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Philippine Science Letters Vol. 13 | No. 02 | 2020
158
Prevention of H5N6 outbreaks in the
Philippines using optimal control
Abel G. Lucido1,2,3, Robert Smith?4, and Angelyn R. Lao2,3
1Department of Science & Technology - Science Education Institute, Bicutan, Taguig, Philippines
2Mathematics and Statistics Department, De La Salle University, 2401 Taft Avenue, 0922 Manila,
Philippines
3Center for Complexity and Emerging Technologies, De La Salle University, 2401 Taft Avenue, 0922
Manila,Philippines
4Department of Mathematics and Faculty of Medicine, University of Ottawa, 150 Louis-Pasteur Pvt
Ottawa, ON K1N 6N5, Canada
ighly Pathogenic Avian Influenza A (H5N6) is a
mutated virus of Influenza A (H5N1) and a new
emerging infection that recently caused an outbreak
in the Philippines. The 2017 H5N6 outbreak
resulted in a depopulation of 667,184 domestic
birds. We incorporate half-saturated incidence and optimal
control in our mathematical models in order to investigate three
intervention strategies against H5N6: isolation with treatment,
vaccination, and modified culling. We determine the direction
of the bifurcation when R0=1 and show that all the models
exhibit forward bifurcation. We apply the theory of optimal
control and perform numerical simulations to compare the
consequences and implementation cost of utilizing different
intervention strategies in the poultry population. Despite the
challenges of applying each control strategy, we show that
culling both infected and susceptible birds is an effective control
strategy in limiting an outbreak, with a consequence of losing a
large number of birds; the isolation-treatment strategy has the
potential to prevent an outbreak, but it highly depends on rapid
isolation and successful treatment used; while vaccination alone
is not enough to control the outbreak.
KEYWORDS
Influenza A (H5N6), half-saturated incidence, isolation-
treatment, culling, vaccination, bifurcation, optimal control
1. INRODUCTION
Avian influenza is a highly contagious disease of birds caused
by infection with influenza A viruses that circulate in domestic
and wild birds (WHO 2020). Some avian influenza virus
subtypes are H5N1, H7N9 and H5N6, which are classified
according to combinations of different virus surface proteins
hemagglutinin (HA) and neuraminidase (NA). This disease is
categorized as either Highly Pathogenic Avian Influenza (HPAI),
which causes severe disease in poultry and results in high death
rates, or Low Pathogenic Avian Influenza (LPAI), which causes
mild disease in poultry (WHO 2020).
As reported by the World Health Organization (WHO), H5N1
has been detected in poultry, wild birds and other animals in over
30 countries and has caused 861 human cases in 16 of these
countries and 455 deaths. H5N6 was reported emerging from
China in early May 2014 (Joob and Viroj 2015). H5N6 has
replaced H5N1 as one of the dominant avian influenza virus
subtypes in southern China (Bi et al. 2016). In August 2017,
cases of H5N6 in the Philippines resulted in the culling of
667,184 chicken, ducks and quails (OIE 2020).
Due to the potential of avian influenza virus to cause a pandemic,
several mathematical models have been developed in order to
test control strategies. Several studies included saturation
H
ARTICLE
*Corresponding author
Email Address: abel_lucido@dlsu.edu.ph
Date received: February 28, 2020
Date revised: August 27, 2020
Date accepted: October 3, 2020
Vol. 13 | No. 02 | 2020 Philippine Science Letters
159
incidence, where the rate of infection will eventually saturate,
showing that protective measures have been put into place as the
number of infected birds increases (Capasso and Serio 1978).
With half-saturated incidence, it includes the half-saturation
constant, which pertains to the density of infected individuals
that yields 50% chance of contracting the disease (Shi et al.
2019). When half-saturated incidence is included, the effect is a
significantly lower peak of the total number of infected humans
compared to the case when half-saturated incidence is not
included (Chong et al. 2013). However, when half-saturation is
included, the disease takes longer to die off. We are thus using
half-saturated incidence to investigate the effects of outbreaks
that may have a long tail. Some intervention strategies employed
to protect against avian influenza are biosecurity, quarantine,
control in live markets, vaccination and culling.
Emergency vaccination, prophylactic or preventive vaccination,
and routine vaccination are the three vaccination strategies
mentioned by the United Nations Food and Agriculture
Organization (UNFAO). In China, A(H5N1) influenza infection
caused severe economic damage for the poultry industry, and
vaccination served a significant role in controlling the spread of
this infection since 2004 (Chen 2009). UNFAO and Office
International des Epizooties (OIE) of the World Organization for
Animal Health suggested vaccination of flocks should replace
mass culling of poultry as the primary control strategy during
outbreak (Butler 2005). For this reason, many mathematical
models focus on how vaccination could prohibit the spread of
infection.
Culling is a widely used control strategy during an outbreak of
avian influenza virus (AIV). During the 2017 outbreak of H5N6
in the Philippines, mass culling was implemented to control the
spread of AIV. Gulbudak and Martcheva (2013) suggested a
modified culling strategy, which involves culling only the
infected birds and high-risk in-contact birds. They utilized a
function to represent the culling rate considering both HPAI and
LPAI. Gulbudak et al. (2014) used half-saturated incidence to
describe the culling of infected birds. The two-host model of Liu
and Fang (2015) showed that screening and culling of infected
poultry is a critical measure for preventing human A(H7N9)
infections in the long term. There is a limited understanding of
the effects of isolation with treatment as a control strategy to
counter the spread of avian influenza. Isolation is also used when
adding a new flock of birds to the poultry farm in order to
prevent the possible transmission of disease to the current flock.
The importance of optimal-control theory in modeling infectious
diseases has been highlighted by several recent studies. Agusto
(2013) used optimal control and cost-effective analysis in a two-
strain avian influenza model. Jung et al. (2009) used optimal
control in modeling H5N1 to prevent an influenza pandemic.
Kim et al. (2018) utilized an optimal-control approach in
modeling tuberculosis (TB) in the Philippines. Okosun and
Smith? (2017) used optimal control to examine strategies for
malaria–schistosomiasis coinfection.
To the best of our knowledge, optimal-control theory has not
been applied to the spread of infectious diseases with
transmission represented by half-saturated incidence. In this
study, we adapt the vaccination model and modify the isolation
model of Lee and Lao (2017). We modify the isolation model by
partitioning the outflow of birds from isolation into two
compartments. A proportion of birds will transfer to the
recovered population, while the remainder will return to the
infected population. We focus on the poultry population and use
half-saturated incidence to describe the transmission of AIV. We
include a modified culling strategy as one of our control
strategies and use half-saturated incidence to depict the modified
culling of susceptible and infected birds. We apply optimal-
control theory to our three strategies — isolation-treatment,
vaccination, and culling — and determine which among these
strategies can inhibit the occurrence of an AIV outbreak.
2. THE MODELS
We examine three control strategies: isolation-treatment, culling,
and vaccination. Our mathematical models are in the form of
half-saturated incidence (HSI), which takes into consideration
the density of infected individuals in the population that yields
50% chance of contracting avian influenza. Mathematical
models with half-saturated incidence are more realistic
compared to models with bilinear incidence (Chong et al. 2013,
Lee and Lao 2018). We present four mathematical models: a
model without control, which describes the transmission
dynamics of avian influenza in bird population (i.e., the AIV
model), and three models obtained by applying the following
intervention strategies: isolation with treatment, vaccination,
and culling. Description of variables and parameters used in the
models are listed in the table in Appendix A.
2.1. AIV model without intervention
Figure 1: Schematic diagram of the AIV model with half-saturated
incidence.
In the AIV model without intervention (shown in Figure 1) the
bird population is divided into subpopulations (represented by
compartments): susceptible birds () and infected birds (). The
total bird population is represented by () at time , where
() = () + (). The number of susceptible birds increases
through the birth rate Λ and reduces through the natural death
rate of birds () which are both constant parameter values.
Infected birds additionally decrease through the disease-specific
death rate ().
The number of susceptible birds who become infected through
direct contact is represented by
, which denotes the transfer
of the susceptible bird population to the infected bird population.
Note that is the rate of transmitting AIV and is the half-
saturation constant, indicating the density of infected individuals
in the population that yields 50% possibility of contracting
avian influenza (Chong et al. 2013). The saturation effect of the
infected bird population indicates that a very large number of
infected may tend to reduce the number of contacts per unit of
time due to awareness of farmers to the disease (Capasso and
Serio 1978). In Figure 1, the dashed directional arrow from to
the arrow from to indicates that
is regulated by .
Based on the AIV model described above, we have the following
system of nonlinear ordinary differential equations (ODEs):
=
+,
=
+(+).
2.2. Confinement with treatment strategy for infected poultry
(isolation-treatment model)
(1)
Philippine Science Letters Vol. 13 | No. 02 | 2020
160
Figure 2: Schematic diagram of isolation-treatment model with
HSI.
Here, we employ the strategy of confining and treating the
infected poultry population (which will be referred as the
isolation-treatment strategy). Several studies concluded that
reducing the contact rate is an effective measure in preventing
the spread of infection into the population (Lee and Lao 2018,
Teng et al. 2018). For the isolation-treatment model (shown in
Figure 2), we have included the compartment representing the
population of isolated birds that undergoes treatment (T) and the
compartment representing the population of recovered birds (R).
We denote the isolation rate of identified infected birds by ψ and
the release rate of birds from isolation by γ.
During isolation, we apply treatment then release birds
afterward. These birds will either recover successfully (transfer
to recovered population) or remain infected (return to the
infected population) depending on the effectiveness of treatment.
The proportion of isolated birds that have recovered is
represented by f, while the proportion of isolated birds that have
not recovered (and so remained infected) are represented by (1-
f). We did not consider natural recovery of poultry in our model,
due to the high mortality rate of HPAI virus infection.
The system of ODEs for the isolation-treatment model is
=
+,
=
++(1)(++),
=(++),
=.
2.3. Immunization of the poultry population (vaccination model)
Figure 3: Schematic diagram of preventi ve vaccination model
with HSI.
We modified the vaccination model of Lee and Lao (2018) by
splitting the birth rate () depending on the proportion of
vaccinated population (), as shown in Figure 3. The poultry
population prone to H5N6 is divided into two compartments:
vaccinated birds represented by and susceptible, unvaccinated
birds denoted by . In our vaccination model, we differentiate
the immunized group (vaccinated) from non-immunized group
(unvaccinated).
We investigate the effectiveness of the vaccine not only through
its reported efficacy (denoted by ) but also based on the waning
rate of the vaccine (denoted by ). To represent the acquired
immunity of the vaccinated group, the infectivity of vaccinated
birds is reduced by a factor
(1). The system of ODEs
representing the vaccination model is
=(1)+
+,
=(1)
+(+),
=
++(1)
+(+).
2.4. Depopulation of susceptible and infected birds (culling
model)
Figure 4: Schematic diagram of depopulation or culling model
with HSI.
We modified the culling model of Gulbudak et al. (2014) by
incorporating the dynamics of half-saturated incidence on the
transmission of infection and on the culling rate for infected
birds and for susceptible birds that are at high risk of infection.
We define the culling function of the infected and susceptible
birds as =
and =
, respectively. The culling
frequency is represented by for susceptible birds and for
infected birds. The following system of ODEs represents the
culling model:
=
+(),
=
+()(+).
3. STABILITY AND BIFURCATION ANALYSIS
We first analyze the AIV model without intervention. The
disease-free equilibrium (DFE) of the AIV model (1) is
=(,)=
, 0.
We denote the basic reproduction number as for the AIV
model and obtain
=
(+).
The DFE
of the AIV model is locally asymptotically stable
if < 1 and unstable if > 1.
The endemic equilibrium for the AIV model is represented by
=(, )=()
,()
()().
(2)
(3)
(4)
(5)
(6)
Vol. 13 | No. 02 | 2020 Philippine Science Letters
161
We can rewrite as
=
+(1).
When 1, it follows that 0, so there is no biologically
feasible endemic equilibrium. For > 1, we have > 0, so
we have an endemic equilibrium. We conclude that the AIV
model has no endemic equilibrium when 1, and has an
endemic equilibrium when > 1. It follows that reducing the
basic reproduction number below one is sufficient to
eliminate avian influenza from the poultry population.
As exhibited in Figure 5A, we have a bifurcation plot between
the infected population and the basic reproduction number .
When the basic reproduction number is below one and the DFE
and the endemic equilibrium coexist, then we have a backward
bifurcation. A forward bifurcation is when the basic
reproduction number crosses one from below and the DFE
becomes unstable while the endemic equilibrium becomes stable.
Clearly, we have a forward bifurcation for the AIV model,
showing that when the basic reproduction number crosses unity,
an endemic equilibrium appears and the DFE continues to exist
but loses its stability.
We continue by investigating different strategies that can reduce
or stop the spread of AIV. From the isolation-treatment model
(2), the DFE is given by
=(,,,)=
, 0,0,0.
The corresponding basic reproduction number
() with
respect to (2) is represented by
=(++)
[(++)(++)(1)].
The DFE (
) of the isolation-treatment model is locally
asymptotically stable if < 1 and unstable if > 1 .
Consequently, we can identify some conditions on how
confinement of infected birds affects the stability of
. The
DFE (
) is locally asymptotically stable whenever
(++)(+)(++)
(++)<.
Figure 5: Bifurcation diagram for the basic reproduction number
for AIV, considering no control (A), isolation-treatment (B),
vaccination (C) and culling (D). Only forward bifurcations occur.
Note the change of scale on the vertical axis in each case. The red
dotted curve illustrates the unstable branch of the bifurcation diagram.
For the endemic equilibrium of the isolation-treatment model
(2), we indicate the presence of infection in the population by
letting
0 and solve for
,
,
, and
. Thus, we have
=(
,
,
)
=(+
)
(+
)+
,
,
++,
++,
where
=(++)[(++)]+(1)
(+)[(++)(++)(1)].
Given the basic reproduction number (7) , we rewrite the
expression
of the isolation-treatment model as
=
+(1).
From (9), it follows that when 1, we have
0 and
there is no endemic equilibrium; however, when > 1, we
have
> 0 and we have an endemic equilibrium. Thus, the
isolation-treatment model (2) has no endemic equilibrium when
1 and has an endemic equilibrium when > 1. Hence
there is no backward bifurcation for the isolation-treatment
model when < 1.
In Figure 5B, we have a forward bifurcation for the isolation-
treatment model, which supports our claim. The bifurcation plot
between the infected population
and the basic reproduction
number for the isolation-treatment model shows that
reducing below unity is enough to eliminate avian influenza
from the poultry population.
Next, we analyze the stability of the associated equilibria of the
AIV model with vaccination strategy
(3). The DFE and the
basic reproduction number are
=(,,)=(+)
(+),
+, 0
and
=(+)
(+)(+).
The DFE
of vaccination model is locally asymptotically
stable if < 1 and unstable if > 1. Moreover, we obtain
some conditions for the proportion of vaccinated poultry
()
and vaccine efficacy
(), which both range from 0 to 1.
is
locally asymptotically stable whenever
+
1(+)
< 1 .
For the endemic equilibrium of the vaccination model (3), we
obtain the following:
=(
,
,
),
where
=(+
)[(1)[(+
)(+)+(1)
]+(H +
)]
[(+
)+
][(+
)(+)+(1)
]
=(+
)
(+
)(+)+(1)
(7)
(8)
(9)
(10)
Philippine Science Letters Vol. 13 | No. 02 | 2020
162
=±4
2,
such that
=(+)[(1)+(+)(+)+(1)],
=(1)+(+)(+)(1)
(+)[(1)+(+)(+)],
=(+)(+)(1).
The vaccination model
(3) has no endemic equilibrium when
1, and has a unique endemic equilibrium when > 1.
Figure 5C illustrates a bifurcation plot between the population
of infected birds and the basic reproduction number ,
showing a forward bifurcation. This bifurcation diagram is in
line with our result in Theorem B.1 in Appendix B, so there is
no endemic equilibrium when < 1 but there is a unique
endemic equilibrium when > 1. In this case, reducing
below one is sufficient to control the disease.
Finally, we analyze the stability of equilibria of the AIV model
with culling (4). The DFE for the culling model is given by
=(,)=
, 0,
and the basic reproduction number is
=
(+).
The endemic equilibria of the culling model is determined as
=(
,
)=(+
)
+(++)
,
where C
=±24
2,
such that
=(++)(++),
=(+)(1)(+)(++),
=(+)(1).
For the culling model
(4), we have shown that a backward
bifurcation does not exist when < 1. Thus, the culling model
(4) has no endemic equilibrium when < 1, and has a unique
endemic equilibrium when > 1.
In Figure 5D, we have a bifurcation diagram showing the
infected population and the basic reproduction number
().
We have a forward bifurcation in the plot, which is similar to the
result stated in Theorem B.2, implying that, when < 1, avian
influenza will be eradicated from the poultry population.
4. OPTIMAL-CONTROL STRATEGIES
We now integrate an optimal-control approach in all our models:
isolation-treatment, vaccination, and culling.
4.1. Isolation-treatment strategy
In applying the isolation-treatment strategy, we identify infected
birds and isolate them at rate . While the birds are isolated, we
apply treatment such that a proportion will successfully
recover. Our first control involves isolating infected birds with
replacing . The second control indicates the effort of the
farmers in choosing a drug that can increase the success of
treatment with replacing . The isolation-treatment model
(2) becomes
=
+,
=
++1()++(),
=()(++),
=().
We represent the rate of isolation of infected birds by control
() that is the rate () transfers from to . The
proportion of successfully treated birds released from isolation
is denoted by ().
The problem is to minimize the objective functional defined by
(,)= ()+()+
2
()+
2
(),
which is subject to the ordinary differential equations in
(12)
and where is the final time. The objective functional includes
isolation control () and treatment control (), while
and are weight constants associated with relative costs of
applying respective control strategies. The quadratic
formulation of the objective functional (,) is popular and
useful to satisfy the convexity property of the cost function
(Agusto 2013, Jung et al. 2009, Kim et al. 2018). Given that we
have two controls () and (), we want to find the optimal
controls
() and
() such that
(
,
)=min
{(,)},
where
=(,)|:0, [,],= 1,2, is Lebesgue integrable
is the control set. We consider the best- and worst-case scenarios
of isolating infected birds and giving treatment by setting the
lower bounds to = 0 and upper bounds to = 1, for = 1,2.
4.1.1. Characterization of optimal control for isolation-treatment
strategy
We generate the necessary conditions of this optimal control
using Pontryagin's Maximum Principle (Pontryagin et al. 1986).
The Hamiltonian is
=()+()+
2
()+
2
()+
+
+
++1()[++()]
+(()(++))+(()),
where ,,, are the associated adjoints for the states
,,,. We obtain the system of adjoint equations by using the
partial derivatives of the Hamiltonian (13) with respect to each
state variable.
Theorem 4.1. There exist optimal controls
()and
() and
solutions ,,, of the corresponding state system
(12)
that minimizes the objective functional (,) over .
Since these optimal solutions exist, there exists adjoint variables
,, and satisfying
=+
+
+,
=1 +
(H + I)
(H + I)+[++()]
(),
=1 [1()]+(++)(),
=,
(11)
(12)
(13)
Vol. 13 | No. 02 | 2020 Philippine Science Letters
163
with transversality conditions = 0 , for = 1, 2, 3, 4 .
Furthermore,
=min ,max ,
and
=min ,max ,
.
Proof. The existence of optimal control (
,
) is given by the
result of Fleming and Rishel (1975). Boundedness of the
solution of our system (2) shows the existence of a solution for
the system. We have nonnegative values for the controls and
state variables. In our minimizing problem, we have a convex
integrand for with respect to
(,). By definition, the
control set is closed, convex and compact, which shows the
existence of optimality solutions in our optimal system. By
Pontryagin's Maximum Principle, we obtain the adjoint
equations and transversality conditions. We differentiate the
Hamiltonian (13) with respect to the corresponding state
variables as follows:
=
,
=
,
=
,
=
,
with = 0 where = 1, 2, 3, 4 . We consider the
optimality condition
=()+= 0 and
=()+= 0,
to derive the optimal controls in (14). We consider the bounds
of the controls and obtain the characterization for optimal
controls as follows:
=min 1, max 0,
and
=min 1, max 0,
.
4.2. Vaccination
For vaccination, the first control represents the effort of the
farmers to increase vaccinated birds, while the other control
describes the efficacy of the vaccine in providing immunity
against H5N6. () and () replace and , respectively,
in the vaccination model (3) to obtain
=1()+
+,
=(t)1()
+(+),
=
++1()
+(+).
We describe the proportion of birds that are vaccinated by the
control () and the immunity of the vaccinated population
against acquiring the disease by (). We have the objective
functional
(,)= ()+
2
()+
2
(),
which is subject to
(15). This objective functional involves
increased vaccination () and the vaccine-efficacy control
(), where and are the weight constants representing
the relative cost of implementing each respective control. We
need to find the optimal controls
() and
() such that
(
,
)=min
{(,)},
where
=(,)|:0, [,],
= 3 ,4, is Lebesgue integrable
is the control set. We consider the lower bound = 0 and upper
bounds = 1, for = 3 , 4.
4.2.1. Characterization of optimal control for vaccination
strategy
In this case, the Hamiltonian is
=()+
2
()+
2
()
+1()+
+
+()(+)1()
+
+
++1()
+(+).
Theorem 4.2. There exist optimal controls
() and
() and
solutions ,, of the corresponding state system
(15) that
minimize the objective functional (,) over . Since
these are optimal solutions, there exists adjoint variables ,
and satisfying
=+
+
+,
=+++1()
+
1()
+,
=1 +
(+)+1()
(+)
(+)+1()
(+)
(+),
with transversality conditions = 0 , for = 1, 2, 3 .
Furthermore,
=min ,max ,
and
=min ,max ,
(+).
The proof is similar to the proof of Theorem 4.1 and can be
found in Appendix C.
4.3. Culling
Finally, we administer optimal control to the culling model (4).
Thus, we have
=
+()
+,
=
+()
+(+).
We represent the frequency of culling the susceptible population
by () and frequency of culling the infected population by
(). We have the objective functional
(15)
(16)
(17)
(18)
Philippine Science Letters Vol. 13 | No. 02 | 2020
164
(,)= ()+
2
()+
2
(),
which is subject to (18). The objective functional includes the
susceptible and infected culling control denoted by () and
(), respectively, with and as the weight constants
representing the relative cost of implementing each respective
control. Hence we have to find the optimal controls
and
such that
(
,
)=min
{(,)},
where
=(,)|:0, [,],
= 5 ,6, is Lebesgue integrable
is the control set. We consider the lower bound = 0 and upper
bounds = 1, for = 5, 6.
4.3.1. Characterization of optimal control for culling strategy
In this case, the Hamiltonian is
=()+
2
()+
2
()
+()
+
+
+
+(+)()
+.
Theorem 4.3. There exists optimal controls
() and
()
and solutions , of the corresponding state system (18) that
minimize the objective functional (,) over . Since
these optimal solutions, there exists adjoint variables and
satisfying
=+()
++
+
+,
=1 + ()
(+)+
(+)
(+)(+)
(2+)()
(+),
with transversality conditions , for = 1, 2. Furthermore,
=min ,max ,
() and
=min ,max ,
(+).
The proof can be found in Appendix D.
5. NUMERICAL RESULTS
The parameter values applied to generate our simulations are
listed in the table in Appendix A. The initial conditions of the
simulations are based on the Philippines' H5N6 outbreak report
(OIE 2020). We set (0)=407 837, (0)=73 360, (0)=
0, (0), and the total population of birds (0)=481 197.
Figure 6: Simulation results showing the transmission dynamics
of H5N6 in the Philippines with no intervention strategy. We use
initial conditions and parameter values as follows: (0)=407 837,
(0)=73 360, =
, = 3.4246 ×10, = 0.025 , =180 000,
= 4 × 10.
Previous studies suggested that the basic reproduction number
for the presence of avian influenza without applying any
intervention strategy was = 3 (Mills et al. 2004, Ward et al.
2009). We consider density-dependent transmission, where the
contact between birds increases as the poultry population
increases (Roche et al. 2009). We have calculated the
transmissibility of the disease (= 0.025) based on
(5) with
= 3, and fixed values of (birth rate), (natural death rate),
(disease induced death rate) and (half-saturation constant).
Without any control strategy, avian influenza will become
endemic in the poultry population as shown in Figure 6. After
50 days, the population of the infected poultry exceeds that of
susceptible poultry, with all birds eventually infected or dead.
5.1 Confinement with treatment strategy
Isolation of infected birds and application of treatment is a
potential strategy to hinder an outbreak and reduce further
spread of infection in the population. Figures 7–9 illustrate the
effects of applying optimal control to the isolation-treatment
strategy under different approaches. In Figure 7, we investigate
the effects of varying the weight constant and , which
represents the relative cost of implementing isolation and
treatment controls, respectively. Figure 8 portrays the difference
between using a constant parameter and optimal control in
describing the spread of infection using the isolation-treatment
strategy. Figure 9 shows the disparity of using both isolation and
treatment to using only one control measure.
As the relative cost of implementing isolation control and
treatment control increases, slightly lower isolation and
treatment rates are utilized, as illustrated in Figure 7. As we
increase and , the population of the susceptible birds
decreases (see Figure 7A) while the population of the infected
birds increases (as shown in Figure 7B). Isolated birds increase
significantly in the first six days, then decline afterward due to
treatment, as portrayed in Figure 7C. Increasing the cost of
treatment leads to slower increase of recovered birds (Figure
7D) and slower decline of isolated birds (Figure 7C). We can
observe that when we have lower values for and , the
susceptible population has a slower decline, there are fewer
infected and isolated birds, and recovered birds increase faster.
Thus, we consider ,=500,000 . Moreover, it can be
observed that cheaper cost controls and (Figures 7A–D)
should be administered at higher rates of and (shown in
Figures 7E–F).
(19)
(20)
Vol. 13 | No. 02 | 2020 Philippine Science Letters
165
Figure 7: Application of isolation-treatment strategy with optimal
control to the population of susceptible (A), infected (B), isolated
(C) and recovered (D) birds along with isolation control (E) and
treatment control (F) for varying values of , for =,, fro m
to birds.
With optimal control, we can possibly prevent the spread of
H5N6 in the poultry population, as demonstrated in Figure 8.
The red dashed line (without optimal control) is a simulation of
the isolation-treatment model (2) where we represent the
isolation rate and the proportion of successfully recovered birds
by a constant parameter. The blue solid line (with optimal
control) is a simulation of isolation-treatment model (12) where
control parameters () and () are included. In Figure 8A,
the susceptible population declines slower under optimal control
compared to using a constant parameter. This is due to rapid
isolation of infected birds triggering the surge in Figure 8C with
78% isolation at the beginning, as seen in Figure 8E. It also has
a faster increase in the recovered population, with 843,600 birds
compared to 73,340 birds without optimal control within 100
days, as portrayed in Figure 8D. Application of optimal controls
() and
() in the susceptible, infected, isolated and
recovered population is clearly better than using constant
parameter (Figure 8). We can observe a slower decline of
susceptible birds, an initial reduction in infected birds and a
delayed increase in infection. More infected birds are isolated
(Figure 8C), and we have a higher number of birds that will
Figure 8: Applying isolation-treatment strategy with optimal
control (blue solid line) and without optimal control or using
constant parameter (red dashed line) in the population of
susceptible (A), infected (B), isolated (C) and recovered (D) birds.
Optimal-control values for isolation control (E) and treatment
control (F) over 100 days.
recover after going through isolation (Figure 8D). Thus, using
optimal control illustrated a more appropriate representation of
implementing isolation-treatment strategy in controlling an
outbreak.
Figure 9: Isolation-treatment strategy with the optimal approach
and with consideration of using both isolation and treatment
control (blue solid line), using isolation control only (red dashed
line), and using treatment only (green dashed line) to the
population of susceptible (A), infected (B), isolated (C) and
recovered (D) birds.
It is evident that using isolation together with treatment showed
better results in all populations compared to implementing
isolation alone or treatment alone, as depicted in Figure 9. In
applying both controls, the susceptible populations decrease
slowly; infected birds are eliminated from the poultry
population; and isolated birds increase within 5 days, and then
decrease afterward, which is due to releasing of birds from
isolation. In reality, treatment can only be applied to birds that
have been identified as infected. In the isolation-treatment
Philippine Science Letters Vol. 13 | No. 02 | 2020
166
Figure 10: Application of vaccination strategy with optimal control to the population of susceptible (A), vaccinated (B) and infected (C) birds
and the increased vaccination coverage (D) and the vaccine-efficacy control (E) with varying values of , for = ,, from to
birds.
Figure 11: Applying the vaccination strategy with optimal control (blue solid line) and without optimal control or using constant parameter
(red dashed line) in the population of susceptible (A), vaccinated (B) and infected (C) birds. Optimal-control values for vaccination p revalence
control (D) and vaccine efficacy control (E) over 300 days.
model, treatment cannot be performed without isolation. Hence,
the continuous increase in the infected population if = 0 and
0 (represented by the green line in Figure 9). Isolated birds
will transfer to either the infected or recovered population,
depending on the effect of treatment. Without treatment (0
and = 0), isolated birds increase continuously then decrease
after 85 days where the birds transfer to the infected population,
as illustrated by the red line in Figures 9B–C. Applying isolation
alone will reduce the infected population and prevent possible
transmission of the disease to the susceptible population.
However, due to the absence of treatment, birds will be released
from isolation even though they are still infectious. This results
in a rapid increase of the infected population after 85 days, as
represented by the red line in Figure 9B. Our result suggests that
isolation of infected birds without applying treatment is not
sufficient to prevent the spread of H5N6 in the population.
5.2 Immunization strategy
Next, we consider immunizing the poultry population via a
vaccine. Figure 10 illustrates the outcome of varying the relative
cost of performing vaccination implementation control and
vaccine efficacy control . In Figure 11, we portray the
comparison using fixed control (red dashed line) and optimal
control (blue solid line).
In Figure 10, we observe that varying the relative costs ( and
) of implementing the controls ( and ) significantly
Vol. 13 | No. 02 | 2020 Philippine Science Letters
167
affects the spread of H5N6 in the vaccinated population. As we
increase the relative costs, the vaccine efficacy decreases
(Figure 10E), and this makes the vaccinated population
vulnerable to acquiring H5N6. As shown in Figure 10D, the
effects of varying the relative costs to vaccination control is very
close to zero (the control ranges from 0 to 0.04), and it has a
minimal effect in the spread of the virus in the population. We
can observe that the changes in the vaccine efficacy (Figure 10E)
greatly affect the curves in the vaccinated and infected
population (Figures 10B–C). As the relative cost of the vaccine
efficacy increases, the value of is lowered. Lower vaccine
efficacy leads to rapid decline in the number of vaccinated birds
and hence an increase in the infected population.
Through the application of optimal control, we can observe that
the diminishing effectiveness of the vaccine results in the spread
of infection in the vaccinated population, as depicted in Figure
11. After 120 days, the vaccine efficacy starts to decline, causing
vaccinated birds to acquire the disease. Simulations shown in
Figures 10–11 contribute to our understanding that immunizing
the poultry population is not sufficient to prevent an outbreak.
In using an optimal-control approach, we see that a successful
immunization strategy highly depends on developing an
effective vaccine. Note that, for the vaccination strategy, the
cheapest vaccination is administered at a higher rate of vaccine
efficacy control (shown in Figure 11).
5.3 Depopulation strategy
We obtain simulations for applying a modified culling strategy
that targets infected birds as well as high-risk susceptible birds
that are in contact with infected birds. Figure 12 compares the
difference in outcomes of applying optimal control versus fixed
control. Figure 13 depicts the effect of changing the relative cost
of implementing the culling strategy for susceptible and infected
populations. In Figure 14, we investigate the discrepancies in
applying the modified culling strategy for culling both
susceptible and infected birds, culling only susceptible birds and
culling only the infected birds.
Integrating optimal control into a culling strategy results in a
lower number of susceptible and infected birds compared to
using a constant value, as portrayed in Figure 12. With optimal
control, intensive culling occurred during the first 30 days of
outbreak then slowed down over time. The decline in the
numbers of both susceptible and infected birds occurs faster
when optimal control is applied. In Figures 12A–B, 88% of
susceptible birds and 63% of infected birds were culled within
30 days to prevent the spread of H5N6 avian influenza virus.
After 100 days, there are only 4% susceptible birds and 11%
infected birds left. Our optimal-control results suggest that
culling of susceptible and infected birds must be implemented
rigorously in the first 30 days of the outbreak to prevent further
spread of the infection.
Even though the relative cost of culling increases for both
susceptible and infected populations, we were able to control the
outbreak and prevent further increase in the number of infected
birds, as illustrated in Figure 13. We have lower values of culling
controls for susceptible and infected populations ( and ,
respectively) when the relative cost of implementation increases,
as depicted in Figures 13C–D. Thus, the higher cost of
implementation of culling will result a higher number of
susceptible birds but also more infected birds. Hence, varying
the relative cost and from 100,000 to 900,000 will not
affect the effectiveness of culling in preventing the spread of the
H5N6 in the poultry population.
Figure 12: Implementing the culling strategy with optimal control
(blue solid line) and without optimal control or using constant
parameter (red dashed line) in the population of susceptible (A)
and infected (B) birds. Optimal-control values of culling
frequency control for susceptible (C) and infected (D) birds over
300 days.
Figure 13: Application of culling strategy with optimal control to
the population of susceptible (A) and infected (B) birds and
susceptible-culling control (C) and infected-culling control (D),
with varying values of , for =,, from to
birds.
Administering a culling strategy for both susceptible and
infected birds is more effective than culling only the infected
birds, as indicated in Figure 14. Looking at the blue dashed line
of Figure 14A, we have more susceptible birds if we cull only
the infected population, but, as shown in Figure 14B, the
infected population increases afterward. This implies that
culling only the infected population is not enough to stop the
spread of infection. We can infer that culling only the infected
population can be successful if we can entirely eradicate the
infected population. Currently, we cannot easily identify
infected birds from the poultry population. Culling both
susceptible and infected birds may lead to near eradication of the
infected population, and due to the low number of susceptible
birds, further spread of H5N6 would not be possible. Thus,
culling both susceptible and infected birds is necessary to
eliminate the spread of infection in the poultry population.
In the 2017 Central Luzon H5N6 outbreak, it cost the country's
poultry industry 2.3 billion pesos with around 160,000 infected
poultry (Simeon, 2017). There is insufficient data for the actual
cost of implementation of each strategy per poultry. Henceforth,
Philippine Science Letters Vol. 13 | No. 02 | 2020
168
Figure 14: Simulation of culling strategy with the optimal
approach and with consideration of using both susceptible
culling control ()and infected culling control () (black solid
line), using susceptible culling control () only (red dotted-
dashed line) and using infected culling control () to the
population of susceptible (A) and infected (B) birds.
in this study, we can only present an abstract concept of the cost
(based on the number of infected birds) and compare the cost
from each strategy. Among the three strategies, we concluded
that the modified culling strategy is the cheapest with the least
number of infected birds after 100 days. For future work,
collaborations with engineers can be established to build the
actual facilities and compute the cost per unit of poultry.
Table 1: Total cost of implementation and the number of
infected birds after 100 days for each strategy.
Strategies Total Cost
Infected birds after
100 days
Isolation-
treatment
5.0x104
3.7x104 (reduced by
50%)
Vaccination 2.8x105
8.9x104 (increased
by 22%)
Modified culling 8.1x103
1.2x104 (reduced by
84%)
6. DISCUSSION
Understanding and learning to control avian influenza is a
crucial issue for many countries, especially in Asia. Avian
influenza virus A (H5N6) is an emerging infectious disease that
was reported in China in early May 2014 (Joob and Viroj 2015).
In 2017, the Philippines reported an outbreak of H5N6 which
resulted in a mass culling of 667,184 birds. After more than two
years H5N6 reemerged, causing the depopulation of 12,000
quails (OIE 2020). Lee and Lao (2017) proposed intervention
strategies against the spread H5N6 virus in the Philippines. They
suggested poultry isolation strategy over vaccination strategy in
reducing the number of infected birds.
There is limited study on the effects of isolation with treatment
as a control strategy against the spread of avian influenza.
Isolation is also used when adding new flocks of birds to the
poultry farm in order to prevent possible transmission of disease
to the current flock. We investigated the effects of isolation-
treatment strategy as a promising policy in controlling an
outbreak. We modified the isolation model of Lee and Lao
(2017) and emphasize the role of treatment in utilizing this
strategy. We focused on the impact of isolation control and
treatment control in applying this strategy. Isolating infected
birds is an effective measure to reduce the spread of H5N6 in the
population, as claimed by Lee and Lao (2017). We followed up
confinement by applying treatment during isolation, which turns
out to have a significant role in applying confinement. Through
our simulation in Figures 7–9, we showed that transmission of
H5N6 virus in the poultry population can be reduced by isolating
at least 78% of the infected birds. In addition, at least 62% of the
isolated birds must successfully recover from the infection
within the first week.
Using optimal-control theory, we showed that the success of
vaccination is highly dependent on the effectiveness of the
chosen vaccine. A less-effective vaccine will make vaccinated
birds vulnerable to acquiring the virus. Vectormune AI is a
rHVT-H5 vaccine which provides 73% protection against AIV
H5 type (Kilany et al. 2014). In the study of Cornelissen and
colleagues (2012), the NDV-H5 vaccine induced 80% immunity
to chicken against H5N1. A fowlpox vector vaccine TROVAC-
H5 protected chickens against avian influenza for at least 20
weeks (Bublot et al. 2006). Despite effective vaccines, there is a
possibility for the effectiveness of the vaccine to decline over
time, so we suggest that vaccination should be implemented
together with other intervention strategies in preventing the
spread of H5N6 in the population.
Mass culling of birds is the current policy used when detecting
an outbreak of avian influenza, which is applied to the infected
farm and a short radius around the infected premises (OIE, 2020).
We considered a modified culling strategy, as suggested in the
study of Gulbudak and Martcheva (2013), which focused on
culling infected birds as well as high-risk susceptible birds that
are in contact with infected birds. We showed that culling only
the infected birds is not enough to contain the spread of H5N6.
Instead, culling 78% of susceptible birds and at least 63% of
infected birds within 30 days can prevent an outbreak and avoid
further transmission of the virus in the poultry population.
The modified culling strategy has the cheapest implementation
cost with the least number of infected birds after 100 days. It
should be implemented if rapid eradication of the outbreak is
necessary, with the understanding that the consequence is losing
a large number of birds in the process. On the other hand, if we
aim to conserve the poultry population, then the isolation with
treatment strategy will potentially prevent the outbreak with
most of the birds recovered from the infection. This strategy can
be achieved through a rapid isolation of infected birds and a
reliable treatment policy. Conversely, vaccination should be
implemented only with other intervention strategies.
Note that we used three different models for each strategy, which
limits our comparison of the three control strategies. Future
work will consider combinations of strategies and conduct
numerical continuation studies to track both stable and unstable
steady states and bifurcation points in the systems in order to
gain better understanding and new discoveries of the overall
dynamics of the epidemiological systems.
Using optimal-control theory gives us a better understanding of
H5N6 outbreak prevention. By applying optimal control to
different strategies against H5N6, we have illustrated the effects
of each policy, together with its respective implementation cost.
Every intervention strategy against H5N6 has advantages and
disadvantages, but proper execution and appropriate application
is a significant factor in achieving a desirable outcome.
Vol. 13 | No. 02 | 2020 Philippine Science Letters
169
ACKNOWLEDGMENTS
Lucido acknowledges the support of the Department of Science
and Technology-Science Education Institute (DOST-SEI),
Philippines for the ASTHRDP Scholarship grant together with
the Career Incentive Program (CIP). Lao holds research
fellowship from De La Salle University. Smith? is supported by
an NSRC Discovery Grant. For citation purposes, please note
that the question mark in “Smith?” is part of his name. We thank
the anonymous reviewers whose comments helped improve and
clarify this manuscript.
CONFLICTS OF INTEREST
Lucido, Smith? and Lao declare that they have no conflict of
interest.
REFERENCES
Agusto FB. Optimal isolation control strategies and cost-
effectiveness analysis of a two-strain avian influenza model.
Biosys2013; 113 (3):155–164.
Bi Y, Chen Q, Wang Q, Shi W, Liu D, Gao GF. Genesis,
Evolution and Prevalence of H5N6 Avian Influenza Viruses
in China. Cell Host Microbe 2016; 20 (6):810–821.
Bublot M, Pritchard N, Swayne DE, Selleck P, Karaca K,
Suarez D, Audonnet J-C, Mickle T. Development and use of
fowlpox vectored vaccines for avian influenza.
Ann N Y
Acad Sci. 2006;1081:193-201.
Butler D. Vaccination will work better than culling, say bird
flu experts. Nature 2005; 434:810.
Capasso V, Serio G. A generalization of the Kermack-
McK
endrick deterministic epidemic model. Math
Biosci1978; 42 (1):43–61.
Chen H. Avian influenza vaccination: the experience in China.
Revue Scientifique et Technique de l’OIE2009; 28 (1):267–
274.
Chong NS, Tchuenche JM, Smith? RJ. A mathematical model
of avian influenza with half-
saturated incidence. Theory
Biosci2014; 133 (1):23–38.
Cornelissen LA, Leeuw OS, Tacken MG, Klos HC, Vries RR,
Boer-Luijtze EA, Zoelen-Bos EA, Rigter A, Rottier PJ,
Moormann RJ, Haan CA. Protective efficacy of Newcastle
disease
virus expressing soluble trimeric hemagglutinin
against highly pathogenic H5N1 influenza in chicken and
mice. PLoS ONE 2012; 7(8):1–11.
FAO. The Global Strategy for Prevention and Control of H5N1
Highly Pathogenic Avian Influenza, 2007.
Fleming W, Rishel R. Optimal deterministic and stochastic
control. Springer, 1975.
Gulbudak H, Martcheva M. Forward hysteresis and backward
bifurcation caused by culling in an avian influenza model.
MathBiosci2013; 246 (1):202–212.
Gulbudak H, Ponce J, Martcheva M. Coexistence caused by
culling in a two-strain avian influenza model. Preprint, J Biol
Dynamics 2014; 367 (1):1–22.
Gumel AB. Global dynamics of a two-strain avian influenza
model. Int J of Comp Math 2009; 86 (1):85–108.
Joob B, Viroj W. H5N6 influenza virus infection, the newest
influenza. Asian Pac J Trop Biomed 2015; 5 (6): 434–437.
Jung E, Iwami S, Takeuchi Y, Jo T-C. Optimal control strategy
for prevention of avian influenza pandemic. J Theor Biol.
2009; 260(2):220–229.
Kilany W, Dauphin G, Selim A, Tripodi A, Samy M, Sobhy H,
VonDobschuetz S, Safwat M, Saad M, Erfan A, Hassan M,
Lubroth J, Jobre Y. Protection conferred by recombinant
turkey herpesvirus avian influenza (rHVT-H5) vaccine in the
rearing period in two commercial layer chicken breeds in
Egypt. Avian Pathol. 2014; 43:6:514-523.
Kim S, de los Reyes AA, Jung E. Mathematical model and
intervention strategies for mitigating tuberculosis in the
Philippines. J Theor Biol. 2018; 443:100–112.
Lee H, Lao A. Transmission dynamics and control strategies
assessment of avian influenza A (H5N6) in the Philippines.
Infec Dis Model2018; 3:35–59.
Liu S, Ruan S, Zhang X. Nonlinear dynamics of avian
influenza epidemic models. Math Biosci 2017; 283:118–
135.
Liu Z, Fang C-T. A modeling study of human infections with
avian influenza A H7N9 virus in mainland China. Int J Infect
Dis 2015; 41:73–78.
Mills CE, Robins JM, Lipsitch M. Transmissibility of 1918
pandemic influenza. Nature 2004; 432 (7019):904–906.
OIE. World Animal Health Information System, 2020.
Okosun KO, Smith? R. Optimal control analysis of malaria–
schistosomiasis co-
infection dynamics. Math BiosciEng
2017; 14:377–405.
Pontryagin LS, Boltyanskii V, Gamkrelidze R, Mishchenko E.
Mathematical Theo
ry of Optimal Processes. CRC Press
Book, 1986.
Roche B, Lebarbenchon C, Gauthier-Clerc M, Chang C-M,
Thomas F, Renaud F, van der Werf S, Guegan J-F. Water-
borne transmission drives avian influenza dynamics in wild
birds: The case of 2005-2006 epidemics in the Camargue
Area. Infect GenetEvol2009; 9:800-805.
Philippine Science Letters Vol. 13 | No. 02 | 2020
170
Shi Z, Zhang X, Jiang D. Dynamics of an avian influenza
model with half-
saturated incidence. ApplMath Comput
2019; 355:399–416.
Simeon, L. (2017, September 08). Avian flu poultry ban
partially
lifted. Retrieved August 23, 2020, from
https://www.philstar.com/headlines/2017/08/22/1731805/av
ian-flu-poultry-ban-partially-lifted.
Teng Y, Bi D, Guo X, Hu D, Feng D, Tong Y. Contact
reductions from live poultry market closures limit the
epidemic of human infections with H7N9 influenza. J Infect
2018; 76 (3):295–304.
van den Driessche P, Watmough J, Reproduction numbers and
sub-threshold endemic equilibria for compartmental models
of disease transmission. Math Biosci2002; 180:29–48.
Ward MP, Maftei D, Apostu C, Suru A. Estimation of the basic
reproductive number (R0) for epidemic, highly pathogenic
avian influenza subtype H5N1 spread. Epi Infect 2009; 137
(2):219–226.
WHO. Influenza (Avian and other zoonotic), 2020.
Vol. 13 | No. 02 | 2020 Philippine Science Letters
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Appendix A. Variables and parameters
Here, we describe each variable and parameter that we used in
each model.
Notation Description or Label
() Susceptible birds
() Infected birds
() Isolated birds
() Recovered birds
() Vaccinated birds
() Total bird population
Constant birth rate of birds
Natural death rate of birds
Rate at which birds contract avian
influenza
Half-saturation constant for birds
Additional disease death rate due to
avian influenza
Proportion of vaccinated poultry
Efficacy of the vaccine
Isolation rate of identified infected birds
Releasing rate of birds from isolation
Proportion of recovered birds from
isolation
Culling frequency for susceptible birds
Culling frequency for infected birds
() Culling rate of susceptible birds
() Culling rate of infected birds
The initial conditions are based on Philippine Influenza A
(H5N6) outbreak report given by the OIE (2020): (0) =
407 837 and (0) = 73 360. We calculated transmissibility of
the disease (= 0.025) using the basic reproduction number
in (5) and equating it to 3, the value of the basic reproduction
number of AIV without intervention (Mills et al. 2004, Ward et
al. 2009).We calculated parameter values that reduce the basic
reproduction number below one and control the spread of AIV
in the poultry population.
Definition Symbol Value Source
Constant birth
rate of birds
per day
(Chong et
al. 2013)
Natural
mortality rate
3.4246 ×
10
per day
(Liu et al.
2017)
Transmissibility
of the disease
0.025per
day
Calculated1
Half-saturation
constant for
birds
180 000birds (Lee and
Lao 2018)
Disease-
induced death
rate of poultry
4×10per
day
(Liu et al.
2017)
Proportion of
vaccinated
poultry
0.50 Calculated1,2
Vaccine
efficacy
0.90 Calculated1,2
Waning rate of
the vaccine
0.00001per
day
Calculated1
Isolation rate of
identified
infected birds
0.01per day Calculated1,2
Release rate of
birds from
isolation
0.09per day Calculated1
Proportion of
recovered
birds from
isolation
0.5 Calculated1,2
Culling
frequency for
susceptible
birds
per day Estimated2
Culling
frequency for
infected birds
per day Estimated2
1Calculated means we compute this value using t he basic reproduction
number
2These values will become the controls when optimal-control theory is
applied.
Philippine Science Letters Vol. 13 | No. 02 | 2020
172
Appendix B. Non-existence of backward bifurcation
Appendix B.1. Vaccination
In showing that a backward bifurcation does not exist for the
vaccination model, we solve for
=±
such that
=(+)[(1)+(+)(+)+(1)],
=(1)+(+)(+)(1)
(+)[(1)
+(+)(+)],
=(+)(+)(1).
Theorem B.1.The vaccination model
(3) has no endemic
equilibrium when 1 and has a unique endemic
equilibrium when > 1.
Proof. We obtain two possible endemic equilibria
and
for the vaccination model. From (. 1), we establish the
relationship between and such that
> 1 > 0 = 1 = 0 < 1 < 0
From (. 1), it is clear that < 0. Consider the cases when >
0, when > 0 and = 0 or 4= 0, and when
< 0, > 0, and 4> 0.
Case 1: > 0
When > 0, we have > 1. Since < 0, it follows
that
=
< 0
=4
2> 0.
When > 1 the infected population
of the
endemic equilibrium
does not exist, and we have a
unique endemic equilibrium
.
Case 2: > 0 and either = 0 or 4= 0
Given that > 0, we consider the case when = 0 and
when4= 0.
Case 2A: = 0
Since = 0, it follows that
= 0 and
> 0. Note that
=
0 leads to the DFE. Hence, if > 0 and = 0, then
> 0, and
we have a unique endemic equilibrium
.
Case 2B: 4= 0
Considering that 4= 0, it follows that
=
and
,
> 0. Thus, if > 0 and 4= 0, then we have a
unique endemic equilibrium
=
.
Case 3: < 0, > 0, and 4> 0.
From the assumption that < 0 and < 0, it follows that
=
> 0 V2
=24
2> 0
Thus, we have two endemic equilibria
and
, which
implies that a backward bifurcation may possibly occur
whenever < 0, > 0, and 4> 0.
However, given the values of and , we can show that when
< 0 , we cannot obtain > 0 , which we prove by
contradiction. Suppose that < 0. By definition of and , the
value of both parameters ranges from 0 to 1. From
(. 1), it
follows that <
, where we define =(+
)and =(+)(+).
Using (. 1) with > 0 , we get (1)+>
2++(+)(1). By simplifying, we
obtain
(+)(1)
>(+)+(+)+(1).
In both extreme values of , it follows that
0 > (+)+(+).
Since ,,0, it implies that (+)+(+)0,
and we have a contradiction. Results above suggest that two
endemic equilibria do not exist when < 1 , since the
condition < 0, > 0, and 4 > 0, cannot be satisfied.
From Cases 1 to 3, it is evident that the vaccination model has
no endemic equilibrium when < 1 and a unique endemic
equilibrium when 1.
Appendix B.2. Culling
To show that a backward bifurcation does not exist for the
culling model, we solve for
=±
such that
=(++)(++),
=(+)(1)(+)(++),
=(+)(1).
Theorem B.2. The culling model (4) has no endemic
equilibrium when < 1 and has a unique endemic
equilibrium when > 1.
Proof. We obtain two possible endemic equilibria,
and
,
for the culling model. From
(. 3), < 0, and we consider
cases where < 1, = 1, and > 1.
Case 1: < 1
When is below unity, it follows that < 0 and < 0. Given
that < 0 and < 0, we have
=+4
2< 0
=4
2< 0.
Thus, in our case when < 1 , we have no endemic
equilibrium.
Case 2: = 1
When = 1 , we have = 0 and < 0 . It follows that
4=. Since < 0, we have
=+
2= 0
=
2< 0.
Hence, when = 1, we have no endemic equilibrium.
Case 3: > 1
When is above the unity, it follows that > 0. Given that
< 0 and > 0, we have
(. 1)
(. 2)
(. 3)
Vol. 13 | No. 02 | 2020 Philippine Science Letters
173
=+4
2< 0
=4
2> 0.
Hence, when > 1, we have
> 0 and a unique endemic
equilibrium
.
Appendix C. Proof of Theorem 4.2
Proof. The existence of optimal control (
,
) is given by the
result of Fleming and Rishel (1975). Boundedness of the
solution of (3) shows the existence of a solution for the system.
We have nonnegative values for the controls and state variables.
In our minimizing problem, we have a convex integrand for
with respect to (,). By definition, the control set is closed,
convex and compact, which shows the existence of optimality
solutions in our optimal system. We use Pontryagin's Maximum
Principle to obtain the adjoint equations and transversality
conditions. We differentiate the Hamiltonian (16) with respect
to the corresponding state variables as follows:
=
,
=
,
=
,
with = 0 where = 1 ,2, 3 . Using the optimality
condition
=()+= 0 and
=()+
+
+= 0,
we derive the optimal controls (17). We consider the bounds for
the control and conclude the characterization for
and
=min 1, max 0,
and
=min 1, max 0,
(+).
Appendix D. Proof of Theorem 4.3
Proof. Analagous to the previous proof, we differentiate the
Hamiltonian (19) with respect to the corresponding state
variables as follows:
=
, and
=
,
with = 0 where = 1, 2. We consider the optimality
condition
=()
= 0 and
=()
= 0,
to derive the optimal controls (20). We consider the bounds of
the controls and get the characterization for
and
=min 1, max 0,
() and
=min 1, max 0,
(+).
