key: cord-324518-a346cjx4 authors: Zhang, Zhibin; Sheng, Chengfa; Ma, Zufei; Li, Dianmo title: The outbreak pattern of the SARS cases in Asia date: 2004 journal: Chin Sci Bull DOI: 10.1007/bf03183407 sha: doc_id: 324518 cord_uid: a346cjx4 The severe acute respiratory syndrome (SARS) caused tremendous damage to many Asia countries, especially China. The transmission process and outbreak pattern of SARS is still not well understood. This study aims to find a simple model to describe the outbreak pattern of SARS cases by using SARS case data commonly released by governments. The outbreak pattern of cumulative SARS cases is expected to be a logistic type because the infection will be slowed down due to the increasing control effort by people and/or due to depletion of susceptible individuals. The increase rate of SARS cases is expected to decrease with the cumulative SARS cases, as described by the traditional logistical model, which is widely used in population dynamic studies. The instantaneous rate of increases were significantly and negatively correlated with the cumulative SARS cases in mainland of China (including Beijing, Hebei, Tianjin, Shanxi, the Autonomous Region of Inner Mongolia) and Singapore. The basic reproduction numberR (0) in Asia ranged from 2.0 to 5.6 (except for Taiwan, China). TheR (0) of Hebei and Tianjin were much higher than that of Singapore, Hongkong, Beijing, Shanxi, Inner Mongolia, indicating SARS virus might have originated differently or new mutations occurred during transmission. We demonstrated that the outbreaks of SARS in many regions of Asia were well described by the logistic model, and the control measures implemented by governments are effective. The maximum instantaneous rate of increase, basic reproductive number, and maximum cumulative SARS cases were also calculated by using the logistic model. ported the transmission epidemic of SARS in Singapore and Hong Kong. Both teams made use of mathematical models based on a system of four subpopulations: susceptible, exposed, infectious, and recovered (immune) individual, also called a SEIR model [3 J . They estimated that the basic reproduction number R o at the early stage without control is of order 2 to 4. The basic reproductive number is a good indicator of the severity of epidemic diseases and effectiveness of contrd 4 -6J . If Ro 1 implies that the disease will persist[6 J . Estimation of R o from disease outbreak data is not an easy job because the actual process of infection is not observable, data are often incomplete and the rate of infection is often nonlinear [7-13 J . In most situations, detailed epidemical data are lacking. It is time consuming and labor consuming to trace the secondary infection cases from a single SARS case. It is necessary to find an alternative model to describe the outbreak pattern of SARS cases by using simple data of number of SARS cases commonly released by governments. In population ecology, the continuous exponential model of population growth is described by where N is the population size, and r is the instantaneous rate of increase. The discrete model is written as where Nt is the population size at time t, A is the finite rate of increase, and A=e r • Because resources are limited, the instantaneous rate of increase usually decreases with the increase in population size, which is called the effect of density dependency. The logistic model has been widely used to describe the population growth under limited resources. The continuous logistic model is written as: where rm is the maximum r, K is the carrying capacity of a population in specific environment. With eq. (1), we can dN prove that when N=K/2, will reach the maximum. dt According to eq. (2), the maximum of cumulative SARS cases (K H ) can be roughly estimated if the outbreak pattern follows the logistic model. Fig. 1 shows how the popula- , h ' tlon sIze an mcrease rate --;jt vary WIt time t, and how r varies with N, as described by eq. (1). When . . Cumulative SARS cases mu1ative SARS cases in a region correspond to the carrying capacity (K) in the logistic model. The "density dependency effect" in this study refers to the fact that the numbers of new SARS cases become smaller over time as the result of the efforts of both control and natural immunization in the whole population. In this paper, the outbreak patterns of cumulative cases of SARS in the regions of Hong Kong, Taiwan and the Mainland of China, and Singapore were investigated by using the logistic model. The original data of the cumulative cases of SARS are collected from the related governmental websites (e.g. http://www.who .inticsr/don/archiveldisease/severe_acute_ respiratory_syndrome/en/; http://www.moh.gov.cn/). .+ iJ. The discrete logistic model of eq. (1) is written as: K From eq. (3), r m and K can be estimated. The equation is also widely used to testify if the population growth is density-dependent or not, depending on whether 1n(Nt+/Nt) is negatively correlated to NP4 J • In fact, except for population growth, many other biological processes, e.g. the growth of body size of organisms, can be well described by using the logistic model. The outbreak pattern of cumulative SARS cases is likely of a logistic type because at the initial stage, it grows exponentially, later due to the increasing control effort by people and/or due to depletion of susceptible individuals, the infection will be slowed down. The increase rate of SARS cases is expected to decrease with the cumulative SARS cases, which corresponds to the density-dependent effect in the logistic model. The total cu- dN reach to the peak at time t p , K can be estimated from dt the N at time t p , i.e. K=2N (Fig. l(a) ). There is a negative linear relationship between rand N ( Fig. 1(b) ). significant and negative linear "density dependency" of the instantaneous rate of increase on the cumulative cases of SARS indicates that the outbreak pattern of SARS can be well described by the logistic model( Fig. 1(a) and (b) ). In Table 1 , according to eq. (3), b o is the estimation of r m and K L =rm/bj, K L is the estimation of the maximum SARS cases K ( Table 2 ). According to eq. (2), K H is also an estimation of K ( Table 2 ). The accuracies of estimation of K by using both methods are very high, further sup porting the conclusion that the growth pattern of SARS cases is generally a logistic curve. The variation of r is larger at the early stage with smaller cumulative SARS cases, probably due to smaller number of cumulative cases. Mathematical models have been widely used to calculate and describe the dynamic evolution of epidemic threshold values and severity [15, [3] . These models are relatively complex, and we often need to consider the susceptible people, exposed people, removed people, etc. when establishing such kind of models. Data collection for estimating model parameters is not only time consuming but also labor consuming. In most situations, it is hard to estimate the basic reproduction number (R o ), and it is often impossible to predict the maximum cumulative cases in the early stage of disease outbreak, which limited its application. We do not consider the different populations when establishing a logistic model; only the number of infectious patients is sufficent. The logistic model is also tolerable to the variation of survey interval. Though simple, the logistic model enables us to easily estimate the maximum relative increase rate (rm) and maximum cumulative SARS cases (K). From rm, R o can be estimated if only the infection period from a single case to the secondary case is given. According to the epidemic studies on SARS in Hong Kong, China and Singapore[I,21, the mean serial interval, defined as the sum of incubation period and the duration of infectiveness of SARS person, is estimated as 8-12 d with an average of 8.4±3.8 din Singapore, and R o is estimated to be from 2.2 to 3.6. Since the incubation period is about 5 or 6.4 d[11, the duration of infectiveness (D 1 ) is about 3 d. Hospitalized SARS persons would also infect front-line doctor staff in hospital, and the average duration of SARS person in hospital (D 2 ) is about 14 d [26] . As shown in Table 1 rate of increase against the cumulative SARS cases in China and in Singapore. This is also a good indication that the control measures, mostly strict isolation of SARS persons and persons with experience of close contact to The maximum cumulative SARS cases in many provinces or cities in the mainland of China were fitted very well byusing eq. (2) based on the peak values of new SARS cases or by using eq. (3) ( Table 2 ). The maximum cumulative SARS cases in Beijing were estimated to be 2398 and 2547 respectively by using the two methods, very close to the observed value 2520; the maximum cumulative SARS cases in whole mainland of China were estimated to be 6212 and 6020 respectively, very close to the observed cases 5328 ( Table 2 ). The prediction accuracy was very high. Therefore, the model could play an important role in prediction of outbreak of SARS accumulative cases. However, prediction using both ways would become difficult if the variation of increase rate is large for a small number of cumulative SARS cases. Our study clearly indicates that the outbreak pattern of the SARS virus in China is of a logistic type, and there is strong negative "density dependency" of the instantaneous SARS persons, implemented by government, are effective, and these measures were also causative factors in the negative correlation between rate of increase and total cumulative SARS cases. 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Davis and Dr. R. 1. Moorhouse for their valuable comments to this manuscript. This work was supported by the Innovation Program of the Chinese Academy of Sciences.