COMBINATION DRUG THERAPY FOR INFLUENZA TREATMENT
by
Kelli Melville
A senior thesis submitted to the faculty of
Brigham Young University - Idaho
in partial fulfillment of the requirements for the degree of
Bachelor of Science
Department of Physics and Astronomy
Brigham Young University - Idaho
April 2017
c 2017 Kelli Melville
Copyright All Rights Reserved
BRIGHAM YOUNG UNIVERSITY-IDAHO
DEPARTMENT APPROVAL
of a senior thesis submitted by
Kelli Melville
This thesis has been reviewed by the research advisor, research coordinator,
and department chair and has been found to be satisfactory.
Date
Kevin Kelley, Advisor
Date
Ryan Nielson, Committee Member
Date
Clair Eckersell, Committee Member
Date
Todd Lines, Senior Thesis Coordinator
Date
Stephen McNeil, Chair
ABSTRACT
COMBINATION DRUG THERAPY FOR INFLUENZA TREATMENT
Kelli Melville
Department of Physics and Astronomy
Bachelor of Science
Influenza virus is a seasonal infection that kills roughly 500,000 people each
year worldwide [1]. Using combination therapy to treat influenza has many
benefits, including reducing the emergence of drug resistant virus strains and
decreasing the cost of antivirals. Mathematical models allow various combinations of efficacies and drug concentrations to be tested. In this work we looked
at the maximum number of dead cells, the production rate of the virus, and
the duration of symptoms to determine the optimal combination of antivirals
for the treatment of influenza. We found that decreasing the virus production
rate and the length of infection make the worst combination for treating influenza. While decreasing the infection rate and increasing the virus clearance
rate produces the optimal combination.
ACKNOWLEDGMENTS
I would like to thank Dr. Hana Dobrovolny and Thalia Rodriguez for all the
help I was given on my research. Thank you to the grant from the National
Science Foundation Research Experience for Undergraduate program PHY1358770. I would also like to thank my thesis committee and advisors, Dr.
Kevin Kelley, Dr. Clair Eckersell, Dr. Todd Lines, and Ryan Nielson. Thank
you to all the professors at BYU-Idaho for teaching me so much.
Contents
Table of Contents
vi
List of Figures
1 Introduction
1.1 Influenza Virus . . . .
1.2 Structure of the Virus
1.3 Replication Process . .
1.4 Classes of Influenza . .
1.5 Combination Therapy
vii
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2 Methods
2.1 Virus Production, Dead Cells, and Duration of
2.2 Mathematical Model . . . . . . . . . . . . . .
2.2.1 Differential Equations . . . . . . . . .
2.2.2 Parameters . . . . . . . . . . . . . . .
2.3 Efficacy and Drug Concentration . . . . . . .
2.3.1 Variables for Combination . . . . . . .
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3 Results and Discussion
3.1 Infection Rate and the Production Rate . . . . . . . . . . .
3.2 Infection Rate and the Time of Infection . . . . . . . . . . .
3.3 Infection Rate and the Clearance Rate . . . . . . . . . . . .
3.4 Infection Rate and the Time of Eclipse . . . . . . . . . . . .
3.5 Production Rate and Clearance Rate with Time of Inflection
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13
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Symptoms
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4 Conclusion
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Bibliography
21
A Extra Plots of Combinations
23
B Code
27
vi
CONTENTS
vii
C Example of Data Points
40
List of Figures
1.1
2.1
2.2
2.3
3.1
3.2
3.3
Viral replication process showing how the virus enters the cell where
the RNA is moved into the nucleus to be copied and replicated. Finally,
the new RNA is placed into new virus structures that will leave the
cell to infect nearby target cells. [2] . . . . . . . . . . . . . . . . . . .
Graphs used to find Vmax from the peak of the graph (labeled Time
of viral peak) and Duration from the width of the graph (labeled
Duration of symptomatic infection). [2] . . . . . . . . . . . . . . . . .
Graph used to find Dmax , the maximum number of dead cells in the
body. Use the slope of the graph to find the rate at which the cell dies
after infection. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simplified model of the virus replication process with variables used in
the differential equations. [2] . . . . . . . . . . . . . . . . . . . . . .
Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
3
6
7
8
14
15
16
LIST OF FIGURES
3.4
3.5
3.6
Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
A.4 Upper left: maximum number of dead cells using efficacy. Upper Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead
cells for drug concentration. Bottom Middle: maximum number of
virus for drug concentration. Bottom Right: Duration of symptoms
for drug concentration. . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
17
18
19
23
24
25
26
Chapter 1
Introduction
1.1
Influenza Virus
A virus is a microscopic infection that survives by replicating itself using a host. [3]
Influenza, more commonly known as the flu, is a seasonal sickness that has symptoms of running nose, cough, fever and an aching body. This disease comes from the
influenza virus which has a high mutation rate, and consequently makes new strains
almost every year. Influenza kills between 250,000 - 500,000 people every year, especially children and the elderly who have weaker immune systems. [1] Because of the
high mutation rate and high mortality rate, a new antiviral is needed almost every
year. However, typical antivirals are expensive for developing countries where the
mortality rate is the highest. [4]
1.2
Structure of the Virus
Influenza virus has a spherical shape with the genetic material surrounded by a protein
layer. The interior of this virus contains the genetic material, RNA, surounded by
1
1.3 Replication Process
2
nucleic acid. Virus lack a nucleus, ribosomes, and mitochondria necessary for survival
independant of a host. The exterior is surrounded by a protein layer containing
spikes of proteins that are connected to sugars on the lipid membrane. Both of these
structures come from the host cell where the virus was made. [5] [6] This outer layer
gives the virus access to the other target cells in the body. The next inner layer is a
matrix protein coat that gives the virus strength and rigidity to protect the RNA of
the virus. [7]
1.3
Replication Process
The replication process (figure 1.1) of Influenza begins with the virus entering the host
cell using the proteins and membrane from the cell where it was made. This is similar
to a lock and key, where the virus made a key from the cell where it was made to open
the locks on our cell walls. Once it has gained entry, the virus will release its RNA
into the cell’s nucleus where the RNA is copied and replicated. Since the virus has no
structures of its own which can perform this task, it hijacks the host cell structures
that copy and replicate our DNA and RNA. Once the RNA has been replicated, the
new virus leaves the cells. As it does so, the virus takes the membrane and protein
spikes from the cell, making the outer layer. Now it can repeat this process in other
target cells until the virus has replicated until it is no longer infectious.
1.4
Classes of Influenza
There are three classes of influenza: A, B, and C. Influenza A’s host species are wild
birds (ducks, geese, and poultry) though it can spread to multiple species. Because of
the various host species, this class has a high tendency to mutate. This mutation can
1.4 Classes of Influenza
3
Figure 1.1 Viral replication process showing how the virus enters the cell
where the RNA is moved into the nucleus to be copied and replicated. Finally,
the new RNA is placed into new virus structures that will leave the cell to
infect nearby target cells. [2]
give rise to pandemics, worldwide outbreaks, including the Spanish Flu, Swine Flu
and the Avian Flu. Influenza A is categorized based on the proteins on the surface,
for example H5N1 has an HA 5 protein and NA 1 protein and will be the class we
focused on for this experiment. This class is the most common infection, however,
both Influenza A and Influenza B have the symptoms previously listed. The host
species of Influenza B are humans, and since it passes from human to human it does
not have a high chance to mutate into new strains. This makes Influenza B less
likely to cause pandemics. However, one third of influenza infections are caused by
Influenza B which is why part of the yearly flu vaccine contains a strain of influenza
B. Almost everyone has been infected with Influenza C virus, since it only causes
mild upper respiratory tract infections and therefore requires no vaccine [8] [9].
1.5 Combination Therapy
1.5
4
Combination Therapy
Our experiment tested whether combination therapy, the use of two or more antivirals,
is beneficial in treating Influenza. [10] In theory, combination therapy is beneficial for
several reasons. First, it reduces the amount of individual antivirals in the body,
making it better for your body and less expensive overall. Combination therapy
utilizes the use of two antivirals in lower doses compared to a single antiviral in a
higher dose. Second, combination therapy decreases the likelihood that the virus will
become drug resistant because it must copy amino acid chains for both drugs. [11]
Third, combination therapy will reduce the duration of the infection, compared with
use of a single antiviral. Because combination therapy has the ability to attack the
virus at multiple points in its lifecycle because multiple antivirals are working together
to decrease the infection rate and increase the virus clearance rate.
Chapter 2
Methods
2.1
Virus Production, Dead Cells, and Duration
of Symptoms
For our computional experiment we tried using combination therapy to determine
which combination of antivirals are the best at treating Influenza. We looked for three
aspects of the virus’s life: 1) the maximum production of virus which we call Vmax ,
2) the duration of the symptoms which we call Duration, 3) the maximum number
of dead cells which we call Dmax . In Figure 2.1, the graph shows the Viral titer vs.
days, which is the the minimum concentration of virus needed to still infect the cells.
The peak of this graph labeled “Time of viral peak”, is the maximum amount of virus
being produced in the body, Vmax . In order to determine when symptoms will begin
to show after virus infection, we used the “Duration of symptomatic infection” line
on the graph. In this graph, one will have symptoms on day 2 and last past day 4. To
model the Duration of symptoms, we only used data at or above this line of about
104 viral titer. In order to model the amount of dead cells, Dmax , we used Figure 2.2.
The amount of dead cells in the body increases at a rate of about 1 dead cell every
5
2.2 Mathematical Model
6
3 hours. In order to save the infected cells, an antiviral must be effective before 3
hours or the cell succumbs to infection and dies.
Figure 2.1 Graphs used to find Vmax from the peak of the graph (labeled
Time of viral peak) and Duration from the width of the graph (labeled
Duration of symptomatic infection). [2]
2.2
Mathematical Model
Figure 2.3 shows the simplified model of virus replication for our mathematical model.
The process begins with a healthy cell, in either plant or animal, called a target cell
(T ). The infection of the virus into the target cell is at a rate of β. Once the virus
is inside the target cell but before the cell becomes infectious, is in the eclipse phase
(E). The amount of time the cell is in the eclipse phase is labeled with τE and lasts
for about 7 hours. The virus’s RNA is reproduced and copied by the host cell’s
own nucleus. Then once the virus’s RNA has been broken down, multiplied, and a
new virus produced, the cell is now in the infectious phase (I). This phase lasts for
about 50 hours, giving influenza a high turnover rate between cells. The virus will be
2.2 Mathematical Model
7
1.0
Dead cells
0.8
0.6
0.4
0.2
0.0
0
6
12
time (hrs)
18
24
Figure 2.2 Graph used to find Dmax , the maximum number of dead cells in
the body. Use the slope of the graph to find the rate at which the cell dies
after infection. [2]
produced at a rate p inside the infected cell and will leave the infected cell and will
spread to infect other target cells. The length of the infectious phase is labeled with
τI . The virus can cause the death of the host cell labeled, D. However, the viruses
also have a clearance rate at which the virus loses its infectivity and that is labeled
c.
2.2.1
Differential Equations
We can also model the life cycle of the virus with differential equations for our mathematical model. To begin we need to model the infection of the healthy target cells
(Equation 2.1). The infection rate (β) shows how quickly the virus (V ) enters the target cells (T ). Equation 2.2 models the eclipse phase which accounts for the amount of
infected cells from Equation 2.1 as well as an intial number of eclipsed cells (nE ) and
the duration of the eclipse phase (τE ). To determine the number of compartments
(Ej ) in the eclipse phase, account for every cell which is found in Equation 2.3. The
infectious phase follows in the same pattern as the eclipse phase. The amount of cells
2.2 Mathematical Model
8
Figure 2.3 Simplified model of the virus replication process with variables
used in the differential equations. [2]
in the infectious phase are seen in Equation 2.4. A certain amount of infected cells
(nI ) and a length of the infectious phase (τI ) are set in place from the beginning of the
experiment. Again in Equation 2.5, account for the number of compartments in the
infectious phase (Ij ). Finally, once the virus leaves the infected cells, the clearance
rate (c) of each virus is modeled in Equation 2.6.
Ṫ = −βT V
Ė1 = βT V −
nE
E1
τE
nE
nE
Ej−1 −
Ej for j = 2, ..., nE
τE
τE
nE
nI
I˙1 =
EnE − I1
τE
τI
n
n
I
I
I˙j = Ij−1 − Ij for j = 2, ..., nI
τI
τI
nI
X
V̇ = p
Ij − cV
Ėj =
j=1
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
2.3 Efficacy and Drug Concentration
2.2.2
9
Parameters
As discussed in the previous paragraph, we assigned initial conditions to various
variables or parameters in the differential equations. These initial values are found in
Table 2.1 and come from previous influenza treatment studies. [12] The h in the table
is hours and the T CID stands for Tissue Culture Infective Dose and is a measure of
infectious viral titer.
Parameter Parameters for Influenza
To
106 cells
c
0.13 /h
τE
6.6 h
τI
49.0 h
nE
30
nI
100
β
4.260 · 10−4
p
176
mL
h · T CID
T CID · h
mL
Table 2.1 Parameter values for our mathematical model from Pinilla et al.
(2012) [12]
2.3
Efficacy and Drug Concentration
To model the effectiveness of the antivirals that treat influenza, we use the efficacy,
, which varies from 0 to 1 found in Equation 2.7. Efficacy will show how effective
the drug applied will be in treating the virus, an efficacy of 0 means the drug was
not effective, while an efficacy of 1 means the drug is completely effective. We model
2.3 Efficacy and Drug Concentration
10
efficacy using the drug concentration (D) and the IC50 of a specific antiviral. An
IC50 is the half maximal inhibitory concentration and it measures the effectiveness
of a substance, in this case an antiviral, at inhibiting a specific function [2]. Each
antiviral is assigned a specific IC50 value and from this we can determine the efficacy
of the antiviral. Using Equation 2.8 to model the drug concentration of the same
antiviral, we assumed max = 1 and solved for D. Since we do not know the IC50
D
→ D. Once D
of the antiviral that works best for our experiment, we made
IC50
is found in our experiment, we can work backwards to find the the associated IC50
value for our antiviral. Our drug concentration of the parameter combinations went
from 10−3 to 103 on a log scale.
=
max · D
D + IC50
(2.7)
1−
(2.8)
D=
2.3.1
Variables for Combination
We apply the same efficacy () to each of the variables discussed below in order to
determine which combination will produce the best efficacy and drug concentration.
By reducing the virus’ rate of infection into healthy cells, found in Equation 2.9, the
body’s immune system is able to destroy the viruses before they enter the cell. One
way to boost your immune sytem against influenza is to receive the yearly flu vaccine
and live a healthy life, through diet and exercise. This is not to say that the vaccine
will completely protect the body against infection, rather it is a good defense against
it.
β ⇒ β(1 − )
(2.9)
In order to prevent the virus from leaving the host cell, we apply an efficacy to
the production rate of the virus, found in Equation 2.10. If less virus leaves the host
2.3 Efficacy and Drug Concentration
11
cell, then fewer healthy target cells will become infected, giving the body’s immune
system time to fight off the infection naturally. However, if antivirals are needed,
there are three to choose from: oseltamivir, zanamivir, and peramivir [13]. All of
these antivirals work by binding to the protein site of neuraminidase, leaving the
virus unable to escape from the host cell. These antivirals can shorten the duration
of influenza by about a day, sometimes more, though they need to be taken when
symptoms first appear.
p ⇒ p(1 − )
(2.10)
Applying an efficacy to the clearance rate, found in Equation 2.11, the rate at
which the virus looses its infectivity or the rate of its decay, decreases the number of
healthy target cells the same virus can infect. With fewer host cells infected by the
virus, the number of cells the virus destroys will decrease.
c⇒
c
(1 − )
(2.11)
Making the eclipse phase last as long as possible prevents the virus’s RNA from
being copied and distributed into new virus which is found in Equation 2.12. This
allows for our immune system to destroy one virus still inside the cell, instead of
having to destroy the many viruses that have been replicated and dispersed out of
the host cell. There are two antivirals, amantadine [14] and rimantadine [15], that
delay the RNA replication process and block the virus’s ion channels once the virus
has entered the cell. These antivirals also assist in increasing the length of time the
cell is infected, found in Equation 2.13, because they eliminate the virus before it can
reach the infectious phase.
τE ⇒
τE
(1 − )
τI ⇒ τI (1 − )
(2.12)
(2.13)
2.3 Efficacy and Drug Concentration
12
After combining all the previous steps, from the differential equations to the parameter values, we were ready to begin running our code to model the virus’s lifecycle
thoughout the body.
Chapter 3
Results and Discussion
Using the variables mentioned above, we combined two at a time to model the efficacy
and drug concentration. In the following plots, Dmax , Vmax , and Duration are modeled
in that order. The top row shows the efficacy while the bottom row model drug
concentration for each combination of parameters. By adding a color bar scale to
each plot, we are able to see the corresponding data for each color. In these plots,
we want to see dark blue, instead of red, because that correlates with few cells dying,
a small quantity of virus produced and a short duration of the symptoms. For Dmax
only, we fixed this scale to be between 0 and 1 in order to truly see whether cells died
or not since the changes in color were so subtle.
3.1
Infection Rate and the Production Rate
The plots for the combination between the infection rate, β, and the rate of virus
production, p, are in Figure 3.1. The left column, shows the maximum number of
dead cells (Dmax ), no matter how much drug is given most of the cells have died
indictated by a fully red plot. This means that an antiviral with this combination
13
3.2 Infection Rate and the Time of Infection
14
Combination of β and p
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1. 0
Dmax for β and p
Drug Concentration for p
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1e1
10 2
10 1
0
2.00
1.92
1.84
1.76
1.68
10 −2
1.52
1.60
1.44
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
0. 3
4.96
4.92
4.88
4.84
4.80
4.76
4.72
4.68
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1. 0
10 3
2.08
10 −1
10 −3 −3
10
0. 45
0
1. 0
10 3
0. 6
0. 15
1.68
0
Duration for β and p
1.86
Efficacy for p
Vmax for β and p
1.92
1.74
0. 15
0
1.98
1.80
0. 3
0. 75
10 3
Duration for β and p
0. 15
0. 45
1. 0
2.04
Drug Concentration for p
0. 3
0. 6
Vmax for β and p
0. 45
1e1 2.10
1. 0
0. 75
Efficacy for p
Efficacy for p
0. 6
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for p
Dmax for β and p
1. 0
0. 75
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
4.98
4.92
4.86
4.80
4.74
4.68
4.62
4.56
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
4.50
Figure 3.1 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
of parameters would not save healthy or infected cells from being killed by influenza.
The middle column shows the maximum number of virus produced (Vmax ) and most
of the bottom half of the plot is red, meaning no efficacy or drug concentration of β
is going to treat the virus. The higher the efficacy and drug concentration of p, the
fewer virus will be produced. Duration of the symptoms are shown in the left column
and similar to the middle column, this combination produces mostly red areas. In the
efficacy plots for both Vmax and Duration, hardly any dark blue can be seen. Such
high levels of antivirals needed to treat the influenza virus can be unsafe for the body,
which makes this combination not optimal for treatment.
3.2 Infection Rate and the Time of Infection
15
Combination of β and τI
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1. 0
Dmax for β and τI
Drug Concentration for τI
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0. 6
1.96
1.92
0. 3
1.88
0. 15
1.84
Duration for β and τI
2.00
Efficacy for τI
Vmax for β and τI
2.04
0. 75
0. 45
0. 3
1.80
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
0
1. 0
10 3
1e1
10 2
10 1
0
10 −1
10 −3 −3
10
2.00
1.92
1.84
1.76
1.60
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
4.88
4.80
4.72
4.64
4.56
4.48
4.32
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1. 0
10 3
2.08
1.68
10 −2
4.96
4.40
0. 15
Duration for β and τI
0. 15
0. 45
1. 0
2.08
Drug Concentration for τI
0. 3
0. 6
Vmax for β and τI
0. 45
1e1
1. 0
0. 75
Efficacy for τI
Efficacy for τI
0. 6
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for τI
Dmax for β and τI
1. 0
0. 75
10 2
10 1
0
10 −1
4.95
4.80
4.65
4.50
4.35
4.20
10 −2
4.05
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
Figure 3.2 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
3.2
Infection Rate and the Time of Infection
The combination of β and τI , found in figure 3.2, has features similar to the first
plot. Again the left column leaves a high number of cells dead, but, in the middle
column for drug concentration, we see a larger area of blue than before. Even the
right column has Duration plots similar to the first combination, which still requires
a high drug dose to see this effect and is unsafe for the body.
3.3
Infection Rate and the Clearance Rate
The plots of the combination between β and c, found in figure 3.3, hold some promise
to being optimal treatments. Once again, the Dmax column does a poor job at saving
3.4 Infection Rate and the Time of Eclipse
16
Combination of β and c
0. 3
0. 15
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1. 0
Dmax for β and c
Drug Concentration for c
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Vmax for β and c
Efficacy for c
0. 45
1. 0
0. 75
0. 6
0. 45
4.9
4.8
4.7
4.6
4.5
0. 3
4.4
0. 15
0
10 3
1e1
Vmax for β and c
Efficacy for c
0. 6
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for c
Dmax for β and c
1. 0
0. 75
10 2
10 1
0
10 −1
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1. 0
2.08
2.00
1.92
1.84
1.76
1.68
1.60
10 −2
10 −3 −3
10
4.3
0
1.52
1.44
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
Figure 3.3 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
infected cells from dying no matter how much drug is given. The middle column of
Vmax is very similar to the previous plots, but the biggest change comes in the right
column. Almost the entire top half of the Duration plot is blue which means this
combination would be successful in shortening the duration of the symptoms without
requiring a large drug concentration of either antiviral. If this combination could save
more cells from dying, it would be effective combination for treatment.
3.4
Infection Rate and the Time of Eclipse
The combination between β and τE , found in figure 3.4, finally shows a blue area on
the Dmax plots. The area is small on the efficacy plot, but quite large on the drug
concentration plot. This shows that with a somewhat high drug concentration of one
3.5 Production Rate and Clearance Rate with Time of Inflection
17
Combination of β and τE
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1. 0
Dmax for β and τE
Drug Concentration for τE
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.2
0.9
0.6
0.3
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
10 1
0
10 −1
10 −3 −3
10
1.6
0.8
0.0
0.8
1.6
2.4
3.2
10 −2
4.0
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
0
0. 15
0. 3
0. 45
0. 6
Efficacy for β
0. 75
1. 0
10 3
1e1
10 2
0. 3
0
1. 0
10 3
0. 6
0. 45
0. 15
0.0
0
Duration for β and τE
1.5
Efficacy for τE
Vmax for β and τE
0. 3
0. 15
0. 75
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1e2
Duration for β and τE
0
0. 45
1e2
1. 0
1.8
Drug Concentration for τE
0. 3
0. 15
0. 6
Vmax for β and τE
0. 45
1e1 2.1
1. 0
0. 75
Efficacy for τE
Efficacy for τE
0. 6
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for τE
Dmax for β and τE
1. 0
0. 75
10 −2
10 −1
0
10 1
Drug Concentration for β
10 2
10 3
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Figure 3.4 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
antiviral cells can be saved. The most noticeable aspect of this combination can be
seen in the Duration plots because the blue area is adjacent to the red area, while the
orange covers most of the plot. This combination would be difficult in order to get
the drug concentration just right so it stays out of the red area of the graph. Since
the efficacy plots show that only with an extremely high drug dose it will be effective
in shortening the duration, it would not be unsafe for treating influenza.
3.5 Production Rate and Clearance Rate with Time of Inflection
18
Combination of p and c
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
1. 0
Dmax for p and c
Drug Concentration for c
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for p
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.7
1.6
Duration for p and c
1.8
Efficacy for c
Vmax for p and c
1.9
1.5
0. 3
0. 75
0. 6
0. 45
0. 3
0. 15
1.3
0. 15
0
1.2
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
1. 0
10 3
10 1
0
2.0
1.8
1.6
1.4
10 −1
1.2
10 −2
1.0
10 −3 −3
10
0.8
10 −2
10 −1
0
10 1
Drug Concentration for p
10 2
10 3
4.8
4.7
4.6
4.5
4.4
4.2
4.1
0
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
1. 0
10 3
1e1
10 2
4.9
4.3
1.4
1e2
Duration for p and c
0. 15
0. 45
1. 0
2.0
Drug Concentration for c
0. 3
0. 6
Vmax for p and c
0. 45
1e1 2.1
1. 0
0. 75
Efficacy for c
Efficacy for c
0. 6
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for c
Dmax for p and c
1. 0
0. 75
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for p
10 2
10 3
1.35
1.20
1.05
0.90
0.75
0.60
0.45
0.30
0.15
0.00
Figure 3.5 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
3.5
Production Rate and Clearance Rate with Time
of Inflection
The combination between p and c, found in figure 3.5, and that of p and τI , found in
figure 3.6, have similar features in their plots. Both have their entire Dmax column
completely red. However, the Vmax columns have a different trend than previously
seen. Instead of needing a high drug dose for only one variable, now we need a high
drug dose for both variables since the dark blue section is in the upper right corners of
the plots. In the Duration columns, we see the same trend from the middle column,
but now we have a larger area of lighter blue and green. These areas mean that the
likelihood of the drug decreasing the duration will not be very likely.
The remaining combinations can be found in the Appendix since those graphs do
3.5 Production Rate and Clearance Rate with Time of Inflection
19
Combination of p and τI
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
1. 0
Dmax for p and τI
Drug Concentration for τI
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for p
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
10 1
0
10 −1
10 −3 −3
10
1.95
1.80
1.65
1.50
1.35
1.20
1.05
10 −2
0.90
10 −2
10 −1
0
10 1
Drug Concentration for p
10 2
10 3
0. 3
0
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
4.6
4.4
4.2
4.0
1e2
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
3.6
1. 0
10 −2
10 −1
0
10 1
Drug Concentration for p
10 2
10 3
Figure 3.6 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
not offer an optimal combination.
4.8
3.8
10 3
1e1 2.10
10 2
0. 45
0
1. 0
10 3
0. 6
0. 15
1.4
0
Duration for p and τI
1.7
Efficacy for τI
Vmax for p and τI
1.8
1.5
0. 15
0
1.9
1.6
0. 3
0. 75
Duration for p and τI
0. 15
0. 45
1. 0
2.0
Drug Concentration for τI
0. 3
0. 6
Vmax for p and τI
0. 45
1e1 2.1
1. 0
0. 75
Efficacy for τI
Efficacy for τI
0. 6
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for τI
Dmax for p and τI
1. 0
0. 75
1.35
1.20
1.05
0.90
0.75
0.60
0.45
0.30
0.15
0.00
Chapter 4
Conclusion
The question about which combination would be optimal for treatment of influenza
virus was answered. Relating a virus’s replication process to a mathematical model
allowed us to see how a virus responds to antivirals. After combining the parameters
of the replication process, we were able to get a better idea of which combination was
the most effective. The optimal combination was β and c, which supports the idea
that the influenza virus is best treated when antivirals prevent the infection from
occurring inside the cell. The large areas of blue in the Vmax and Duration plots
were greater than any other combination including those with blue sections in the
left column. Even though our model killed any cell infected by the virus, resulting
in an entire red Dmax plot, this combination had the best results for both Vmax and
Duration. Not only have we found the optimal combination, we also found that
every combination had some dark blue section in the Vmax and Duration plots, which
makes them successful as a antiviral combination.
20
Bibliography
[1] “Influenza (Seasonal),”, 2016.
[2] H. Dobrovolny, personal communication.
[3] E. V. Koonin, T. G. Senkevich, and V. V. Dolja, “The ancient Virus World and
evolution of cells,” Biology direct 1, 29 (2006).
[4] M. Jit, A. T. Newall, and P. Beutels, “Key issues for estimating the impact and
cost-effectiveness of seasonal influenza vaccination strategies,” Human vaccines
& immunotherapeutics 9, 834–840 (2013).
[5] C. S. HARBOR and N. LI, “Cold Spring Harbor symposia on quantitative biology,” In Cold Spring Harb Symp Quant Biol, 43, 1197–1208 (1979).
[6] M. Cushion, A. Balows, and M. Sussman, “Topley and Wilson’s microbiology and
microbial infections,” Topley and Wilson’s Microbiology and Microbial Infections
(1998).
[7] N. J. Dimmock, A. J. Easton, and K. N. Leppard, Introduction to modern virology
(John Wiley & Sons, 2016).
[8] V. R. Racaniello and P. Palese, “Isolation of influenza C virus recombinants.,”
Journal of virology 32, 1006–1014 (1979).
21
BIBLIOGRAPHY
22
[9] M. Hatta and Y. Kawaoka, “The NB protein of influenza B virus is not necessary
for virus replication in vitro,” Journal of virology 77, 6050–6054 (2003).
[10] I. Bozic et al., “Evolutionary dynamics of cancer in response to targeted combination therapy,” Elife 2, e00747 (2013).
[11] M. N. Prichard and C. Shipman, “A three-dimensional model to analyze drugdrug interactions,” Antiviral research 14, 181–205 (1990).
[12] L. T. Pinilla, B. P. Holder, Y. Abed, G. Boivin, and C. A. Beauchemin, “The
H275Y neuraminidase mutation of the pandemic A/H1N1 influenza virus lengthens the eclipse phase and reduces viral output of infected cells, potentially compromising fitness in ferrets,” Journal of virology 86, 10651–10660 (2012).
[13] A. Moscona, “Neuraminidase Inhibitors for Influenza,” New England Journal of
Medicine 353, 1363–1373 (2005), pMID: 16192481.
[14] C. Wang, K. Takeuchi, L. Pinto, and R. Lamb, “Ion channel activity of influenza
A virus M2 protein: characterization of the amantadine block.,” Journal of virology 67, 5585–5594 (1993).
[15] X. Jing, C. Ma, Y. Ohigashi, F. A. Oliveira, T. S. Jardetzky, L. H. Pinto, and
R. A. Lamb, “Functional studies indicate amantadine binds to the pore of the
influenza A virus M2 proton-selective ion channel,” Proceedings of the National
Academy of Sciences 105, 10967–10972 (2008).
Appendix A
Extra Plots of Combinations
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for c
0. 75
1. 0
Dmax for c and τI
Drug Concentration for τI
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10
−2
10
−1
0
10 1
Drug Concentration for c
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0. 3
1.92
1.84
1.76
1.68
1.60
0. 15
0
0. 75
0
0. 15
0. 3
0. 45
0. 6
Efficacy for c
0. 75
10 1
0
10 −1
0
1.95
1.80
1.65
1.50
1.35
1.05
10
−2
10
−1
0
10 1
Drug Concentration for c
10 2
10 3
0. 3
1.44
1.20
10 −2
0. 45
0. 15
4.4
4.0
3.6
3.2
2.4
0
0. 15
0. 3
0. 45
0. 6
Efficacy for c
0. 75
1e2
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
2.0
1. 0
10 −2
10 −1
0
10 1
Drug Concentration for c
10 2
10 3
Figure A.1 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
23
4.8
2.8
10 3
1e1 2.10
10 2
0. 6
1.52
1. 0
10 3
10 −3 −3
10
Efficacy for τI
0. 45
1. 0
2.00
Duration for c and τI
0. 15
0. 6
2.08
Drug Concentration for τI
0. 3
Vmax for c and τI
0. 45
0. 75
Vmax for c and τI
0. 6
1e1
1. 0
Efficacy for τI
Efficacy for τI
0. 75
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for τI
Dmax for c and τI
1. 0
Duration for c and τI
Combination of c and τI
1.35
1.20
1.05
0.90
0.75
0.60
0.45
0.30
0.15
0.00
24
Combination of c and τE
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for c
0. 75
1. 0
Dmax for c and τE
Drug Concentration for τE
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for c
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0. 75
1.2
0. 6
0.8
0.4
0.0
0. 15
0
0. 15
0. 3
0. 45
0. 6
Efficacy for c
0. 75
10 1
0
10 −1
0.8
0.0
0.8
1.6
2.4
3.2
10 −2
10 −3 −3
10
1.6
4.0
10 −2
10 −1
0
10 1
Drug Concentration for c
10 2
10 3
4.8
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
0. 15
0. 3
0. 45
0. 6
Efficacy for c
0. 75
1e2
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
0.0
1. 0
10 3
1e1
10 2
0. 3
0
1. 0
10 3
0. 45
0. 15
0.4
0
1e2
Duration for c and τE
0. 3
1.6
Efficacy for τE
0. 45
1. 0
Duration for c and τE
0. 15
0. 6
2.0
Drug Concentration for τE
0. 3
Vmax for c and τE
0. 45
0. 75
Vmax for c and τE
0. 6
1e1
1. 0
Efficacy for τE
Efficacy for τE
0. 75
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for τE
Dmax for c and τE
1. 0
10 −2
10 −1
0
10 1
Drug Concentration for c
10 2
10 3
Figure A.2 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
25
Combination of p and τE
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
1. 0
Dmax for p and τE
Drug Concentration for τE
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10
−2
10
−1
0
10 1
Drug Concentration for p
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.2
0. 6
0.8
0.4
0.0
0. 15
0
0. 75
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
10 1
0
10 −1
10 −2
10 −3 −3
10
10
−2
10
−1
0
10 1
Drug Concentration for p
10 2
10 3
1.6
0.8
0.0
0.8
1.6
2.4
3.2
4.0
4.8
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
0. 15
0. 3
0. 45
0. 6
Efficacy for p
0. 75
1e2
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
0.0
1. 0
10 3
1e1
10 2
0. 3
0
1. 0
10 3
0. 45
0. 15
0.4
0
1e2
Duration for p and τE
0. 3
1.6
Efficacy for τE
0. 45
1. 0
Duration for p and τE
0. 15
0. 6
2.0
Drug Concentration for τE
0. 3
Vmax for p and τE
0. 45
0. 75
Vmax for p and τE
0. 6
1e1
1. 0
Efficacy for τE
Efficacy for τE
0. 75
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for τE
Dmax for p and τE
1. 0
10 −2
10 −1
0
10 1
Drug Concentration for p
10 2
10 3
Figure A.3 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
26
Combination of tauI and τE
0
0
0. 15
0. 3
0. 45
0. 6
Efficacy for τI
0. 75
1. 0
Dmax for τI and τE
Drug Concentration for τE
10 3
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
10 −2
10 −1
0
10 1
Drug Concentration for τI
10 2
10 3
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0. 75
1.2
0. 6
0.8
0.4
0.0
0. 15
1e2
Duration for τI and τE
0. 3
1.6
Efficacy for τE
0. 45
1. 0
0. 45
0. 3
0
0. 15
0. 3
0. 45
0. 6
Efficacy for τI
0. 75
10 3
10 1
0
10 −1
10 −2
10 −2
10 −1
0
10 1
Drug Concentration for τI
10 2
10 3
0
0. 15
0. 3
0. 45
0. 6
Efficacy for τI
0. 75
1
0
1
2
3
4
5
6
7
1.4
1.2
1.0
0.8
0.6
1e2
10 2
10 1
0
10 −1
10 −2
10 −3 −3
10
0.0
1. 0
10 3
1e1 2
10 2
10 −3 −3
10
0
1. 0
1.6
0.2
0.4
0
1.8
0.4
0. 15
Duration for τI and τE
0. 15
0. 6
2.0
Drug Concentration for τE
0. 3
Vmax for τI and τE
0. 45
0. 75
Vmax for τI and τE
0. 6
1e1
1. 0
Efficacy for τE
Efficacy for τE
0. 75
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Drug Concentration for τE
Dmax for τI and τE
1. 0
10 −2
10 −1
0
10 1
Drug Concentration for τI
10 2
10 3
Figure A.4 Upper left: maximum number of dead cells using efficacy. Upper
Middle: maximum number of virus using efficacy. Upper Right: Duration
of symptoms using efficacy. Bottom Left: maximum number of dead cells
for drug concentration. Bottom Middle: maximum number of virus for drug
concentration. Bottom Right: Duration of symptoms for drug concentration.
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Appendix B
Code
This code was given to me by Dr. Hana Dobrovolny and Thalia Rodriguez which I
then modified for my experiment.
from numpy import *
import numpy as np
import math
import matplotlib.pyplot as plt
from scipy.integrate import odeint
plt.ion()
################
# parameters based on influenza treatment
################
nE= 30
nI= 100
b= 4.260/10**4 #
p= 176.0 #
tdIi= 49.0 #
tdEj= 6.6 #
27
28
gamma= 0.0 #
c= 0.13 #
Vmax = np.empty([100,100])
Dmax = np.empty([100,100])
Duration = np.empty([100,100])
data = np.empty([10000,5])
#################
#loops for measuring the combination of two different drugs including the
differential eqs.
#################
for i in range (100):
Eps=i*0.01
tdI=tdIi*(1-Eps)
for j in range (100):
Eps2=j*0.01
tdE=tdEj/(1-Eps2)
#print(Eps,Eps2)
# solve the system dy/dt = f(y, t)
def f(Z,t):
#
global par
#
b = par[0]
#
prod = par[1]
#
clear = par[2]
#
nI = int(par[3])
#
nE = int(par[4])
d = nI / tdI
k = nE / tdE
29
N = 1
#
T, E, I, D, N, V = 0, 1, 2, np.arange(3, 3 + nE), np.arange(3
+ nE, 3 + nE + nI), np.sum(np.arange(T, D + 1))
f = []
f.append(-(b / N) * Z[1+nE+nI] * Z[0])
f.append((b / N) * Z[1+nE+nI] * Z[0] - (k * Z[1]))
for i in range(1,nE):
f.append(k * (Z[i]-Z[i+1]))
f.append(k * Z[nE] - (d * Z[1+nE]))
for i in range(1, nI):
f.append(d * (Z[nE+i]-Z[nE+i+1]))
f.append(p * np.sum(Z[1+nE:nE+nI]) - (c * Z[1+nE+nI]))
f.append(d*Z[nE+nI])
return f
# initial conditions
#T0 = y0[0]
y0 = [(10**6)-50]
#E0 = y0[2]= 0
y0.append(50)
for iii in range (2,nE+1):
y0.append(0)
#I01 = y0[3]= 0
for jjj in range (1,nI+1):
y0.append(0)
#V0 = y0[1]=0
y0.append(0)
#D0 = 0
30
y0.append(0)
time = np.linspace(0, 240., 100) # time grid
# solve the DEs
soln = odeint(f, y0, time,mxstep=50000000) #found this on
google, it increased the max allowed step size. needed
this in order to get rid of the lsoda warning, which was
creating spots on the contour plots.
T = soln[:, 0]
V = soln[:, -2]
E1 = soln[:, 1]
I1 = soln[:, nE+1]
D= soln[:, -1]
#plt.plot(log(V))
#plt.show(True)
Dmax[i,j]=max(D)
Vmax[i,j]=max(V)
if len(np.where(V>10**4)[0])==0:
Duration[i,j]=0
else:
Duration[i,j] =
(np.where(V>10**4)[0][-1]-np.where(V>10**4)[0][0])*2.4
#Tmax=max(T)
#V12=V[120]
#V48=V[480]
#D36=D[360]
#auc=sum(V)
data[100*i+j,:]=[Eps,Eps2,Dmax[i,j],Vmax[i,j],Duration[i,j]]
31
#,Dmax,V12,V48,D36,auc
savetxt("Eps1_Eps2.dat",data)
################
# contour plots of Dmax, Vmax, and Duration
################
#plt.figure(4)
#plt.contour(log(Vmax)) #Dmax, Duration
#Vmax
plt.figure(1)
plt.imshow(log(Vmax))
plt.ylabel(’Eps1 for tau E’)
plt.xlabel(’Eps2 for tau I’)
plt.title(’Vmax for Combination Therapy for tau E and tau I’)
plt.colorbar()
#Dmax
plt.figure(2)
plt.imshow((Dmax))
plt.ylabel(’Eps1 for tau E’)
plt.xlabel(’Eps2 for tau I’)
plt.title(’Dmax for Combination Therapy for tau E and tau I’)
plt.colorbar()
#Duration
plt.figure(3)
plt.imshow((Duration))
plt.ylabel(’Eps1 for tau E’)
32
plt.xlabel(’Eps2 for tau I’)
plt.title(’Duration for Combination Therapy for tau E and tau I’)
plt.colorbar()
plt.show(block=True)
from numpy import *
import numpy as np
import math
import matplotlib.pyplot as plt
from scipy.integrate import odeint
plt.ion()
################
# parameters based on influenza treatment
################
nE= 30
nI= 100
b= 4.260/10**4 #
p= 176.0 #
tdIi= 49.0 #
tdEj= 6.6 #
gamma= 0.0 #
c= 0.13 #
Vmax = np.empty([100,100])
Dmax = np.empty([100,100])
Duration = np.empty([100,100])
data = np.empty([10000,5])
#################
33
#loops for measuring the combination of two different drugs including the
differential eqs.
#################
i = 0
j = 0
for D1 in np.logspace(-3.,3,num=100):
#Eps=i*0.1
Eps = D1/(1+D1)
tdI=tdIi*(1-Eps)
j = 0
for D2 in np.logspace(-3.,3, num=100):
#Eps2=j*0.1
Eps2=D2/(1+D2)
tdE= tdEj/(1-Eps2)
#print(Eps,Eps2)
# solve the system dy/dt = f(y, t)
def f(Z,t):
#
global par
#
b = par[0]
#
prod = par[1]
#
clear = par[2]
#
nI = int(par[3])
#
nE = int(par[4])
d = nI / tdI
k = nE / tdE
N = 1
#
T, E, I, D, N, V = 0, 1, 2, np.arange(3, 3 + nE), np.arange(3
34
+ nE, 3 + nE + nI), np.sum(np.arange(T, D + 1))
f = []
f.append(-(b / N) * Z[1+nE+nI] * Z[0])
f.append((b / N) * Z[1+nE+nI] * Z[0] - (k * Z[1]))
for i in range(1,nE):
f.append(k * (Z[i]-Z[i+1]))
f.append(k * Z[nE] - (d * Z[1+nE]))
for i in range(1, nI):
f.append(d * (Z[nE+i]-Z[nE+i+1]))
f.append(p * np.sum(Z[1+nE:nE+nI]) - (c *
Z[1+nE+nI]))
f.append(d*Z[nE+nI])
return f
# initial conditions
#T0 = y0[0]
y0 = [(10**6)-50]
#E0 = y0[2]= 0
y0.append(50)
for iii in range (2,nE+1):
y0.append(0)
#I01 = y0[3]= 0
for jjj in range (1,nI+1):
y0.append(0)
#V0 = y0[1]=0
y0.append(0)
#D0 = 0
y0.append(0)
35
time = np.linspace(0, 240., 100) # time grid
# solve the DEs
soln = odeint(f, y0, time, mxstep = 500000000)
T = soln[:, 0]
V = soln[:, -2]
E1 = soln[:, 1]
I1 = soln[:, nE+1]
D= soln[:, -1]
#
plt.plot(log(V))
#
plt.show(True)
Dmax[i,j]=max(D)
Vmax[i,j]= max(V)
if len(np.where(V>10**4)[0])==0:
Duration[i,j]=0
else:
Duration[i,j] =
(np.where(V>10**4)[0][-1]-np.where(V>10**4)[0][0])*2.4
#Tmax=max(T)
#V12=V[120]
#V48=V[480]
#D36=D[360]
#auc=sum(V)
data[100*i+j,:]=[D1,D2,Dmax[i,j],Vmax[i,j],Duration[i,j]]
#,Dmax,V12,V48,D36,auc
j += 1
#print j
i += 1
36
print i
savetxt("D1_D2.dat",data)
#print T
#print V
#print time
#print Eps
#print Eps2
################
# contour plots of Dmax, Vmax, and Duration
#plt.xlabel(’E1 for beta’)
#plt.ylabel(’E2 for rho’)
#plt.title(’Vmax for Combination Therapy’)
################
#plt.figure(1)
#plt.contour(log(Vmax)) #Dmax, Duration
#Vmax
plt.figure(1)
plt.imshow(log(Vmax))
plt.ylabel(’D1 for tau I’)
plt.xlabel(’D2 for tau E’)
plt.title(’Vmax for Combination Therapy for tau I and tau E’)
plt.colorbar()
#plt.show(block=True)
#Dmax
plt.figure(2)
plt.imshow((Dmax))
37
plt.ylabel(’D1 for tau I’)
plt.xlabel(’D2 for tau E’)
plt.title(’Dmax for Combination Therapy for tau I and tau E’)
plt.colorbar()
#plt.show(block=True)
#Duration
plt.figure(3)
plt.imshow((Duration))
plt.ylabel(’D1 for tau I’)
plt.xlabel(’D2 for tau E’)
plt.title(’Duration for Combination Therapy for tau I and tau E’)
plt.colorbar()
plt.show(block=True)
from optparse import OptionParser
from pylab import *
import numpy as np
parser = OptionParser()
parser.add_option("-f", "--filename", metavar="FILE", default="combined",
help="read data file FILE [combined]")
(options, args) = parser.parse_args()
del parser, args
data = np.loadtxt("tauI_tauE.dat")
rb = data[:,0] # Find y data
rp = data[:,1] # Find x data
38
dst = data[:,2] # Find z data. Change from 2-4 depending on dmax, vmax,
and duration.
dose = compress( rp == rp[0], rb ) # Find unique x values
time = compress( rb == rb[0], rp ) # Find unique y values
z = reshape( dst, (len(dose), len(time)) ) # Reshape z into matrix
fig = plt.figure()
ax = fig.add_axes([0.15, 0.15, 0.7, 0.7])
#plt.plot(log10(dose), log10(0.046*3.2*10**(-5)*4.*10**8/5.2+dose-5.2),’k’)
#print(dose)
#print(0.046*3.2*10**(-5)*4*10**8/5.2-dose-5.2)
#plt.imshow( np.transpose(abs(z)/10000),
aspect=’auto’,vmin=min(abs(dst)/10000), vmax=max(abs(dst)/10000)
,origin=’lower’, cmap=cm.jet, extent=(-3,3,-3,3), axes=ax)
plt.imshow( np.transpose((z)), aspect=’auto’,vmin=0, vmax=1 ,
origin=’lower’, cmap=cm.jet, extent=(-3,3,-3,3), axes=ax)
yticks( arange(-3,4,1), (’$10^{-3}$’,’$10^{-2}$’,
’$10^{-1}$’,’$0$’,’$10^1$’,’$10^2$’,’$10^3$’), fontsize=15 )
xticks( arange(-3,4,1), (’$10^{-3}$’,’$10^{-2}$’,
’$10^{-1}$’,’$0$’,’$10^1$’,’$10^2$’,’$10^3$’), fontsize=15 )
xlabel(r’Drug Concentration for $\tau_I$’, fontsize=18) #Diffusion
ylabel(r’Drug Concentration for $\tau_E$’, fontsize=18) #Advection
#title(’Duration of Drug Concentration’, fontsize=20)
39
#text(-1.75, 10.25, r’Peak viral titer (upper)’, fontsize=24)
ax = fig.add_axes([0.89, 0.15, 0.03, 0.65])
cb =colorbar(cax=ax ,orientation=’vertical’)
cb.formatter.set_powerlimits((0, 0))
cb.update_ticks()
#labels = [’$ $’, ’50’, ’$ $’, ’100’, ’$ $’, ’150’, ’$ $’, ’200’ ]
#ax.set_yticks([-4, -2, 0, 2, 4, 6])
#ax.set_yticklabels(labels)
text(-1.3, 0.9, r’Dmax for $\tau_I$ and $\tau_E$’, fontsize=18,
rotation=90)
yticks(fontsize=13)
show()