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

Dy Dynamics of of Dru Drug g Resistance: Op Optimal al c control o of 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 Drug-resistant strain 𝑠 𝑄 ) (𝑢) Susceptible to infection Infected 𝑄(𝑢) 𝛾 1 − 𝑄(𝑢) 𝑄 * (𝑢) 𝑠 Drug-susceptible strain Infected 𝑄 𝑢 = 𝑄 ) 𝑢 + 𝑄 * 𝑢

̇ 𝛾 SI SIS S Model Drug-resistant strain Susceptible 𝑠 𝑄 ) (𝑢) to infection 𝛾 𝑄 * (𝑢) 1 − 𝑄(𝑢) Drug-susceptible strain 𝑠 1. Drug resistance doesn’t affect System of equations: infectiousness (one 𝛾) 𝑄 . 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑠 𝑄 . 𝑢 , 𝑢 ≥ 0 2. Infection rate is constant (not influenced by disease prevalence) For an initial condition 𝑄 . 0 = 𝑞 . : 3. Drug resistance doesn’t affect 𝑞 . (𝛾 − 𝑠)𝑓 567 8 virulence (one 𝑠) 𝑄 . 𝑢 = 𝑞 . 𝛾 𝑓 567 8 − 1 + (𝛾 − 𝑠) 4. Disease is not self-limiting: 𝑠 < 𝛾

̇ ̇ ̇ ̇ 𝛾 SIS SI S Model Drug-resistant strain + Treatm + tment Susceptible 𝑠 𝑄 ) (𝑢) to infection 𝛾 𝑄 * (𝑢) 1 − 𝑄(𝑢) Drug-susceptible strain Treatment Policy: 𝑠 + 𝑿 W 𝑢 ∈ [0,1]: % of the infected population that will receive treatment 𝑄 * (𝑢) Efficacy of treatment normalized to 1. Drug “Quality”: 𝑅 𝑢 = 𝑄 ) 𝑢 + 𝑄 * 𝑢 System of equations: Re-write the system of equations: For 𝑢 ≥ 0 , For 𝑢 ≥ 0 , 𝑄 ) 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑠 𝑄 ) 𝑢 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 − 𝑠 𝑄 𝑢 𝑄 * 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑋 𝑢 − 𝑠 𝑄 * 𝑢 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1

̇ ̇ ̇ ̇ Problem formulat Pr ation • For an initial condition, (𝑞, 𝑟) , identify the treatment policy, 𝑋, that minimizes the total cost of the disease I 1.0 (𝑑 * 𝑋 𝑢 𝑄 𝑢 + 𝑑 K 𝑄(𝑢))𝑓 6L8 𝑒𝑢 D:ℝ F →[),*] H inf 0.9 0.8 Prevalence (p) ) 0.7 S.t. Discount rate 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 − 𝑠 𝑄 𝑢 , 𝑢 ≥ 0 0.6 Cost of Cost of 0.5 treatment illness 0.4 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1 , 𝑢 ≥ 0 0.3 s.t. 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 − 𝑠 𝑄 𝑢 , 0.2 0.1 0 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1 , 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 Quality (q) 𝑢 ≥ 0 (Prop. of infections that are susceptible)

̇ ̇ Problem formulat Pr ation • For an initial condition, (𝑞, 𝑟) , identify the treatment policy, 𝑋, that minimizes the total cost of the disease I 1.0 (𝑑 * 𝑋 𝑢 𝑄 𝑢 + 𝑑 K 𝑄(𝑢))𝑓 6L8 𝑒𝑢 D:ℝ F →[),*] H inf 0.9 0.8 Prevalence (p) ) 0.7 0.7 0.6 Cost of Cost of 0.6 0.5 treatment illness 0.5 0.4 0.4 0.3 0.3 s.t. 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 − 𝑠 𝑄 𝑢 , 0.2 0.2 0.1 0.1 0 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1 , 0 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 Quality (q) 𝑢 ≥ 0 (Prop. of infections that are susceptible)

Suffici Su cient optimality condition (HJ HJB equation) Proposition 1 (HJB Equation) 0.7 Let 𝑤 ∗ : 0,1 ×[0,1] → ℝ be a bounded function that 0.6 Prevalence (p) is almost everywhere differentiable on any 0.5 trajectory (𝑄 𝑢; 𝑞, 𝑟, 𝑋 , 𝑅(𝑢; 𝑟, 𝑋)) , 𝑢 ≥ 0 , and that 0.4 satisfies the Hamilton-Jacobi-Bellman equation: 0.3 0.2 O∈[),*] {𝑑 * 𝑥𝑞 + 𝑑 K 𝑞 0 = min 0.1 0 +𝜖 * 𝑤 ∗ 𝑞, 𝑟 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 𝛾 1 − 𝑞 − 𝑥𝑟 − 𝑠 𝑞 Quality (q) (Prop. of infections that are susceptible) +𝜖 K 𝑤 ∗ 𝑞, 𝑟 𝑥𝑟 𝑟 − 1 − 𝜍𝑤 ∗ (𝑞, 𝑟)}

Suffici Su cient optimality condition (HJ HJB equation) Proposition 1 (HJB Equation) 0.7 0.7 0.7 Let 𝑤 ∗ : 0,1 ×[0,1] → ℝ be a bounded function that 0.6 0.6 0.6 Prevalence (p) is almost everywhere differentiable on any 0.5 0.5 0.5 trajectory (𝑄 𝑢; 𝑞, 𝑟, 𝑋 , 𝑅(𝑢; 𝑟, 𝑋)) , 𝑢 ≥ 0 , and that 0.4 0.4 0.4 satisfies the Hamilton-Jacobi-Bellman equation: 0.3 0.3 0.3 0.2 0.2 0.2 O∈[),*] {𝑑 * 𝑥𝑞 + 𝑑 K 𝑞 0 = min 0.1 0.1 0.1 0 0 0 +𝜖 * 𝑤 ∗ 𝑞, 𝑟 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.7 0.8 0.8 0.9 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 𝛾 1 − 𝑞 − 𝑥𝑟 − 𝑠 𝑞 Quality (q) (Prop. of infections that are susceptible) +𝜖 K 𝑤 ∗ 𝑞, 𝑟 𝑥𝑟 𝑟 − 1 − 𝜍𝑤 ∗ (𝑞, 𝑟)}

Op Optimal al po policy Prevalence (p) • Bang-bang with a single switching time • Treat everyone until it is not economical to treat anyone Quality (q) (Prop. of infections that are susceptible) • Withholding treatment to preserve the drug’s effectiveness is not optimal

0.7 0.7 Pr Properties of the value 0.6 0.5 0.4 0.3 0.2 0.1 0.6 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Prevalence (p) fu function 0.5 0.4 0.3 0.2 • When 𝜍 ≤ 𝛾 − 𝑠 0.1 Low discount rate 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 à Long-term focused planner 0.7 0.7 0.6 0.5 Value function is not Lipschitz continuous 0.4 0.6 0.3 Prevalence (p) 0.2 0.1 à Traditional numerical methods may 0 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 not be computationally stable 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Quality (q) (Prop. of infections that are susceptible)

̇ ̇ Non Non-co constant disease transmission rat ate NEW System of equations: For 𝑢 ≥ 0 , • Protective behaviours in 𝑄 𝑢 = 𝛾 1 − 𝑄 𝑢 − 𝑅 𝑢 𝑋 𝑢 + 𝑠 𝑄 𝑢 response to high prevalence 𝑄 𝑢 𝑅 𝑢 = 𝑋(𝑢)𝑅(𝑢) 𝑅 𝑢 − 1 B(p ) Consider Drug-resistant strain 𝐶 𝑞 = 𝛾𝑞 6*.\ Susceptible 𝑠 𝑄 ) (𝑢) to infection 𝑄 * (𝑢) B(p ) 1 − 𝑄(𝑢) Drug-susceptible strain 𝑠 + 𝑿

Op Optimal al po policy with h a a no non-co constant disease tr transmission r rate 0.7 0.6 • Bang-bang with a singular arc 0.5 0.4 0.3 • Treat everyone until the boundary, 0.2 0.1 then treat a fraction of the population 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (stay on the boundary)

Op Optimal al po policy with h a a no non-co constant disease tr transmission r rate 0.7 0.6 • Bang-bang with a singular arc 0.5 0.4 0.3 • Treat no one until it is economical to 0.2 treat a fraction of the population 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 • Withholding treatment to preserve the drug’s effectiveness may be optimal (e.g., strategic stockpile of antivirals)

Ex Extensively r resistant g t gonorrhea • Prevalence: 0.27% (general pop’n) • Resistance to ceftriaxone & Prevalence (%) azithromycin: 0.3% 12 years 9.5 years 30 years 6-25 years 5-20 years • Current cost: $1.50/person 14-62 years • Expected value of optimal policy: $28.19/person Quality (%) (% of infections that are susceptible)