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The Effects of Isoniazid and Rifapentine on Tuberculosis

Mathematics 19a: Modeling and Differential Equations for the Life Sciences

Joshua M. Elkington

May 2, 2012

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Background

Tuberculosis has the second highest mortality rate of any infectious agent. A type of

bacteria called mycobacterium tuberculosis causes tuberculosis. Tuberculosis (TB) is an

infectious disease that commonly attacks the lungs, and the disease is spread through the

air via phlegm. In many people, infection by mycobacterium tuberculosis causes no

symptoms, but the disease is still detectable by a tuberculin test. People infected with

tuberculosis are non-infectious (latent) or infectious (active). Approximately, one-third of

the world’s population has latent tuberculosis; however, these people only have a 10%

chance of becoming active throughout their lifetime, but people with weak immune

systems due to HIV or other diseases have a higher risk of becoming active.  

Over 95% of the cases and deaths of tuberculosis are in developing countries

especially in Africa and Asia. Twenty-two countries account for around 80% of the TB

cases reported with TB declining in some countries such as China (5). Due to

tuberculosis’s global prevalence in poor countries, a treatment that is cheap and effective is

needed.

In The New England Journal of Medicine, researchers show that a combination

therapy of tuberculosis is more effective in treating latent Mycobacterium tuberculosis than

using a single drug. The researchers determined that use of rifapentine and isoniazid

(combination therapy) for three months was jut as effective as nine months of treatment

with only isoniazid. From this, I determined that the rate of recovery due to combination

therapy is twice the rate due to isoniazid (2).

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The paper will investigate the effects of quarantine and drug treatments on TB. A

SIR model will initially be used to model TB; however, the model will be modified to

include a latent infected population. The inclusion of this latent population will change

the dynamics of the SIR model, and the effects of rifapentine and isoniazid on the latent

infectious will be modeled in order determine how long a treatment program needs to be

used in order to eradicate this disease. In order to provide estimates for the various rates in

the model, I used documented rates and reasonable assumptions.

Ultimately, the model shows that combination therapy is an effective way to reduce

the infectious population when compared to other methods. However, the quarantine of

infected individuals should still be used. Furthermore, this treatment is a cheaper way to

treat TB because the time frame for treatment is one-third the time needed for treatment

with only isoniazid.

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Mathematical Model

In this paper two models are used: a SIR model and a model with two classes of

infected people (3).

In the models, S(t) represents the population of susceptible individuals, I(t)

represents the infectious population, and R(t) represents the recovered population. In the

SIR model, I use the assumption that once an individual has recovered, the person cannot

be re-infected. The model with two classes of infected individuals,  !!  represents the

population of active infectious people and !! represents the population of latent infectious

!"  !"

= -­‐βS! !"!"

= βS!  -­‐ α  ! !"!"

= α!  

!"  !"

= -­‐βS!!

!"!!"

= βS!!-­‐  γ!!

!"!!"

= γ!! -­‐α  !!

!"!"

= α!!

 

Terms: β – infection rate α – recovery rate µ - birth rate ε - death rate ω - death rate due to TB γ – rate of changing from latent to active t – time (in years) ϕ – effects of Isoniazid on latent recovery 2ϕ – effects of Isoniazid and Rifapentine on latent recovery η – relapse rate Q – Quarantine population q – rate at which infected population is quarantined α*- recovery rate of quarantined population

Terms Values: β = .002 (using N=300 [units in millions] and !! = 12) α = .05 (assuming that within a year 1/20 of infected recover within a year) µ = .02 ε  =.01 ω = .00045 (4) γ = .001 (using that within a lifetime a latent individual has a 10% chance of becoming active) ϕ = .28 (assuming within a year one fourth of infected recover due to treatment) η = .05 (using that within 2 years 10% of recovered people relapse) (1) q = .1 α* = .16 (assuming that within a year 3/20 of infected recover within a year)  

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people. The constant rate β represents the rate of infection, and α represents the rate of

recovery. The rate γ represents the rate at which a latent infected person becomes active.

For the models, I make the following assumptions:

(1) The rate of infection and recovery is constant for the population

(2) The initial population is healthy and disease free

For the model with two classes of infectious, I make the assumption that eventually

all individuals with latent TB become active. In reality, most people with latent TB do not

develop active TB so a more realistic model would have a death rate.

Using these two models, I make adjustments by adding a relapse rate (η), a death rate

due to TB (ω), a net birth and death rate (µ), a rate of recovery due to drug treatment (ϕ),

and a rate of quarantine (q).

!"  !"

= -­‐βS!! + µ(S+!!+  !!+R) –  εS !"!!"

= βS!!-­‐  (γ+  ε)!!

!"!!"

= γ!! -­‐ (α+ε)  !!

!"!"

= α!! –  εR

 

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Analysis

   

In both models if the susceptible population initially begins at a point to the right

of the nullcline S=!  !  

then the infected population initially increases. Then the infected

population decreases once the curve crosses the S-nullcline and approaches an equilibrium

solution on the S-axis. In the SIR model, the infected population begins to decrease at

S=25, and in comparison with quarantine the infected population decreases at S=75.

Therefore quarantine is a tool to prevent an epidemic.

 SIR  

 SIR  with  quarantine  

!"  !"

= -­‐βS! !"!"

= βS!  -­‐ α  ! !"!"

= α!  

 

!"  !"

= -­‐βS! !"!"

= βS!  -­‐ α  ! -­‐ qI

!"!"

= α! +α*Q

!"!"

= qI -­‐ α*Q

   

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People with Tuberculosis (Latent and Active with effects of Isoniazid) !"  !"

= -­‐βS!!

!"!!"

= βS!!-­‐(γ+ϕ)!!

!"!!"

= γ!! -­‐α  !!

!"!"

= α!! + ϕ!!

People with Tuberculosis (Latent and Active with effects of combination therapy) !"  !"

=  -­‐βS!!

!"!!"

= βS!!-­‐  (γ+2ϕ)!!

!"!!"

= γ!! -­‐α  !!

!"!"

= α!!+2ϕ!!    

 Graph with no treatment

Blue-Susceptible Red-Latent Infected  

 Graph with Isoniazid

 Graph with Combination Therapy

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The graphs above show the susceptible and latent infectious populations with

respect to time. Without treatment, overtime the latent infectious population grows, while

the susceptible population decreases. However, when TB is treated the latent infectious

population dies out. Moreover, the individuals with latent tuberculosis recover sooner

when treated with combination therapy than with only Isoniazid treatment.

                   

The graphs above show the active infectious population with respect to time.

Without treatment, the active infectious population grows overtime. The active infectious

population initially increases when treated with drugs, but the increase is more

pronounced when the active infectious individuals are only treated with Isoniazid. These

 Active  Infected  Population  with  Isoniazid  

treatment  

 Active  Infected  Population  with  

Combination  therapy    

 Active  Infected  Population  with  no  

treatment    

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results imply that the latent population will initially grow as well, and if the infected

population grows initially more with only Isoniazid treatment then TB will survive longer

within the population compared to a Combination treatment program.

Finally, I determined the trace and determinant of the Jacobian matrix for the

model given above in order to determine the stability of certain points. There is a stable

equilibrium point at R=N and S=!!=!!=0, which makes sense because everyone recovers

from TB. Also, there is an unstable equilibrium point at !!=!!=0, S=K, and R=N-K where

K is an element of [0,N], which makes sense because the disease will not die out when

there is a susceptible population.

     

Latent and Active TB with birth, death, and relapse rates !"  !"

= -­‐βS!! + µ(S+!!+  !!+R)  –  εS !"!!"

= βS!! - (γ+  ε)!!+ηR

!"!!"

= γ!! -­‐ (α+ω+  ε)  !!+ηR

!"!"

= α!! – (2η+  ε)R

 

 Graph with No Treatment

Blue-Susceptible Red-Latent Infected Yellow-Active Infected Green-Recovered  

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    The graphs above show a more realistic depiction of the dynamics of tuberculosis

because the system of equations account for birth, death, and relapse. When TB is not

treated, latent TB remains present within the population. However, when TB is treated

with drugs, individuals with latent TB decrease in a population. Furthermore, individuals

with latent TB approach zero more quickly when treated with combination therapy in

comparison with only Isoniazid treatment. When TB is treated with drugs, active infectious

individuals decrease more slowly in comparison with no treatment because more people

Latent and Active TB with birth, death, and relapse rates treated with Isoniazid !"  !"

= -­‐βS!! + µ(S+!!+  !!+R)  –  εS !"!!"

= βS!!-­‐ (γ+  ε +ϕ)!!+ηR

!"!!"

= γ!!  -­‐ (α+ω+  ε)  !!+ηR

!"!"

= α!! – (2η+  ε)R+ϕ!!  

Latent and Active TB with birth, death, and relapse rates treated with Combination Therapy !"  !"

= -­‐βS!!  + µ(S+!!+  !!+R)  –  εS !"!!"

= βS!!-­‐  (γ+ε+2ϕ)!!+ηR

!"!!"

= γ!! -­‐ (α+ω+  ε)  !!+ηR

!"!"

= α!! – (2η+  ε)R+2ϕ!!  

 Graph with Combination Therapy

 

 Graph with Isoniazid

 

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recover and have a greater chance of relapsing and becoming active or latent infectious. So

in reality, treatment will lead to a slow reduction in the active infectious population.

Conclusion

The models used show that quarantine is an effective tool to prevent an epidemic.

Furthermore, my analysis shows that combination therapy is a more effective and quicker

treatment than only using Isoniazid. Ultimately, the models prove that a treatment that has

a shorter time span is more effective and naturally cheaper. Creating treatment program for

tuberculosis that is cheap is important because TB is concentrated in developing countries.

Therefore, combination therapy would be a more effective way to deal with this disease

when compared to a single drug treatment.

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References

1. Lambert Marie-Laurence, Epco Hasker, Armand Van Deun, Dominique Roberfroid,

Marleen Boelaert, Patrick van der Stuyft. Recurrence in tuberculosis: relapse or reinfection? The

Lancet Infectious Diseases 3 (2003): 282–287.

2. Sterling Timothy R., M. Elsa Villarino, Andrey S. Borisov, et al. Three months of

rifapentine and isoniazid for latent tuberculosis infection. The New England Journal of Medicine

365.6 (2011): 2155–2166.

3. Young Douglas, Jaroslav Stark, Denise Kirschner. Systems biology of persistent infection:

tuberculosis as a case study. Nature Reviews Microbiology 6 (2008): 520–528.

4. “TB Deaths per 100,000 population 2010.” globalhealthfacts. The Henry J. Kaiser Family

Foundation, n.d. Web. 26 Apr. 2012.

5. World Health Organization. World Heath Organization, 26 Apr. 2012. Web. 26 Apr.

2012.

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Appendix

Total Population: N=S(t)+!(!)!"#$%#+!(!)!"#$%&+R(t)

S – Susceptible I – Infected R – Recovered SIR Model !"  !"

= -­‐βS! !"!"

= βS!  -­‐ α  ! !"!"

= α! SIR with quarantine

SIR with relapse !"  !"

= -­‐βS! !"!"

= βS!  -­‐ α  ! +ηR

!"!"

= α!-­‐ηR

Terms: β – infection rate α – recovery rate µ - birth rate ε -death rate ω - death rate due to TB γ – rate of changing from latent to active t – time (in years) ϕ – effects of Isoniazid on latent recovery 2ϕ – effects of Isoniazid and Rifapentine on latent recovery η – relapse rate Q – Quarantine population q – rate at which infected population is quarantined α*- recovery rate of quarantined population Terms Values: β = .002 (using N=300 [units in millions] and !! = 12) α = .05 (assuming that within a year 1/20 of infected recover within a year) µ = .02 ε  =.01 ω = .00045 (4) γ = .001 (using that within a lifetime a latent individual has a 10% chance of becoming active) ϕ = .28 (assuming within a year one fourth of infected recover) η = .05 (using that within 2 years 10% of recovered people relapse) (1) q = .1 α* = .16 (assuming that within a year 3/20 of infected recover within a year)  

!! = !!!

 

!"  !"

= -­‐βS! !"!"

= βS!  -­‐ α  ! -­‐ qI

!"!"

= α! +α*Q

!"!"

= qI -­‐ α*Q  

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Latent and Active TB

Latent and Active TB with effects of combination therapy !"  !"

=  -­‐βS!!

!"!!"

= βS!!-­‐  (γ+2ϕ)!!

!"!!"

= γ!! -­‐α  !!

!"!"

= α!!+2ϕ!! Latent and Active TB with birth and death rates !"  !"

= -­‐βS!! + µ(S+!!+  !!+R) –  εS !"!!"

= βS!!-­‐  (γ+  ε)!!

!"!!"

= γ!! -­‐ (α+ω+  ε)  !!

!"!"

= α!! –  εR

!"  !"

= -­‐βS!!

!"!!"

= βS!!-­‐  γ!!

!"!!"

= γ!! -­‐α  !!

!"!"

= α!!  

Latent and Active TB with effects of Isoniazid !"  !"

= -­‐βS!!

!"!!"

= βS!!-­‐(γ+ϕ)!!

!"!!"

= γ!! -­‐α  !!

!"!"

= α!! + ϕ!!  

Latent and Active TB with birth, death, and relapse rates !"  !"

= -­‐βS!! + µ(S+!!+  !!+R)–  εS !"!!"

= βS!! - (γ+  ε)!!+ηR

!"!!"

= γ!! -­‐ (α+ω+  ε)  !!+ηR

!"!"

= α!! – (2η+  ε)R

 

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Latent and Active TB with birth, death, and relapse rates treated with Isoniazid !"  !"

= -­‐βS!! + µ(S+!!+  !!+R)  –  εS !"!!"

= βS!!-­‐ (γ+  ε +ϕ)!!+ηR

!"!!"

= γ!!  -­‐ (α+ω+  ε)  !!+ηR

!"!"

= α!! – (2η+  ε)R+ϕ!! Latent and Active TB with birth, death, and relapse rates treated with Combination Therapy !"  !"

= -­‐βS!!  + µ(S+!!+  !!+R)  –  εS !"!!"

= βS!!-­‐  (γ+ε+2ϕ)!!+ηR

!"!!"

= γ!! -­‐ (α+ω+  ε)  !!+ηR

!"!"

= α!! – (2η+  ε)R+2ϕ!!