Using Mathematical Models To Understand Transmission Of Infection In Vaccinated Populations

January 22, 2021

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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.