1. Compartmental Models for H1N1(Swine flu)
Disease
By
ALBERT SINGYE ANGBOURABA
M.Sc. Environmental Science

2. Background to H1N1 Disease
• Influenza A (H1N1) virus is the subtype of influenza A
virus that was the most common cause of
human influenza (flu)
• It is an Orthomyxovirus that contains the glycoproteins
Haemagglutinin and Neuraminidase. For this reason, they are
described as H1N1, H1N2 etc. depending on the type of H or
N antigens they express
• Some strains of H1N1 are endemic in humans and cause a
small fraction of all influenza-like illness and a small fraction
of all seasonal influenza.
• In June 2009, the World Health Organization (WHO)
declared the new strain of swine-origin H1N1 as a pandemic

3. Transmission of H1N1
• Evidence suggests that all influenza viruses in mammals,
including humans, derived from viruses in wild ducks and
other waterfowl.
• Influenza is transmitted from person to person through large
respiratory droplets; expelled directly through coughing or
sneezing, indirectly through contact with respiratory droplets
or secretions, followed by touching the nose or the mouth and
one needs not to be more than one meter to be infected.

4. transmission cont’d
• Preventing transmission requires removing one or more of
the conditions necessary for transmission: e.g. blocking and
or minimizing the ways by which the virus can get to a
susceptible host, inhibiting or killing the virus.
• People infected with H1N1 first pass through latent and
incubation period where they are not infectious and do not
have the symptoms
• The symptoms of influenza are: cough, nausea, diarrhea,
fever, headache, sore throat, muscle aches, runny nose,
shortness of breath, joint pains
• People of all age groups are susceptible to this new virus.
In addition considering the virus high contagiousity, it is
transmitted rapidly from an infected to a susceptible person

5. • People of all age groups are susceptible to this new virus. In
addition considering the virus high contagiousity, it is
transmitted rapidly from an infected to a susceptible person.

6. SIS Model
• The simplest epidemiological model in which recovery does
not give immunity is the SIS model. In this model, the
population is divided into two groups of people (see figure
3.1), those that have been infected by the disease and are
infectious, and those that are susceptible to being infected by
the disease. The SIS models are appropriate for some bacterial
agent diseases such as meningitis, plague, and sexually
transmitted diseases and for protozoan agent diseases such as
malaria and sleeping sickness.
Figure 1. SIS model

7. The S-I-R Model
• This model is the same as the SIS model except that once a
person has recovered from the disease, they would receive
lifelong immunity.
• The SIR Model has been proven a relatively good predictor for
infectious diseases such as measles, mumps, and rubella.
Under this model, when an individual becomes infected,
he/she becomes immediately infectious and is able to infect
other individuals. However, the infected individuals may
recover from the disease and therefore move to a recovered
class, where they will be no longer infectious while acquiring
immunity to the disease.
Figure 2. SIR Model

8. The SIRS Model
• An extension of the SIR model is the SIRS model
• The only difference between the SIR and the SIRS is that the
SIRS model allows members of the recovered class to be free
of infection and rejoin the susceptible class.
• Thus the infected population can acquire immunity for a
period before they become susceptible again.
Figure 3. SIRS Model

9. The SEIR Model
• The only difference between the SEIR and the SIR is the
introduction of an ‘exposed’ compartment.
• The exposed phase or latency period is where the person is
infected but not infectious (thus, symptoms of the disease have
not been shown and the person cannot communicate the
disease either).
• According to Uhavax (2001), though the SIS, SIR, SIRS and
other epidemiological models have been effective in
describing the dynamics of other diseases, they cannot be used
to model influenza because there is some form of latency or
exposed phase of the Influenza virus.

10. The SEIR Model Cont’d
• The SEIR model has the following assumptions;
• The population has constant size N, which is sufficiently large
so that the sizes in each class can be considered as continuous
variables.
• Births and deaths occur at equal rates and that all newborns are
susceptible (no inherited immunity).
• The population is homogeneously mixing, with no restriction
of age, mobility or other social factors.
• We assume once infected you become exposed to the
environment before becoming infectious.
• The transmission coefficient β>0 , the latency coefficient α>0,
the recovery coefficient γ>0 and the capital death rate µ>0.

11. SEIR Model Cont’d
• A flow diagram for the SEIR model can be represented as;
Figure 4. SEIR Model

12. The SEIR Model Cont’d
• From the diagram the following ordinary differential equations can be
deduce
• ..…………………….……………...1
………..……………………………...…..2
………..……………………………...…..3
………..……………………………...4
Where S(t)=Susceptible, E(t)=Exposed, I(t)=Infected and R(t)=
Recovered individuals with time; µ=average birth and death rate;
µN=rate at which individuals are born into the susceptible class with
no passive immunity; µS= rate at which they leave it via death; βSI=
rate at which susceptible enters the Exposed class without been
infectious αE=the rate at which an exposed person becomes
infectious and γI is the rate at which an infected individual may
recover

13. Treatment and Control
Epidemic controls:
H1N1 epidemic controls:
• Reduce s(t): Through vaccination
• Reduce : By wash hands, isolate sick persons,
shut down public events, close schools
• Increase : better/faster acting medicine, antivirals
Increase : better/faster acting medicine, antivirals
Epidemic controls:
 Reduce s(t): vaccination
 Reduce : wash hands, isolate sick persons,
shut down public events, close schools
 Increase : better/faster acting medicine, antivirals

14. Effects of Mass Vaccination
• For a population with sufficiently high vaccine coverage,
a disease can be eradicated without vaccinating everyone.
• Therefore, as coverage increases, there is a greater
individual incentive not to vaccinate, since non-
vaccinators can gain the benefits of herd immunity
(population-level immunity) without the risk of vaccine
complications
Therefore at any time (preferably before outbreak), if we can
sufficiently reduce the density of susceptible individuals (by
vaccinating), the epidemics will die out
isi
dt
tdi
si
dt
tds




)(
)(
0)(
)(
 is
dt
tdi
