Vul People

Network Dynamics & Simulation Science Laboratory Optimal Implementable Policies Unfortunately it is not implementable to directly assign vaccines to most vulnerable people. We can identify them in our synthetic population through EpiFast simulations. But in real population, it is difficult to find them. Can we make use of vulnerability measure and assign vaccines based on it?

Network Dynamics & Simulation Science Laboratory Optimal Vaccine Assignment: idea 1 Partition population into groups. Allocate vaccines to groups based on their average vulnerability. Various ways for grouping; most naturally by age. 5 age groups: [0,19], [20,39], [40,59],[60,79], [80,∞).

Network Dynamics & Simulation Science Laboratory Idea 1: allocation matters Fair allocation: give same amount of vaccines to each group. Weighted allocation: fraction of vaccinated people in each group is proportional to “group vulnerability”.

Network Dynamics & Simulation Science Laboratory Idea 1: In-Group Assignment also matters With same between-group allocation (weighted), it matters how to assign vaccines within each group: randomly, to most vulnerable, or to least vulnerable.

Network Dynamics & Simulation Science Laboratory Idea 1: Grouping by Multi-Dimension There are other natural variables: household size, household income, degree in social network. Does it help to further partition the groups by using more and more dimensions? DimensionNumber of groups How A (age)5see previous slides S (household size)51, 2, 3, 4, 5 and above I (household income)3low, medium, high; evenly D (degree in social network)3low, medium, high; evenly

Network Dynamics & Simulation Science Laboratory Idea 1: Grouping by Multi-Dimension Further grouping does not help much.

Network Dynamics & Simulation Science Laboratory Idea 1: Limited Effectiveness “Lower bound”: assigning to most vulnerable people. Weighted allocation performs much less effectively than “lower bound”.

Network Dynamics & Simulation Science Laboratory Idea 2: Winner-Takes-All Allocation Assign all vaccines to most vulnerable age group. Which age group is most vulnerable? age group 1

Network Dynamics & Simulation Science Laboratory Idea 2: Winner-Takes-All Allocation Assign all vaccines to age group 1: outperforms weighted allocation. Can we do better?

Network Dynamics & Simulation Science Laboratory Idea 2: Better Proxy for Vulnerability Contact of each person is sum of durations of all his contacts. (weighted degree) Contact has strong correlation with vulnerability. Divide people into 3 contact groups (C): low, medium, high. Or combine contact and age for grouping. Assign all vaccines to most vulnerable contact group, or most vulnerable (age+contact) group.

Network Dynamics & Simulation Science Laboratory Idea 2: All Vaccines to Most Vul Group

Network Dynamics & Simulation Science Laboratory Why Winner-Takes-All Works Better? Same grouping by age; different allocation schemes: fair, weighted, winner-takes-all.

Network Dynamics & Simulation Science Laboratory Why Winner-Takes-All Works Better: Age Groups

Network Dynamics & Simulation Science Laboratory Why Winner-Takes-All Works Better: Age Groups Large variance of vulnerability within each age group: under random assignment vaccines often do not go to most vulnerable people in each group. Age group 1 is much more vulnerable than other groups: in both Miami and Seattle, about 80% of people in age group 1 has vulnerability larger than average of any other age group. Giving all vaccines randomly to age group 1: vaccines very likely go to highly vulnerable people.

Network Dynamics & Simulation Science Laboratory Why Winner-Takes-All Works Better: Contact Groups

Network Dynamics & Simulation Science Laboratory Why Winner-Takes-All Works Better: Contact Groups Even more obvious for contact grouping: when all vaccines are given randomly to contact group 3, more than 99% of the recipients have vulnerability larger than average vulnerability of any other contact group. Coefficient of correlation between vulnerability and contact is more than 0.95 for either Miami or Seattle!!

Network Dynamics & Simulation Science Laboratory CDC Recommendations Pre-2008 vaccination recommendation for seasonal flu: age 6mo to 5yr and 50yr and above. For seasonal flu (after 2008): age 6mo to 18yr and 50yr and above. Vaccination guideline for H1N1 flu (July 2009): age 6mo to 5yr then 5yr to 18yr. Subsequent guideline for H1N1 flu vaccination (Oct. 2009): age 6mo to 24yr.

Network Dynamics & Simulation Science Laboratory Compare CDC and our vaccination schemes CDC is improving. All its strategies are outperformed by ours.

Network Dynamics & Simulation Science Laboratory Compare CDC and our vaccination schemes The best CDC vaccination strategy decreases epidemic peak by 52%, and delays the peak by 25 days. Our optimal vaccination strategy decreases epidemic peak by 70%, and delays the peak by 46 days! The lower the peak, the better our logistics can handle the worst case scenario. The more we delay the outbreak, the better we can get prepared and come up with other measures.

Network Dynamics & Simulation Science Laboratory Optimal Vaccine Assignment: A Solution Group people by their total contact time with others, or by age, or by both. Social contact network + EpiFast: tells which group is most vulnerable. Assign all vaccines randomly to people in most vulnerable group.