# Mathematical modeling a tool for decision making in situations

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Mathematical modeling: et traumatismes

a tool for decision making in situations of uncertainty
Infectious
Diseases               Memorandum on the development by the Institut de veille sanitaire
of three projects related to the modeling of the impact of immunization
against chickenpox, cervical cancer and rotavirus infections

Introduction                                                                A multi-disciplinary approach
Vaccines against diphtheria, tetanus and pertussis were
initially integrated into the immunization schedule in the hope               "Mathematical models allow us to establish
of avoiding hundreds of deaths per year, compared to only                     scenarios and to virtually test the impact of
dozens for the majority of vaccines now commercialized in                     public health actions, and more specifically
industrialized countries.                                                     immunization."
Their intended public health objectives are less clear, while their           Daniel Lévy-Bruhl, Department of Infectious Diseases, InVS
cost is constantly increasing due to research and development
investments and stricter security requirements. It is therefore
becoming more and more difficult to determine if a new vaccine              In 1760, mathematician Daniel Bernouilli was already
should be integrated or not into the immunization schedule.                 calculating probabilities that led him to "geometrically" assert
That decision is based on an expertise process that each country            that smallpox vaccination would be beneficial to all.
should undertake once appropriate authorities have allowed the
commercialization of a new vaccine (in France, that authority is            But mathematical model concepts, as used nowadays in
the French Health Products Safety Agency or Agence française                epidemiology, only appeared in early 20th Century, following
de sécurité sanitaire des produits de santé, Afssaps, and in                the introduction of the mass action principle. According to
Europe, the EMEA or European Medicines Agency). This process,               this principle, the spread of an infectious agent within a given
which involves various partners including the French Institute              population depends on the proportion of infected individuals,
for Public Health Surveillance (Institut de veille sanitaire, InVS),        and of susceptible non-immunized people (either through the
is carried on under the aegis of the Immunization Technical                 disease or through vaccination). Spreading of the infectious
Committee (Comité technique des vaccinations, CTV).                         agent can only occur above a "threshold" ratio of susceptible
All immunization-related fields (and more specifically                      people, immunized individuals representing a natural obstacle.
immunology, epidemiology, sociology and economy) are called                 This led in the late 1920s to the concept of critical threshold,
upon to assess the costs and benefits of a given immunization               which completed the mass action principle.
strategy.
Epidemic spread is also determined by the R0 parameter,
Within that framework, mathematical modeling represents an                  which represents the number of cases induced in a susceptible
extremely valuable decision support tool, which is being used               population by an infected individual. R0 is contingent upon
on an increasingly regular basis.                                           not only the inherent characteristics of the infectious agent
When should mathematical models be used? What do they                       (transmissibility), but also on the characteristics of the population
entail? How do they facilitate the immunization decisional                  affected. Age distribution, educational level, and individual
process? What are their limits?                                             behavior are some of the factors influencing the "contact rate"
These are some of the issues covered in this executive summary,             amongst contagious and susceptible individuals, and therefore
based on three mathematical models implemented by the InVS                  the R0 value. In a population where only one S ratio of the
for commercializing three types of vaccines against respectively            population is susceptible, the number of induced secondary
chickenpox, cervical cancer and rotavirus infections.                       cases is reduced proportionally, R0 being then replaced by R,
where R equals R0 x S.

Infectious disease control measures intend to reduce this
R value in order to limit and even stop the spread of the
infectious agent.

Mathematical modeling: a tool for decision making in situations of uncertainty – French Institute for Public Health Surveillance / p. 1
While quarantine measures are empirically aimed at reducing                   InVS, between modellers and
R0 by limiting contacts between contagious and susceptible                    health professionals
individuals, immunization has a direct impact on R by reducing
the S population ratio.                                                       Three models have recently been developed through the InVS
to support decision-making in terms of vaccination strategies to
If through control measures R can drop below 1, each case leads               be implemented for three vaccines (against chickenpox, cervical
to less than one new case and the number of cases declines                    cancer and rotavirus infections).
by vaccinating a critical part of the population, to limit the                In developing these models, the InVS was able to take full
spread of an infectious agent and possibly to stop it. This is                advantage of external expertise in the matter, using its extensive
the herd immunity concept on which are based immunization-                    network of partners, which includes research institutes as well
related disease eradication strategies, one of which helped                   as French and international government agencies.
Predicting the impact of chickenpox
Nowadays, infectious disease control relies on more advanced                  vaccination for infants in France
strategies, eventually combining several prevention methods,
sometimes targeting very specific population subgroups, based                 The upcoming commercialization of a new vaccine combination,
on age or risk of exposure to the disease (associated for example             which includes the trivalent vaccine against measles, mumps,
to a professional occupation or to a specific behavior).                      rubella (MMR) and the chickenpox vaccine, makes the
immunization of infants against chickenpox feasible. Modeling
Hence the use of mathematical models that help simulate                       allowed to assess the impact of such vaccination on the spread
transmission of a disease among the population, based on                      of chickenpox within the population.
a multi-disciplinary approach and integrating all R0 and S
determining factors within a given population. The benefits of
specific preventive strategies can also be virtually assessed for
"The benefits of smallpox vaccination must be
optimal adaptation of actions undertaken to remedy a public
assessed against the potentially greater number
health issue.
Isabelle Bonmarin, Department of Infectious Diseases, InVS
How is a mathematical model constructed?

The first step in constructing a mathematical model is to                    Widespread vaccination of children against some early childhood
reproduce, using equations, the transmission pattern of                      diseases may result in an age shift towards adult cases, knowing
the disease within the targeted population, based on the                     that the risk of serious complications, including death, is higher
disease’s characteristics (incubation period, contagiosity                   in adults than in children.
period, transmission methods, etc.), as well as the
population’s socio-demographic parameters, which define                      This phenomenon, which has been observed in France, stems
contact rates between infected and susceptible individuals                   from inadequate vaccination coverage against measles that
of each age group.                                                           limited the spread of the virus, but failed to eradicate it. The
virus being less and less frequent, the odds of being infected in
Calibrating the model is the second step. Data pertaining to                 early childhood become lower. Yet, non-immunized individuals
the current context, objectivized through epidemiological                    may become infected by the virus much later in life. Thus, a
monitoring, must be integrated to validate the results it                    sufficiently broad vaccination of infants must be implemented
generates.                                                                   to limit this age shift.

Should these results coincide with monitoring data, the
model may be used for prediction purposes. For example, the
impact of various intervention strategies can be assessed. As
for immunization, parameters associated with the vaccine
(efficacy, protection term, etc.) and the immunization
programme (target population, vaccination coverage, etc.)
must be taken into account. Besides the health impact
(number of prevented cases), the models can also be used
to assess the economic impact of measures by comparing
the cost of treating the disease (before and after taking
action) and that of implementing measures.

Finally, these models facilitate optimal use of existing data:
they provide a virtual visualization of areas for which data
is unavailable. Once collected, this data will be integrated
into the model for greater prediction capability.

p. 2 / Mathematical modeling: a tool for decision making in situations of uncertainty – French Institute for Public Health Surveillance
Figure            Age shift in cases based on the spread of the infectious agent

Age of incidence of the disease

The less the virus spreads in a population, the more likely it is to be infected for the first time in adulthood.

Modeling efforts to help solve this issue were the result of                What role does immunization play
a dynamic collaboration between the InVS and its British                    in preventing cervical cancer?
counterpart, the Health Protection Agency (HPA). The
mathematical model already used to develop chickenpox                       The planned role of immunization versus screening had to be
vaccination strategies in Canada, Australia and Great Britain               assessed following the commercialization of a new vaccine
was modified to reflect French specificities.                               against cervical cancer (15th cause of cancer amongst French
women).
"Contact matrixes" were modified to take into account
children’s participation in community life (day care, school,               Screening is recommended by the World Health Organization
etc.), which occurs much earlier in France than in Great Britain.           (WHO), the International Agency for Research on Cancer (IARC)
Furthermore, predictions were made to test lower vaccination                and the European Union (EU), as a cost effective strategy,
coverage rates than those put to the test in other countries.               especially if it is designed to monitor the sexually active female
A survey of intentions conducted amongst French general                     population. Seven (7) of the twenty-seven (27) EU countries
practitioners and pediatricians indicates a vaccination coverage            have currently established a national cervical cancer screening
no higher than 70%, even if vaccination against chickenpox                  procedure. In France, the beneficiaries are primarily women
was recommended in infants and the cost of the vaccine                      seen by their gynecologist.
was reimbursed. The impact of immunizing infants against
chickenpox was assessed based on these predictions, and a                   Although it cannot replace screening, the new vaccine could
30 to 90% vaccination coverage.                                             contribute to the prevention effort by targeting types 16 and
18 Human Papilloma Virus (HPV), linked to approximately 70%
or cervical cancer cases.
"To develop these models, InVS was used as an                               A CTV Working Group has been set up, with the mandate of
interface between modellers and health care                                 assessing this contribution and to submit recommendations
professionals."                                                             to the Health Department. Data pertaining to the disease
Daniel Lévy-Bruhl, Department of Infectious Diseases, InVS                  (epidemiology, clinical aspects, surveillance), screening (test
sensitivity, implemented coverage, costs), and to vaccination
(efficacy, costs) was thus collected and analyzed prior to
developing a mathematical model.

Mathematical modeling: a tool for decision making in situations of uncertainty – French Institute for Public Health Surveillance / p. 3
Modeling of HPV immunization using                                           Collecting new data to achieve
clinical data                                                                greater certainty
The structure of the model used to assess the impact of HPV                  The simulated impact of chickenpox immunization in France
vaccination is based on that of a pre-existing model. Three                  was a contributing factor in CTV opting not to integrate this
stages of the disease are taken into account: infection by the               new vaccine into the 2007 infant immunization schedule, but
HPV virus, onset of precancerous lesions, and cancer growth,                 to recommend it for teenagers and women of child-bearing
from which the number of cancer cases and of cancer deaths                   age, with no chickenpox history.
can be measured in the absence of any prevention.
Based on expected vaccine coverage in France (below 90%),
Four strategies are then assessed:                                           vaccination would incur an age shift and consequently
- individual screening alone (actual reference)                              a greater number of adult cases (particularly amongst
- individual screening associated with the vaccination                       pregnant women), the amplitude of which cannot easily be
of girls from the age of 14                                                predicted. Results seem influenced by two factors, for which
- screening structure at the national level                                  there is only limited data: the selection of contact matrixes
- screening structure linked to vaccination                                  and the duration of immunity granted by the vaccine. More
detailed data on the expected vaccine coverage must also
be collected, as results provided by the model vary based
How cost-efficient is immunization                                            on that parameter’s value.
against rotaviruses?
Furthermore, according to preliminary results from the United
Notwithstanding their expected advantages in preventing                       States, an immunization schedule of two doses in the second
acute diarrhea in children, new antirotavirus vaccines cost a                 year of life (as recommended in the US since 2007) would limit
lot more than other childhood vaccines (roughly 30 € for two                  or prevent the increase of adult cases.
MMR doses vs. 100 to 150 € for the 2 to 3 doses of antirotavirus
vaccines).                                                                    New data from epidemiological surveillance in countries of
widespread vaccination, pertaining to the efficacy and the
What would the cost-benefit ratio be for immunizing infants                   protection term of this new immunization schedule, will be
against rotaviruses in France? The Tourcoing Hospital Center,                 integrated to the model in order to periodically reevaluate the
the Economic and Social Research Laboratory (CNRS URA 362),                   relevance of recommending that infants be vaccinated against
the Tours Regional University Center and the InVS had to                      chickenpox.
Immunization combined with structured
The model used relied on virtual monitoring of a cohort of                    screening for cervical cancer
750,000 children (annual average number of births in
France between 1997 and 2001) over their first three years                    In regards to cervical cancer prevention policy, modeling has
of life, to assess both the health and economic impact of                     confirmed the primary role of organized screening in preventing
immunization.                                                                 cervical cancer in France, as well as the potential benefits of
immunizing teenage girls prior to the onset of sexual activity.
Various parameters were assessed, including gained quality-                   It has also shown that vaccination of women who are not yet
adjusted life years (QALY), by measuring the number of avoided                sexually active would increase the impact of immunization with
deaths and the impact of the disease on the quality of life of                no effect on the economic analysis.
affected children and their parents (anxiety, absence from work,
time spent in medical consultation, etc.).                                    Considering these results, the CTV has recommended that from
2007 onward all 14-year old French girls should be vaccinated,
and that a catch-up programme should be implemented for
From modeling to decision-making                                              some girls and young women up to the age of 23. At the same
time, the CTV has recommended that screening of cervical pre-
cancerous and cancerous lesions be set up and promoted.
"Modeling is used more and more frequently to
facilitate the health decision-making process.                               Setting up epidemiological surveillance would also help monitor
It requires a close collaboration between                                    the impact of this new policy on the incidence of HPV infections,
surveillance epidemiologists and biomathematics                              their epidemiology (flow of HPV genotypes) and behaviors
researchers."                                                                associated with sexually transmitted diseases. Additional data
Jean-Claude Desenclos, Scientific Director, InVS                             on tolerance, efficacy and protection term of the vaccine will
also be collected.

It will thus be possible to adapt the vaccination strategy in
order to maximize its benefits. More specifically, the relevance of
vaccinating young men should be assessed in the near future.

p. 4 / Mathematical modeling: a tool for decision making in situations of uncertainty – French Institute for Public Health Surveillance
Economic issues of antirotavirus                                           infections (rotavirus contribution to acute diarrhea, health
vaccination                                                                system imputed costs, contribution of nosocomial infections).
Additional studies will have to be conducted to determine if
Vaccination against rotaviruses has not been deemed cost                   these variances stem from true differences between countries
efficient in France. According to the model, rotaviruses                   or from methodological biases. These studies should also
are responsible for 182,000 episodes of acute diarrhea,                    provide a clearer view of the situation, by evaluating more
18,000 hospitalizations, and 13 deaths annually in France.                 precisely indirect costs (such as work-related absenteeism)
Immunization could help prevent respectively 89,000,                       associated with rotavirus infections.
10,500 and 8 such cases at a cost of 138,690 euro/QALY. This
number is much higher than threshold values below which
a public health strategy is usually considered cost efficient              Conclusion
(50,000\$/QALY in the USA, 30,000€/QALY in the Netherlands,
and 20,000 to 30,000£/QALY in the United Kingdom). For                     Mathematical models, which generally speaking provide a
immunization to be cost efficient, the cost of the vaccine would           simplified interpretation of reality, do help clarify increasingly
have to be significantly lowered.                                          complex public health issues. According to Nigel J. Gay,
biomathematician from the HPA (Health Protection Agency,
Results are in agreement with those of American and                        London), "they act as a focus" that help eliminate gray areas
British studies. But other countries have determined that                  within a complex epidemiological context. Resulting predictions,
immunization could be cost efficient. That different point                 uncertain by nature, must be reevaluated as the situation evolves
of view can be explained by the various methodologies                      and knowledge broadens. Yet, generated arguments help steer the
used (more specifically economic analyses that take into                   decision-making process as well as expertise while highlighting
account health system or social costs) and differences in                  data shortcomings. They are precision tools that can be used to
the estimation of the incidence and impact of rotavirus                    realign public health strategies for optimal efficiency.

Mathematical modeling: a tool for decision making in situations of uncertainty – French Institute for Public Health Surveillance / p. 5
This memorandum was coordinated by the Cellule de valorisation éditoriale (CeVE - Editorial Support Unit,
InVS), and written in French by Avicenne (Catherine Mary). Translation in English by Alpha Omega Translations.

The mathematical models in which InVS has contributed to
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Mars 2007. www.invs.sante.fr

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Suggested citation:
Mathematical modeling: a tool for decision making in situations of uncertainty. Saint-Maurice (Fra): French Institute for Public Health Surveillance,
November 2008, 6 p. Available at: www.invs.sante.fr

Institut de veille sanitaire, 12 rue du Val d’Osne 94 415 Saint-Maurice Cedex France - Tél. : 33 (0)1 41 79 67 00 - Fax : 33 (0)1 41 79 67 67
www.invs.sante.fr - ISSN : 1956-6956 - ISBN : 978-2-11-098347-3 - ISBN-NET : 978-2-11-098449-4 - Réalisation : DIADEIS - Paris - Imprimé par : France Repro -
Maisons-Alfort - Tirage : 120 exemplaires - Dépôt légal : novembre 2008

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