Dy Dynamics of of Dru Drug g Resistance: Op Optimal al c - - PowerPoint PPT Presentation

Dy Dynamics of

• f Dru

Drug g Resistance: Op Optimal al c control o

• f an

f an i infectious d diseas ase

Naveed Chehrazi Lauren E. Cipriano Eva A. Enns

Fundi Funding ng & & D Disclosur ure

• Natural Sciences and Engineering Research Council of Canada (PI: Cipriano)
• National Institute for Allergy and Infectious Diseases at the National Institutes of

Health [Grant K25AI118476 (PI: Enns)].

• The funding agencies had no influence on the design and conduct of the study;

collection, management, analysis, and interpretation of the data; or in the preparation or review of the manuscript.

• The content is solely the responsibility of the authors and does not necessarily

represent the official views of the National Institutes of Health.

An Antimicrobial resistance

• Treatment resistant bacteria, parasites, viruses, and fungi
• De novo resistant genes
• Genes that confer resistance transferred between species and strains
• Previously easy-to-treat infections are now difficult, intensive, and expensive

to treat

• Significant threat to public health
• Antibiotic resistance in the US: 2 million infections and 23,000 deaths annually
• In many cases, few effective treatment options remain

Re Research questions

• What treatment policy minimizes the cost of disease to society in the

presence of resistance?

• Should the last remaining effective treatment be withheld to preserve

the drug for a potentially more serious future outbreak?

• Restricting access for general medical use
• Stockpiling

In Infectio ious s dise isease se models ls wit ith resis sistance

• Majority of literature uses detailed disease models and numerical

methods to evaluate and compare controls

• i.e., vaccination vs. quarantine; prevention vs. treatment
• Few generalizable insights and sometimes contradictory results
• Models of pandemic influenza with resistance have found prophylaxis,

a mix of prophylaxis and treatment, and no prophylaxis to be optimal

Mo Modeling approach

• Focus on SIS-type infectious diseases e.g., gonorrhea, H. pylori, TB
• Assume that there is one remaining effective treatment

𝛾 𝑠 Susceptible to infection 1 − 𝑄(𝑢) 𝑄)(𝑢) Drug-resistant strain Drug-susceptible strain 𝑄

*(𝑢)

𝛾 𝑠 Infected 𝑄(𝑢) Infected 𝑄 𝑢 = 𝑄) 𝑢 + 𝑄

* 𝑢

SI SIS S Model

̇ 𝑄. 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑠 𝑄. 𝑢 , 𝑢 ≥ 0 𝛾 𝑠 Susceptible to infection 1 − 𝑄(𝑢) 𝑄)(𝑢) 𝛾 𝑠 Drug-resistant strain Drug-susceptible strain 𝑄

*(𝑢)

• 1. Drug resistance doesn’t affect

infectiousness (one 𝛾)

• 2. Infection rate is constant

(not influenced by disease prevalence)

• 3. Drug resistance doesn’t affect

virulence (one 𝑠)

• 4. Disease is not self-limiting: 𝑠 < 𝛾

𝑄. 𝑢 = 𝑞.(𝛾 − 𝑠)𝑓 567 8 𝑞.𝛾 𝑓 567 8 − 1 + (𝛾 − 𝑠)

For an initial condition 𝑄. 0 = 𝑞.: System of equations:

SI SIS S Model + + Treatm tment

Treatment Policy: W 𝑢 ∈ [0,1]: % of the infected population that will receive treatment Efficacy of treatment normalized to 1. System of equations: ̇ 𝑄) 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑠 𝑄) 𝑢 ̇ 𝑄

* 𝑢 = 𝛾 1 − 𝑄 𝑢

− 𝑋 𝑢 − 𝑠 𝑄

* 𝑢

Drug “Quality”:

For 𝑢 ≥ 0, 𝑅 𝑢 = 𝑄

*(𝑢)

𝑄) 𝑢 + 𝑄

* 𝑢

Re-write the system of equations:

For 𝑢 ≥ 0,

̇ 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 − 𝑠 𝑄 𝑢 ̇ 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1 𝛾 𝑠 Susceptible to infection 1 − 𝑄(𝑢) 𝑄)(𝑢) 𝛾

𝑠 + 𝑿

Drug-resistant strain Drug-susceptible strain 𝑄

*(𝑢)

Pr Problem formulat ation

• For an initial condition, (𝑞, 𝑟), identify the treatment policy, 𝑋, that

minimizes the total cost of the disease inf

D:ℝF→[),*] H ) I

(𝑑*𝑋 𝑢 𝑄 𝑢 + 𝑑K𝑄(𝑢))𝑓6L8𝑒𝑢

̇ 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 − 𝑠 𝑄 𝑢 , 𝑢 ≥ 0 ̇ 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1 , 𝑢 ≥ 0 S.t.

Cost of treatment Cost of illness Discount rate

̇ 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 − 𝑠 𝑄 𝑢 , ̇ 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1 , s.t.

0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 1.0 0.9 0.4 0.2 0.1 0.3 0.5 0.6 0.8 0.9 0.7 1.0

Prevalence (p) Quality (q)

(Prop. of infections that are susceptible)

𝑢 ≥ 0

0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 1.0 0.9 0.4 0.2 0.1 0.3 0.5 0.6 0.8 0.9 0.7 1.0

Pr Problem formulat ation

• For an initial condition, (𝑞, 𝑟), identify the treatment policy, 𝑋, that

minimizes the total cost of the disease inf

D:ℝF→[),*] H ) I

(𝑑*𝑋 𝑢 𝑄 𝑢 + 𝑑K𝑄(𝑢))𝑓6L8𝑒𝑢

Cost of treatment Cost of illness

̇ 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 − 𝑠 𝑄 𝑢 , ̇ 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1 , s.t.

0.4 0.2 0.1 0.3 0.5 0.6 0.7

Prevalence (p) Quality (q)

(Prop. of infections that are susceptible)

𝑢 ≥ 0

Su Suffici cient optimality condition (HJ HJB equation)

0 = min

O∈[),*]{𝑑*𝑥𝑞 + 𝑑K𝑞

+𝜖K𝑤∗ 𝑞, 𝑟 𝑥𝑟 𝑟 − 1 − 𝜍𝑤∗(𝑞, 𝑟)} +𝜖*𝑤∗ 𝑞, 𝑟 𝛾 1 − 𝑞 − 𝑥𝑟 − 𝑠 𝑞

0.4 0.2 0.1 0.3 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 0.9 0.6 0.7

Proposition 1 (HJB Equation)

Let 𝑤∗: 0,1 ×[0,1] → ℝ be a bounded function that is almost everywhere differentiable on any trajectory (𝑄 𝑢; 𝑞, 𝑟, 𝑋 , 𝑅(𝑢; 𝑟, 𝑋)), 𝑢 ≥ 0, and that satisfies the Hamilton-Jacobi-Bellman equation:

Prevalence (p) Quality (q)

(Prop. of infections that are susceptible)

0.4 0.2 0.1 0.3 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 0.9 0.6 0.7 0.4 0.2 0.1 0.3 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 0.9 0.6 0.7 0.4 0.2 0.1 0.3 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 0.9 0.6 0.7

Su Suffici cient optimality condition (HJ HJB equation)

0 = min

O∈[),*]{𝑑*𝑥𝑞 + 𝑑K𝑞

+𝜖K𝑤∗ 𝑞, 𝑟 𝑥𝑟 𝑟 − 1 − 𝜍𝑤∗(𝑞, 𝑟)} +𝜖*𝑤∗ 𝑞, 𝑟 𝛾 1 − 𝑞 − 𝑥𝑟 − 𝑠 𝑞 Proposition 1 (HJB Equation)

Let 𝑤∗: 0,1 ×[0,1] → ℝ be a bounded function that is almost everywhere differentiable on any trajectory (𝑄 𝑢; 𝑞, 𝑟, 𝑋 , 𝑅(𝑢; 𝑟, 𝑋)), 𝑢 ≥ 0, and that satisfies the Hamilton-Jacobi-Bellman equation:

Prevalence (p) Quality (q)

(Prop. of infections that are susceptible)

Op Optimal al po policy

• Bang-bang with a single switching time
• Treat everyone until it is not economical

to treat anyone

• Withholding treatment to preserve the

drug’s effectiveness is not optimal

Prevalence (p) Quality (q)

(Prop. of infections that are susceptible)

Pr Properties of the value fu function

• When

Low discount rate à Long-term focused planner Value function is not Lipschitz continuous à Traditional numerical methods may not be computationally stable

0.4 0.2 0.1 0.3 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 0.9 0.6 0.7

0.4 0.2 0.1 0.3 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 0.9 0.6 0.7

𝜍 ≤ 𝛾 − 𝑠

Prevalence (p) Prevalence (p) Quality (q)

(Prop. of infections that are susceptible)

Non Non-co constant disease transmission rat ate

• Protective behaviours in

response to high prevalence Consider

NEW System of equations: For 𝑢 ≥ 0, ̇ 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 + 𝑠 𝑄 𝑢 ̇ 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1

𝐶 𝑞 = 𝛾𝑞6*.\ B(p)

𝑠 Susceptible to infection 1 − 𝑄(𝑢) 𝑄)(𝑢)

B(p) 𝑠 + 𝑿

Drug-resistant strain Drug-susceptible strain 𝑄

*(𝑢)

Op Optimal al po policy with h a a no non-co constant disease tr transmission r rate

• Bang-bang with a singular arc
• Treat everyone until the boundary,

then treat a fraction of the population (stay on the boundary)

0.4 0.2 0.1 0.3 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 0.9 0.6 0.7

Op Optimal al po policy with h a a no non-co constant disease tr transmission r rate

• Bang-bang with a singular arc
• Treat no one until it is economical to

treat a fraction of the population

• Withholding treatment to preserve the

drug’s effectiveness may be optimal (e.g., strategic stockpile of antivirals)

0.4 0.2 0.1 0.3 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.7 0.9 0.6 0.7

Ex Extensively r resistant g t gonorrhea

• Prevalence: 0.27% (general pop’n)
• Resistance to ceftriaxone &

azithromycin: 0.3%

• Current cost: $1.50/person
• Expected value of optimal policy:

$28.19/person

Prevalence (%) Quality (%)

(% of infections that are susceptible)

9.5 years 5-20 years 12 years 6-25 years 30 years 14-62 years

Con Contribution

• n and Key findings
• SIS infectious disease with constant disease transmission rate
• Closed form solution to optimal control problem with two continuous state variables
• Analysts need to check value function properties before applying traditional numerical

methods

• Optimal policy: Bang-bang with a single switching time

àWithholding treatment to preserve the drug’s effectiveness is not optimal

• Applied model to problem of extensively resistant gonorrhea to identify value of research
• SIS infectious disease with non-constant disease transmission rate
• Dramatically changes the form of the optimal policy

à Withholding treatment to preserve the drug’s effectiveness may be optimal

• Approximating with a constant rate may indicate a suboptimal policy

Li Limitations & & Future Research ch

• SIS model
• Expand to consider R and V state (natural or vaccine immunity)
• Limited to the last drug
• Expand to consider the optimal use of a set of n drugs
• Do not consider the effect of drug misuse on resistance pressure
• i.e., antibiotic use for viral infections
• Do not consider innovation
• How much should be invested in new antimicrobial research?
• How to balance investment in stewardship vs. research?

Th Thank k you!

Lauren: LCipriano@ivey.uwo.ca Naveed: Chehrazi@utexas.edu Eva: EEnns@umn.edu