Jun, 2002 MTBI Cornell University TB Cluster Models, Time Scales and Relations to HIV Carlos Castillo-Chavez Department of Biological Statistics and Computational Biology Department of Theoretical and Applied Mechanics Cornell University, Ithaca, New York, 14853

Jun, 2002 MTBI Cornell University Outline A non-autonomous model that incorporates the impact of HIV on TB dynamics. Model to test CDC’s TB control goals. Casual versus close contacts and their impact on TB. Time scales and singular perturbation approaches in the study of the dynamics of TB.

Jun, 2002 MTBI Cornell University TB in the US ( )

Jun, 2002 MTBI Cornell University Reemergence of TB New York City and San Francisco had recent outbreaks. Cost of control the outbreak in NYC alone was estimated to be about 1 billion. Observed national TB case rate increase. TB reemergence became an international issue. CDC sets control goal in 1989.

Jun, 2002 MTBI Cornell University Basic Model Framework N=S+E+I+T, Total population F(N): Birth and immigration rate B(N,S,I): Transmission rate (incidence) B`(N,S,I): Transmission rate (incidence)

Jun, 2002 MTBI Cornell University Model Equations

Jun, 2002 MTBI Cornell University CDC Short-Term Goal: 3.5 cases per 100,000 by Has CDC met this goal? CDC Long-term Goal: One case per million by Is it feasible? TB control in the U.S.

Jun, 2002 MTBI Cornell University Model Construction Since d has been approximately equal to zero over the past 50 years in the US, we only consider Hence, N can be computed independently of TB.

Jun, 2002 MTBI Cornell University Non-autonomous model ( permanent latent class of TB introduced)

Jun, 2002 MTBI Cornell University Effect of HIV

Jun, 2002 MTBI Cornell University Upper Bound and Lower Bound For Epidemic Threshold If R  <1, L 1 (t), L 2 (t) and I(t) approach zero; If R  >1, L 1 (t), L 2 (t) and I(t) all have lower positive boundary; If  (t) and d(t) are time-independent, R  and R  are Equal to R 0.

Jun, 2002 MTBI Cornell University Parameter estimation and simulation setup ParameterEstimation  0.22 c10 k0.001 r1r r2r2 r3r p0.1 InitialValues I(0)87423 L 1 (0)  10 6 L 2 (0)  10 6

Jun, 2002 MTBI Cornell University N(t) is from census data and population projection Parameter estimation and simulation setup

Jun, 2002 MTBI Cornell University RESULTS

Jun, 2002 MTBI Cornell University CONCLUSIONS

Jun, 2002 MTBI Cornell University CONCLUSIONS

Jun, 2002 MTBI Cornell University CDC’s Goal Delayed Impact of HIV. Lower curve does not include HIV impact; Upper curve represents the case rate when HIV is included; Both are the same before Dots represent real data.

Jun, 2002 MTBI Cornell University Regression approach A Markov chain model supports the same result

Jun, 2002 MTBI Cornell University Cluster Models Incorporates contact type (close vs. casual) and focus on the impact of close and prolonged contacts. Generalized households become the basic epidemiological unit rather than individuals. Use natural epidemiological time-scales in model development and analysis.

Jun, 2002 MTBI Cornell University Close and Casual contacts Close and prolonged contacts are likely to be responsible for the generation of most new cases of secondary TB infections. “A high school teacher who also worked at library infected the students in her classroom but not those who came to the library.” Casual contacts also lead to new cases of active TB. WHO gave a warning in 1999 regarding air travel. It announced that flights of more than 8 hours pose a risk for TB transmission.

Jun, 2002 MTBI Cornell University Transmission Diagram

Jun, 2002 MTBI Cornell University Basic epidemiological unit: cluster (generalized household) Movement of kE 2 to I class brings nkE 2 to N 1 population, where by assumptions nkE 2 (S 2 /N 2 ) go to S 1 and nkE 2 (E 2 /N 2 ) go to E 1 Conversely, recovery of  I infectious bring n  I back to N 2 population, where n  I (S 1 /N 1 )=  S 1 go to S 2 and n  I (E 1 /N 1 )=  E 1 go to E 2 Key Features

Jun, 2002 MTBI Cornell University Basic Cluster Model

Jun, 2002 MTBI Cornell University Basic Reproductive Number where is the expected number of infections produced by one infectious individual within his/her cluster. denotes the fraction who survives the latency period and become active cases.

Jun, 2002 MTBI Cornell University Diagram of Extended Cluster Model

Jun, 2002 MTBI Cornell University  (n) Both close casual contacts are included in the extended model. The risk of infection per susceptible, , is assumed to be a nonlinear function of the average cluster size n. The constant p measures the average proportion of the time that an “individual spends in a cluster.

Jun, 2002 MTBI Cornell University R 0 Depends on n in a non-linear fashion

Jun, 2002 MTBI Cornell University Role of Cluster Size E(n) denotes the ratio of within cluster to between cluster transmission. E(n) increases and reaches its maximum value at The cluster size n * is defined as optimal as it maximizes the relative impact of within to between cluster transmission.

Jun, 2002 MTBI Cornell University Hoppensteadt’s Theorem (1973) Full system Reduced system where x  R m, y  R n and  is a positive real parameter near zero (small parameter). Five conditions must be satisfied (not listed here) to apply the theorem. In addition, it is shown that if the reduced system has a globally asymptotically stable equilibrium then the full system has a g.a.s. equilibrium whenever 0<  <<1.

Jun, 2002 MTBI Cornell University Two time Scales Latent period is long and roughly has the same order of magnitude as that associated with the life span of the host. Average infectious period is about six months (wherever there is treatment, is even shorter).

Jun, 2002 MTBI Cornell University Rescaling Time is measured in average infectious periods (fast time scale), that is,  = k t. The state variables are rescaled as follow: Where    /  is the asymptotic carrying capacity.

Jun, 2002 MTBI Cornell University Rescaled Model

Jun, 2002 MTBI Cornell University Rescaled Model

Jun, 2002 MTBI Cornell University Dynamics on Slow Manifold Solving for the quasi-steady states y 1, y 2 and y 3 in terms of x 1 and x 2 gives Substituting these expressions into the equations for x 1 and x 2 lead to the equations of motion on the slow manifold.

Jun, 2002 MTBI Cornell University Slow Manifold Dynamics Where is the number of secondary infections produced by one infectious individual in a population where everyone is susceptible

Jun, 2002 MTBI Cornell University Theorem If R c 0  1,the disease-free equilibrium (1,0) is globally asymptotically stable. While if R c 0 > 1, (1,0) is unstable and the endemic equilibrium is globally asymptotically stable. This theorem characterizes the dynamics on the slow manifold

Jun, 2002 MTBI Cornell University Dynamics for Full System Theorem : For the full system, disease-free equilibrium is globally asymptotically stable whenever R 0 c 1 there exists a unique endemic equilibrium which is globally asymptotically stable. Proof approach: Construct Lyapunov function for the case R 0 c 1, we use Hoppensteadt’s Theorem. A similar result can be found in Z. Feng’s 1994, Ph.D. dissertation.

Jun, 2002 MTBI Cornell University Bifurcation Diagram Global bifurcation diagram when 0<  <<1 where  denotes the ratio between rate of progression to active TB and the average life-span of the host (approximately). 1

Jun, 2002 MTBI Cornell University Numerical Simulations

Jun, 2002 MTBI Cornell University Conclusions from cluster models TB has slow dynamics but the change of epidemiological units makes it possible to identify non-traditional fast and slow dynamics. Quasi steady assumptions (adiabatic elimination of parameter) are valid (Hoppensteadt’s theorem). The impact of close and casual contacts can be study using this approach as long as progression rates from the latently to the actively-infected stages are significantly different.

Jun, 2002 MTBI Cornell University Conclusions from cluster models Singular perturbation theory can be used to study the global asymptotic dynamics. Optimal cluster size highlights the relative impact of close versus casual contacts and suggests alternative mechanisms of control. The analysis of the system for the case where the small parameter  is not small has not been carried out. Simulations suggest a wider range.