A mathematical model about human infections of H7N9 influenza in China with the intervention of live poultry markets closing

This paper develops a deterministic differential equations model that captures the H7N9 virus transmission from live poultry to human via poultryhuman contacts in live poultry markets (LPMs). The virus circulation among live poultry, which happens but is hard to be detected (since contaminated live poultry appear to be asymptomatic), is also incorporated in the model. The time-dependent contact rate between human and live poultry based on LPMs closing information can be estimated. From data of LPMs closing news, the contact rate function can be easily estimated. This model could serve as a rational basis for public health authorities to evaluate the effectiveness of LPM closing, as well as other interventions according to simple modifications. Without data about daily cases, I also provide suggestions for some of the basic parameters that would be a useful fitness parameter set for future simulation. 1 Background and Methods The dynamic of the H7N9 virus transmission is similar to that of a nosocomial bacteria infection [5], from which was the modeling idea originated. Figure 1 shows the dynamics of H7N9 transmission from live poultry to human and among live poultry. We divide human population into three classes: susceptible people who frequently go to LPMs, H7N9 infected people, and people identified of H7N9 infection. Since the total number of live poultry is hard to be estimated, we use fraction to measure contaminated and uncontaminated live poultry. For simplicity, we assume homogeneous mixing among live poultry, as well as homogeneous contact between susceptible people and live poultry in the LPMs. Based on the observation of H7N9 population outbreak, the circumstance of constant poultry-poultry transmission rate can be ruled out. The ordinary differential equations system of the model (2) is presented in Appendix. With a constant poultry-poultry transmission rate α, the basic reproduction number of the ∗Department of Mathematics, Ryerson University, Toronto, ON M5B 2K3, Canada †Center for Disease Modelling, York University, Toronto, ON M3J 1P3, Canada 1 ar X iv :1 40 9. 55 26 v1 [ qbi o. PE ] 1 9 Se p 20 14 S (t) susceptible population β (t)C1 (t) I (t) infected population μ R (t) identified population C0 (t) uncontaminated poultry α C1 (t) contaminated poultry ν Figure 1: Compartmental model of transmission dynamics. S(t) is the number of susceptible individuals at time t, I(t) is the number of infected people at time t, R(t) is the number of people who become symptomatic at time t. C0(t) is the fraction of uncontaminated live poultry at time t, C1(t) is the fraction of contaminated live poultry at time t. Hence, C0(t) +C1(t) = 1. β (t) is the transmission rate of H7N9 virus from live poultry to human, which is a product of poultry-human contact rate and the possibility for a transmission to occur during contact. live poultry system is <0 = αν . Depending on initial conditions, one of the following dynamics must have happened: (1) if <0 ≤ 1, the fraction of contaminated live poultry decreases to zero and stabilize after sufficiently long time; if <0 > 1, the fraction (2) either decreases to a non-zero value and stabilize after sufficiently long time, or (3) increases to a non-zero value and stabilize after sufficiently long time. With a decreasing poultry-human transmission rate function [1] estimated for the outbreak in spring, 2013, the above three possibilities would lead to either of the following happens: (1) and (2) would both give a decreasing number of daily symptomatic people through out the outbreak, which does not match the outbreak result reported by WHO [9]; (3) would possibly match the outbreak [9] until the resume of LPMs, but could not explain the fact of no new cases appeared after the resume of LPMs in late May, 2013. Therefore, we assume that the poultry-poultry contamination rate, α, to be a time-dependent parameter, then the reduced system follows: dS (t) dt = −β (t)C1 (t)S (t) dI (t) dt = β (t)C1 (t)S (t)− μI (t) dC1 (t) dt = C1 (t) [(α (t)− ν)− α (t)C1 (t)] (1) With the time-dependent contamination rate α (t), if α(t) ν < 1 after a certain time, the epidemic will finally stop; otherwise, if α(t) ν > 1 after a certain time, new cases would consistently appear. α (t) would fluctuate due to impacts such as seasonality and some unknown aspects, hence in the simulation, people should focus more on the system behavior in relatively short terms that are no longer than 6 months.