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Applied Mathematical Sciences, Vol. 10, 2016, no. 26, 1269 - 1276
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2016.511694
A Simulation Model Including Vaccination and
Seasonality for Influenza A-H1N1 Virus
Maria E. Cardenas P., Irma Pérez C., C. L. V. Pérez
Facultad de Medicina
Benemérita Universidad Autónoma de Puebla
Puebla, Puebla, México
Gonzaga Ospina P. and Anibal Muñoz L.
Facultad de Educación
Grupo de Modelación Matemática en Epidemiologı́a (GMME)
Universidad del Quindı́o
Armenia, Colombia
c 2015 Maria E. Cardenas P. et al. This article is distributed under the
Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
We have described a SIR simulation model for the influenza AH1N1 virus including variable population, periodic transmission coefficient and constant vaccination rate for any age and time. Also, it has
been considered the natural mortality rate due to the infection. We
analyze the model by means of simulations using several scenarios. The
epidemic threshold, that is time and seasonality dependent, has been
obtained.
Keywords: SIR model, variable population, epidemic threshold, seasonality
1
Introduction
A-H1N1 flu is a severe respiratory disease caused by some of the three known
influenza viruses: A, B, and C. Type A has a sub-classification depending
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on their surface proteins (hemagglutinin (HA) and neuraminidase (N)), from
which, it is dependent their capacity to generate severe forms of the disease.
From a public health care point of view the most important virus is type A.
This one may drive to pandemics associated with high mobility, high number
of deaths due to the virus, and social and economic disruptions.
This infection causes substantial morbidity and mortality around the world.
Also, it is extremely difficult to find a solution non-limited to prevention with
vaccination or medication to control the symptoms. For the prophylaxis are
employed some antivirals and annual vaccination, that are adjusted each year
to the circulating virus. In special circumstances antivirals are prescribed to
enhance the vaccination efficacy. The OMS recommends trivalent or inactivated vaccines that are constituted by purified superficial antigens. These ones
are the most important prevention method for influenza virus, being of particular importance for infections with highest risk of complication [7].
Vaccines against influenza virus are essential components of an adequate response to the pandemic. The scientific and technological advances in the creation of vaccines motivate to set and model the A-H1N1 transmission dynamics. These models are applied to the susceptible and vaccinated populations
as an optimal measure to the disease control.
Aron and Schwartz (1984) reported a SEIR model including a periodic incidence rate, limit cycles and bifurcations [5]. Barnes, Glass and Becker (2007)
have determined the role of the healthcare workers and the antiviral drugs in
the influenza pandemic control, using the basic reproduction number of the
dynamics [1]. Also, Becker, Glass, Li and Aldis (2005) have studied the emergency control of infectious diseases (as example, the severe acute respiratory
syndrome) by using the basic reproduction number [6].
On the other hand, Browman, Gumel, Driessche, Wu and Zhu (2005) have
proposed and analyzed a mathematical model to establish control strategies
for the virus del Nilo [2]. Iwami, Takeuchi, Liu and Nakaoka (2009) have
realized studies on the resistance to the vaccine in the geographic propagation
of the influenza epidemic. Also, it has been described a deterministic model
structured in patches for the avian influenza in heterogeneous zones and their
control using a vaccination program in the context of the resistance-chains existence [9]. Klinkenberg, Van der Wind, Graat, and Jong (2003) have been
applied an ordinary differential equations system including stochastic variables
to the quantification of the effect on the control strategies in classic-fever epidemics [3].
A simulation model including vaccination and seasonality for ...
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Model
We propose a simulation model based on coupled nonlinear differential equations that interpret the A-H1N1 virus transmission dynamics, including constant preventive control by using vaccination, periodic transmission coefficient
due to the stationarity of the virus appearance, and death caused by infection.
To construct this model we have considered the following assumptions: it has
been modeled the A-H1N1 virus transmission with stationarity, considering
a periodic transmission coefficient. The human population is variable, since
the birth rate is constant but different to the natural death rate. Also, it is
considered the death rate due to the infection, immunity to a virus serotype
and constant vaccination rate for any age and time.
The variables and parameters of the model are: x average number of susceptible people, y is the average number of infected people, z average number
of immune people to a serotype of the influenza virus by medic treatment and
vaccination, N is the variable-total population at time t. ∆ is the susceptible
people mobility, µ the natural mortality rate, σ the vaccinated people rate at
any age and time, β(t) is the periodic transmission coefficient, θ is the infected
people recovery rate, ω is the death rate due to the infection, and is the
stationarity degree [5, 10, 8].
The following diagram represents the dynamics’ behavior.
δ
x
µx
σx
y
β(t) x+y+z
x
/
y
θy
(µ + ω)y
%
/
z
µz
The variation of each population as function of the time is interpreted by means
of a differential equation, this one results of the difference between inflows and
outflows. So then, dx
indicates the susceptible population variation as function
dt
of the time. This one is defined as the susceptible people mobility δ minus
the vaccinated population at any age and time σx minus the incidence term
y
β(t) x+y+z
x minus the population death due to natural causes µx. Namely, we
have the following differential equation:
dx
y
= δ − σx − β(t) x − µx
dt
N
(1)
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is defined as: the
The variation of the infected population in the time dy
dt
y
presence of new cases (incidence) β(t) x+y+z x minus the infected population
that recovers and becomes immune to an A-H1N1 virus serotype θy minus the
death people due to the disease and by natural causes (ω + µ)y. Such that,
dy
y
= β(t) x − θy − (ω + µ)y
dt
N
(2)
dz
dt
indicates the variation of an immune population to an A-H1N1 virus serotype
as function of time, this one having the following form. The positive term of
the vaccinated people against A-H1N1 σx plus people that becomes immune
to an A-H1N1 virus serotype θy minus the death of immune people by natural
causes µz. In mathematical therms we have the following equation:
dz
= σx + θy − µz
dt
(3)
N (t) = x(t) + y(t) + z(t) drives to the following differential equation
dN
= δ − µN − ωy
dt
(4)
In a compact way, the A-H1N1 virus transmission dynamics with constant preventive control by vaccination is interpreted by the following dynamic system
of nonlinear differential equations:
dx
dt
dy
dt
dz
dt
dN
dt
y
x − µx
N
(5)
y
x − (µ + θ + ω)y
N
(6)
= δ − σx − β(t)
= β(t)
= σx + θy − µz
(7)
= δ − µN − ωy
(8)
All considered parameters are constant and positive, δ, µ, σ, β0 , θ > 0. The
initial conditions are: x(0) = x0 , y(0) = y0 , z(0) = z0 , N (0) = N0 and
β(t) = β0 (1 + cos 2πt)
where 0 ≤ ≤ 1.
2.1
Epidemic threshold
The basic reproduction number represents the average number of cases in which
an infected person, in the infectious period, produces a susceptible population
A simulation model including vaccination and seasonality for ...
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[4]. This epidemic threshold interprets the infection evolution. If R0 < 1, the
A-H1N1 virus is extinguished, when R0 ≥ 1, there is presence of the A-H1N1
virus in the population.
R0 (T ) =
β̄µ
(σ + µ)(µ + θ + ω)
where,
1ZT
β(t)dt
β̄ =
T 0
being β̄ the average transmission coefficient in the interval of time [0, t] [5].
So then,
β0 Z T
β0
β̄ =
(1 + cos 2πt)dt = (T −
sin 2πT )
(9)
T 0
T
2π
namely,
β0 µ
(T − 2π
sin 2πT )
T
R0 (T ) =
(σ + µ)(µ + θ + ω)
3
Results and conclusions
Table 1: Variables and parameters
Parameters
Simbol
Initial susceptible population
x(0)
Initial infected population
y(0)
Initial immune population
z(0)
Total population
N (0)
Susceptible people flux (mobility)
δ
Natural mortality rate
µ
β(t) coefficient
β0
Stationarity degree
Infected people recovery rate
θ
Death rate due to the infection
ω
Value
1000
5
0
1005
2
0.003
1
0.9
0.5
0.1
According to the hypothetic assigned values to the parameters, see Table
1., we have realized plots of the R0 (T ) and simulations of the model using the
MAPLE code.
In figure 1(A) we plot the basic reproduction number behavior in a time period without vaccination, where we can observe an oscillatory decrement in
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Maria E. Cardenas P. et al.
Figure 1: Behavior of basic reproduction number R0 (T ) without vaccination
σ = 0 and with vaccination σ = 0.1.
the range from 1.4 to 1.9. This fact indicates that the disease remains endemic
and periodic over the time. On the other hand, figure 1(B) includes results
with a 10% of vaccination. In this image is observed an oscillatory decrement
but with a threshold in the range from 0.4 to 0.6, this result implies that the
disease remains in an endemic and periodic way, showing an effect of efficient
control by vaccination as a measure of primary prevention.
In figure 2 we depict the behavior of the infected population in a period
Figure 2: Behavior of infected people y(t)(−), with σ = 0, σ = 0.1 and σ = 0.7.
of 50 days, without and with different percentage of vaccination (10%, 70%).
As it is observed from the simulations with σ = 0, there exist an epidemic
with periodic and variable infection, having their maximum value around 15
days. Comparing these results with the ones for 10% of vaccination, we can
observe a less number of infected persons in less days. Finally, for a 70% of
A simulation model including vaccination and seasonality for ...
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vaccination the infected persons are even less in number in around 2 days. In
general, the impact of the vaccination is efficient for small coverages.
Acknowledgements. We thank to Facultad de Medicina of the Benemérita
Universidad Autónoma de Puebla, México. Also, thank to the Grupo de Modelación Matemática en Epidemiologı́a (GMME) from Facultad de Educación
(Universidad del Quindı́o, Colombia).
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Received: November 22, 2015; Published: April 2, 2016