This paper studies a billiards-like microscopic heat conduction model, which describes the dynamics of gas molecules in a long tube with thermalized boundary. We numerically investigate the law of energy exchange between adjacent cells. A stochastic energy exchange model that preserves these properties is then derived. We further numerically justified that the stochastic energy exchange model preserves the ergodicity and the thermal conductivity of the original billiard model.

We propose a mathematical model to evaluate the effect of China's "Four-Free-One-Care Policy" in MSM population in Beijing. We divided the drug resistant HIV patients into two sub-populations: primary drug resistance and secondary drug resistance. Uncertainty and sensitivity analysis based on Latin Hypercube Sampling (LHS) were used for these thresholds of our model. We find that drug-resistant HIV will spread fast in MSM population under China's current treatment policy. Especially, primary-resistant strain is very likely to dominate the HIV positive MSM individuals after 10 years. The conclusions hint that, China's outlook on HIV infections is not optimistic if sufficient kinds free second-line drugs in China cannot be put into use in the near future.

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