Using Mathematical Models To Understand Transmission Of Infection In Vaccinated Populations
January 22, 2021Information
Virginia E. Pitzer, ScD, associate professor of epidemiology (microbial diseases), Yale School of Public Health, discussed how mathematical models can be used to understand the transmission of infection in vaccinated populations, including direct vs. indirect (herd immunity) vaccine protection.
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- 00:00For the introduction.
- 00:03And hopefully this screen is sharing now.
- 00:09And thanks for the opportunity to talk.
- 00:11As Melissa said,
- 00:12I'm going to be talking about how we
- 00:15can use mathematical models to really
- 00:18better understand transmission of
- 00:20infection in vaccinated populations.
- 00:22And really, what I'm focusing on
- 00:24here is the distinction between
- 00:26the direct and indirect protection
- 00:28that is confirmed by vaccination.
- 00:31We've heard a lot so far,
- 00:34mostly about the direct protection
- 00:36that can be confirmed by vaccination.
- 00:39Two individuals receiving the vaccine
- 00:41and preventing them from having
- 00:43kind of severe consequences of
- 00:45infections such as hospitalization,
- 00:47and this is generally estimated from
- 00:50randomized control trials or estimates of.
- 00:53Effectiveness from case control studies.
- 00:56But vaccines can also confer
- 00:59indirect protections by preventing
- 01:02individuals who may be UN vaccinated
- 01:06from becoming infected and shedding.
- 01:09This pathogens and infecting other
- 01:12individuals in the population who,
- 01:14for example,
- 01:16maybe too young to receive vaccines.
- 01:19And thereby preventing those individuals
- 01:21who are exposed from developing severe
- 01:24disease and being hospitalised themselves.
- 01:26And so this indirect protection,
- 01:29which is also been also referred
- 01:31to us herd immunity,
- 01:33is something that really can only be
- 01:36estimated from very specific cluster
- 01:38randomized trial design or something
- 01:41that can be estimated and predicted
- 01:44using dynamic mathematical models.
- 01:47And the way that these models work is
- 01:49to follow generally what's considered
- 01:51the basic sirf type model design,
- 01:54where we assume that when
- 01:56individuals are born,
- 01:57they might be susceptible to
- 01:59infection with a particular pathogen,
- 02:01an ask they are exposed to
- 02:04that pathogen overtime,
- 02:05they may become infected and
- 02:07in turn these individuals are
- 02:09infectious to other individuals,
- 02:11and So what we're most concerned
- 02:13about with these models is tracking
- 02:15infectious individuals as opposed to.
- 02:18The cases of disease within the population.
- 02:21And when once that infection resolves and
- 02:23individuals and no longer infectious,
- 02:25we can assume that they may have
- 02:27any bodies and be recovered and
- 02:29be immune from further infection,
- 02:31at least for some period of time.
- 02:34And there also is death that can occur
- 02:37from all of these compartments and
- 02:39then all of this really gets described
- 02:42by series of differential equations,
- 02:45which are mathematical expressions
- 02:47just showing how the rate of.
- 02:50Or the number of individuals in
- 02:53each state changes overtime in
- 02:55relation to these various rates.
- 02:58And when it comes to modeling
- 03:00vaccination kind of,
- 03:01the simplest way to do it within
- 03:04the sort of basic sirf type model
- 03:06framework is to assume that some
- 03:09fraction of individuals which is
- 03:11considered be here who are vaccinated
- 03:14and affectively protected by the
- 03:16vaccine might be moved from the
- 03:18suseptable compartment into the
- 03:20recovered and immune compartment while
- 03:22bypassing the infectious compartment,
- 03:24and so this reduces the number of
- 03:27currently infectious individuals.
- 03:28Within the population.
- 03:29And if you implement this in
- 03:32a very simple sirf type model,
- 03:36assuming and are not a 5 or the
- 03:38an average number of secondary
- 03:41infections produced by an infectious
- 03:43individual on a fully susceptible
- 03:46population and a 50%
- 03:47vaccine coverage with 100%
- 03:49effective vaccine, or vice versa,
- 03:51100% coverage with a 50% effective vaccine,
- 03:54then on left in blue here is plotted
- 03:58what the epidemic would look like.
- 04:01Kind of each week through time.
- 04:03If there was no vaccination
- 04:05while on the right,
- 04:06the blue line represents the total
- 04:09number of cases cumulatively
- 04:11through time with no vaccination.
- 04:13And the dashed redline presents what
- 04:15you would expect if the vaccine were
- 04:18really just providing the direct
- 04:20protection to vaccinated individuals
- 04:22and preventing them from getting sick,
- 04:25which would just be a same epidemic
- 04:28but 50% smaller through time.
- 04:30The solid red line here represents what we
- 04:34actually see within the model framework.
- 04:37If you vaccinate 50% of the population before
- 04:40the vaccine or the epidemic takes off,
- 04:43which is an epidemic,
- 04:45which is considerably delayed and
- 04:47blunted compared to the epidemic that
- 04:50you see without any vaccination.
- 04:52And if you look at the cumulative
- 04:55number of cases occurring overtime
- 04:57by the end of the epidemic.
- 05:00With vaccination you see.
- 05:01Lower cumulative number of cases
- 05:03that occur within the population,
- 05:05then would be expected just based
- 05:07on this direct protection from
- 05:09the vaccine alone,
- 05:10and this difference between what
- 05:12you'd expect from the direct
- 05:14protection alone versus what you
- 05:16actually get from this reduction in
- 05:18transmission that occurs is generally
- 05:20measured as the indirect effect.
- 05:22But you'll note that if you
- 05:24measured the indirect effect,
- 05:26say back on week seven here,
- 05:28you would estimate a considerably
- 05:30stronger indirect effect,
- 05:31and so that's one aspect of this
- 05:34indirect effect is that it's inherently
- 05:37dynamic and changing overtime.
- 05:39And Furthermore it changes with
- 05:40coverage within the population.
- 05:42So for example,
- 05:43if you had 80% coverage in this scenario,
- 05:46according to the direct protection,
- 05:47you just expect to see an epidemic
- 05:50that's 20% the size of the epidemic.
- 05:52But in this case,
- 05:54if you have an 80% effective vaccine,
- 05:56you would be able to eliminate the
- 05:59pathogen altogether and prevent the
- 06:01epidemic from occurring in the first place.
- 06:03More generally,
- 06:04these models can be adapted to
- 06:07account for the waning vaccine,
- 06:09induced immunity and leaky protection,
- 06:11for example,
- 06:12by including a separate compartment
- 06:14for vaccinated individuals,
- 06:15which can then kind of Wayne back
- 06:18into this acceptable state or have
- 06:20a differential rate of infection
- 06:23occurring from this compartment.
- 06:25Or can be modified to allow for infected
- 06:28individuals to be somehow different
- 06:31from UN vaccinated infected individuals.
- 06:34So, for example,
- 06:35less infectious than unvaccinated
- 06:37infected individuals.
- 06:38And in reality,
- 06:40these vaccines models get much more
- 06:42complicated when you take into
- 06:45account the specifics of natural
- 06:47immunity and the Natural History of
- 06:50infection of different pathogens.
- 06:52These are just two different
- 06:55models for rotavirus.
- 06:56One in which we don't account for
- 06:59the different strains of rotavirus
- 07:00and one in which we do,
- 07:02which get considerably kind of
- 07:04more increasingly complicated and
- 07:06in reality what I spend my time
- 07:08looking at is a whole bunch of code
- 07:11that is used to implement these
- 07:13models in a computer program.
- 07:15And so the ways in which these
- 07:19different models can be used,
- 07:21including explaining observed
- 07:22patterns in data. So, for example,
- 07:25models have helped us to understand how the
- 07:29seasonality of rotavirus epidemics changed
- 07:32following vaccine introduction in the US,
- 07:35Anan why this change of curd.
- 07:38Furthermore, we've also used models
- 07:40to evaluate cost effectiveness.
- 07:42Specifically, my group has looked at
- 07:44the cost effectiveness of different
- 07:47vaccination strategies against typhoid fever.
- 07:50Asking is it cost effective to introduce
- 07:53typhoid vaccines in various low income
- 07:56countries and kind of under what conditions?
- 08:00And then finally,
- 08:01these models can be used to address
- 08:04issues around future trends,
- 08:05like for example,
- 08:07can we possibly eliminate COVID-19
- 08:09through vaccination?
- 08:10And so I just want to briefly into
- 08:13a couple examples from my own work.
- 08:16So back in 2009 I was involved in a
- 08:19study looking at trying to understand
- 08:21the early impact of rotavirus
- 08:24vaccine introduction in the West,
- 08:26where what was observed following the
- 08:29introduction of rotavirus vaccines
- 08:31in 2006 isn't at the first season
- 08:33following vaccine introduction,
- 08:34which occurs kind of only among
- 08:37infants in the US.
- 08:39The rotavirus epidemic in the US was
- 08:41really kind of similar in size to previous.
- 08:45Pre vaccination epidemics,
- 08:46so in blue is the 2006 2007 rotavirus
- 08:50season and number of rotavirus positive
- 08:54specimens in the US surveillance system,
- 08:58whereas the grey in the black is the
- 09:01mean pre vaccination rotavirus season.
- 09:05Whereas in the second season following
- 09:08vaccine introduction plotted in red here,
- 09:11the epidemic was considerably smaller than.
- 09:15Free vaccinate vaccination,
- 09:16epidemics and pizza around 10 weeks after
- 09:20the usual peak in the rotavirus season.
- 09:23So current kind of quite a bit
- 09:26later in kind of late winter.
- 09:28Early spring time.
- 09:30Um?
- 09:30And initially this was not really well
- 09:33understood because it wasn't really
- 09:36known that introducing rotavirus vaccine
- 09:39would prevent a lot of transmission,
- 09:42since only infants were getting vaccinated.
- 09:45But based on models that we have fitted
- 09:48to pre vaccination data in the US,
- 09:51in particularly the spatio temporal
- 09:53pattern of epidemics in the US,
- 09:55We were able to retrospectively
- 09:57predict this delay in the timing
- 10:00of rotavirus epidemics,
- 10:01particularly in the second season
- 10:03following the introduction of the
- 10:06vaccine based solely on the idea that
- 10:08infants seem to be the ones who are most
- 10:12infectious when infected with rotavirus.
- 10:14And this provided an important
- 10:16form of that model.
- 10:18Validation for kind of future prediction,
- 10:20although of course models aren't
- 10:22perfect and we didn't do a great
- 10:24job of reproducing kind of the
- 10:26relative size of the epidemics
- 10:28that a curd in in 2008 2009,
- 10:30compared to 2000 seven 2008.
- 10:32Although we did again get the
- 10:34timing quite similar,
- 10:35but one of the things that we
- 10:37predicted with this model was that
- 10:39ask the coverage within the under five
- 10:41children population kind of increased.
- 10:43You would start to see.
- 10:45Lower our sort of later and later,
- 10:48at epidemics of rotavirus occurring
- 10:50each year until you reach this region
- 10:53in which you will see kind of 80
- 10:56to 90% coverage among the eligible
- 10:58infant population and what was happening
- 11:00here was that you actually the model is
- 11:03predicting that you actually start to
- 11:05get epidemics occurring every two years,
- 11:08or these biennial epidemics
- 11:09of rotavirus happening.
- 11:10And this was something that
- 11:12we predicted back in 2009,
- 11:14just after the vaccines have been rolled out.
- 11:17And sure enough,
- 11:18if you look at more recent data,
- 11:21this for example from New York City,
- 11:24in which we have rotavirus hospitalizations
- 11:27encoded in blue here and lab confirmed
- 11:30rotavirus cases within NYC in gold.
- 11:32Here you can see that sure enough,
- 11:35beginning around 2011, 2013,
- 11:37you really are starting to see this pattern,
- 11:41in which you're getting rotavirus
- 11:43epidemics happening primarily in
- 11:44the odd numbered years and much
- 11:47lower rotavirus activity happening.
- 11:49In even numbered winter seasons,
- 11:51and this was very consistent with
- 11:53the predictions that we had for
- 11:56our model in a shift in the in
- 11:59the disease to potentially kind
- 12:01of slightly older age groups.
- 12:04So another policy question that we've
- 12:06addressed using these mathematical
- 12:08models is the question around in
- 12:11which Gabby eligible countries.
- 12:12Would it be potentially cost
- 12:14effective to introduce these novel
- 12:17typhoid conjugate vaccines which have
- 12:19just been developed and licensed
- 12:21and approved recommended by WHL
- 12:23within the past couple of years,
- 12:26and in particular when introducing
- 12:28these vaccines,
- 12:29which would be the best strategy
- 12:31just to routinely vaccinate
- 12:33infants at nine months of age?
- 12:36Or to potentially also include a
- 12:38catch up campaign among children
- 12:40up to five or 15 years of age.
- 12:42And so we evaluated these three
- 12:45different strategies using a dynamic
- 12:47model of typhoid transmission in
- 12:49order to predict vaccine impact
- 12:51over a 10 year period,
- 12:53and generally found reductions that
- 12:55vary slightly from country to country,
- 12:57based primarily on the age structure
- 13:00of the country,
- 13:01but that overall routine vaccination
- 13:03plus a catch up campaign to 15 years of
- 13:07age was predicted potentially decrease
- 13:09typhoid fever incidents by around 58%.
- 13:11Overall Gabby eligible countries.
- 13:13And when we consider the costs
- 13:15associated with vaccine introduction as
- 13:18well as the illness itself generally,
- 13:21what we found was that routine vaccination.
- 13:23Plus this catch up campaign to 15
- 13:26years of age was always the preferred
- 13:29strategy whenever introducing typhoid
- 13:31vaccines in the 1st place was cost
- 13:34effective and the strategy was
- 13:36likely to be cost effective based
- 13:38on willingness to pay thresholds
- 13:40or willingness to kind of adopt
- 13:43A health strategy.
- 13:44That's somewhat reasonable in in 38
- 13:46out of 54 Gabby eligible countries.
- 13:49For example,
- 13:50when the threshold set to 25% of the
- 13:53GDP per capita, which is relatively low.
- 13:58Threshold.
- 13:59Um?
- 13:59And one of the things that we've
- 14:02done is to then take some of these
- 14:04results and to put it on a website
- 14:07that take on Typhoid website,
- 14:09which is one of the main advocacy
- 14:11websites around typhoid vaccine
- 14:12information that can be then used to
- 14:15allow potential decision makers to
- 14:16explore some of these results on their own.
- 14:19And then finally,
- 14:20when it comes to this question
- 14:22of what's going to happen with COVID-19
- 14:25and the question of herd immunity around
- 14:29introducing COVID-19 vaccination.
- 14:31And will we eventually be
- 14:33able to eliminate COVID-19?
- 14:35My take on this is that the
- 14:37answer is maybe that is really,
- 14:39but that's really going to require
- 14:41massive undertaking in the way
- 14:43models help us is to consider this
- 14:45critical proportion to vaccinate,
- 14:47which is going to be equal to 1 -- 1
- 14:50over are not an for for COVID-19 if you
- 14:53consider and are not of around 3:00,
- 14:56this is where you get some of
- 14:58these estimates of 60 to 70% of
- 15:00people needing to be vaccinated.
- 15:02But this is not just the coverage
- 15:04that's needed, it's really.
- 15:06The coverage plus the efficacy
- 15:08against transmission,
- 15:09which we don't really know and so For
- 15:12these reasons I think it's really
- 15:14going to be be difficult to eliminate
- 15:17infection altogether with vaccination.
- 15:19And so with that I'm going to end and thank
- 15:23you again for the invitation to speak.