Data collection

Data on varicella outbreak came from a public primary school in a central city in China reported by the Center for Disease Control and Prevention (CDC) of Changsha in the year 2016. The school contained six grades, eight classes, a total of 360 people. Students attend classes 5 days a week (from Monday to Friday), and boiled water was unified supplied by the school. Varicella cases were diagnosed clinically based on the history of infectious agent exposure and clinical features, such as pruritic macules, papules, blisters and respiratory infection symptoms.

Models

Model of varicella outbreak without interventions

An SEIR model [Reference Horn, Damm and Greiner11, Reference Hethcote, Zhien and Shengbing12] was established to simulate the transmission of varicella in school without any intervention. Population in this model was divided into four categories according to the disease status (Fig. 1a): susceptible (S), exposed (E), infected (I) and recovered (R). The model was developed based on the following facts or assumptions, which assumed that some individuals moved among categories because of infection or recovery: (1) The population was defined as closed and stable, because the varicella outbreak usually occurred in school in which birth, death and migration were negligible; (2) susceptible person was assumed to have an equal infected rate (β) with the disease; (3) after infection, the exposed person (E) would turn to infected individuals (I) after a certain exposed period (1/ω), the number of newly infected individuals per unit time was ωE; (4) γ represented the recovery rate, and 1/γ meant the infective period, the number of newly recovered individuals per unit time was γI; (5) the fatality rate of both varicella and asymptomatic infection was ignored. The corresponding model equations are as follows, dS/dt, dE/dt, dI/dt and dR/dt denoted the number of individuals (n) at time t in the corresponding categories:

(1)$$\left\{ \matrix{dS/dt = -\beta SI \hfill \cr dE/dt = \beta SI-\omega E \hfill \cr dI/dt = \omega E-\gamma I \hfill \cr dR/dt = \gamma I \hfill} \right.$$

Fig. 1. Flow chart of models of varicella outbreak in school.

Model of varicella outbreak with case isolation

Isolation would be adopted when the infected individuals (I) were identified. We assumed that the isolation rate was ψ, ψI individuals have been isolated and turned to quarantine (Q), which were not infectious. Population in the SEIQR model was divided into five categories (Fig. 1b), and the corresponding model equations are as follows:

(2)$$\left\{ \matrix{dS/dt = -\beta SI \hfill \cr dE/dt = \beta SI-\omega E \hfill \cr dI/dt = \omega E-\psi I-(1-\psi )\gamma I \hfill \cr dQ/dt = \psi I-\gamma Q \hfill \cr dR/dt = \gamma Q + (1-\psi )\gamma I \hfill} \right.$$

Model of varicella outbreak with vaccination

The SEIRV model was constructed by vaccinating susceptible people, who were separated from susceptible people and turned to vaccination (V) directly because of their immunity to varicella. Assuming that the vaccination coefficient was δ, the number of susceptible (S) individuals turned to exposed (E) at t time were (1–δ)SβI. Population in this model was divided into five categories (Fig. 1c), and the corresponding model equations are as follows, the epidemic trend under different vaccination coefficients would be simulated by adjusting δ.

(3)$$\left\{ \matrix{dS/dt = -\delta S-(1-\delta )\beta SI \hfill \cr dE/dt = (1-\delta )\beta SI-\omega E \hfill \cr dI/dt = \omega E-\gamma I \hfill \cr dR/dt = \gamma I \hfill \cr dV/dt = \delta S \hfill} \right.$$

Model of varicella outbreak with ventilation and disinfection

Ventilation and disinfection (spraying or wiping the classroom, corridor floor and public facilities with ‘84’ disinfectants or disinfecting the air through oxyacetic acid and ultraviolet lamp) were adopted to reduce the infected rate (β) of varicella virus [Reference Zhao13, Reference Liu14]. Assuming that m was the efficiency of ventilation and disinfection to eliminate the varicella virus, the number of individuals in some categories in this model would change. The model is shown in Fig. 1d and the corresponding equations are as follows:

(4)$$\left\{ \matrix{dS/dt = -\beta S(1-m)I \hfill \cr dE/dt = \beta S(1-m)I-\omega E \hfill \cr dI/dt = \omega E-\gamma I \hfill \cr dR/dt = \gamma I \hfill} \right.$$

Modelling combinations of the intervention strategies

For the varicella outbreak in a school, we simulated the combinations of intervention strategies (isolation, vaccine, ventilation and disinfection) to explore the influence on the control of varicella outbreak in school.

(5)$$\left\{ \matrix{dS/dt = -\delta S-(1-\delta )S(1-m)\beta I \hfill \cr dE/dt = (1-\delta )S(1-m)\beta I-\omega E \hfill \cr dI/dt = \omega E-\psi I-(1-\psi )\gamma I \hfill \cr dQ/dt = \psi I-\gamma Q \hfill \cr dR/dt = \gamma Q + (1-\psi )\gamma I \hfill \cr dV/dt = \delta S \hfill} \right.$$

Parameter estimation

There were six parameters in all models in this study, which were infected rate (β), latency coefficient (ω), removal rate (γ), isolation rate (ψ), vaccination coefficient (δ), disinfection and ventilation efficiency (m). The parameters ω, γ, ψ and δ could be obtained by combining the actual situation with literatures, while β could be obtained by fitting the actual data of model (1) and typical outbreak cases. Literature analysis showed that the exposure period of varicella is 10–21 days [Reference Marin15]; in this study, we assumed the exposure period (1/ω) as 21 days, so ω would equal to 1/21. Usually, the infected period of varicella was from 1 to 2 days before the eruption to scab formation; however, students would take medical measures once they have fever, rash or other symptoms; so, the infected period of varicella in school was usually shorter than the natural infection period. In this study, we assumed the isolation rate ψ was equal to 100%, the infection period (1/γ) was 3 days and the recovery rate γ was equal to 1/3 [Reference Suo, Lu and Chen16Reference Liu, Mu and Lv17]. Combining with the actual situation, in order to explore the influence of vaccination coefficient (δ) and the efficiency of ventilation and disinfection (m) on the control of epidemic situation of varicella outbreak in school, we have set δ as 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and m as 0.1, 0.3, 0.5, 0.7, 0.9, respectively. All parameters and initial values of each category are listed in Table 1.

Table 1. List of parameters and initial values of each category in models

Simulation methods

We fitted the data from the varicella outbreak in a school in a central city of China to an SEIR model curve to estimate β and then simulated the effects of isolation, vaccine, ventilation and disinfection in the outbreak. Berkeley Madonna 8.3.18 and Microsoft Office Excel 2010 software were employed for the model simulation and data management, respectively. Graphpad prism 5 was used for the figure development, while the curve fitting problem was solved by the Runge–Kutta fourth-order method, with a tolerance of 0.001. A goodness-of-fit test (χ2 test) was performed using the IBM-SPSS software, in which the significance level was α = 0.05.

Intervention assessment

The total attack rate (TAR), cumulative number of cases and duration of outbreak (DO, duration in days from index case to last case) were used to assess the effectiveness of control interventions in the varicella outbreak. Epidemic curves were used to evaluate the effectiveness using the strategies that are compared with the actual situation.