| Chapter 1 Comparative estimation of Case fatality Risk and an applicationBackgroudIn recent decades, a series of infectious disease have taken place which coursed devastating impact to the human society and remained remarkable concerns and warnings, such as avian flu in 1957, Ebola in Congo in 1976, SARS in 2003, H7N9 in 2013, Ebola in West Africa in 2014 and so on. All of these diseases created the huge economic loss and the society affects. Moreover, the emergence of new strains will still continue to pose threats to the public health and the scientific communities.When an emerging influenza virus appears in humans, the early concerns are whether the virus has the potential to be fatal in a substantial number of people. It is widely known that the severity of a virus can be measured by case fatality risk (CFR), which is defined as the proportion of cases of a specified condition that are fatal within a specified time. The CFR can help to predict the number of potential deaths and reflects the medical quality, and has been used by the Centers for Disease Control (CDC) of United States of America for determining the Pandemic Severity Index (PSI), which is used to determine detail public-health actions in accordance to the categories of severity of an influenza pandemic. Therefore an accuracy estimate of CFR is extremely urgent. However, the censoring in the epidemic is a serious problem. Many authors constructed different models to overcome the problem, and Chen-Nakamura constructed a model included covariate. But no comparisons among different CFRs estimated by different models which not included covariate are performed. In order to have an accuracy estimation of CFR, the comparisons among them relatively necessary when a new epidemic is coming.ObjectiveMonte-Carlo simulations are used to simulate the scenarios of an epidemic, and we applied the methods for estimating CFR to the simulated epidemic. Then comparisons among different estimations are performed.MethodsWe considered two different scenarios when using Monte Carlo simulations, reflected the different patterns of disease progress. In scenario I, the observed timely CFR was constant at 0.2. This reflects the strain of disease virus remains unchanged. In scenario II, the decreasing observed timely CFR was considered, the CFR at original (i.e., day 0) was set to 0.8, decreased with a daily rate 0.01. The expression is written as CFR(t)= (0.8-0.01t)I(t≤ 80), where I(x) is an indicator function. Both of the illness onset time are generated by gamma distribution with a standard deviations (sd) of 31 days and a mean of 65days. In order to determine the terminal subject’s survival status (either dead or cured at the end of the epidemic), a binary indicator was generated by a uniform distribution on the interval from 0 to 1 which the cut-off points was set to the observed timely CFR (0.2 in scenario I and CFR at time t in scenario II). Both of the time distribution from illness onset to death and cure obey gamma distributions motivated by the parametric approximation of SARS data. The mean of time from illnesses onset to cure is set to 10,15, and 20 days while the mean of time from illness onset to death was set to 15 days. The standard deviations of both are set to 10days. Criteria of evaluationobserved timely CFR the crude estimate (crude) the estimated CFR with the 95% confidence interval empirical variance (Var.) the average of the estimated within simulation variance (Mvar) bias (the deviation in an estimate from the true cCFR) the mean square error (MSE, defined as squared bias plus the empirical variance)ResultsIn scenario I where CFR was constant, i.e.CFR= 0.2. Obviously, crude method underestimates the CFR overall an epidemic. Among methods based on daily case notification data, the MD2 method gave a reasonable estimate in the early stage of an epidemic because this method was proposed for the early stage. However, when the epidemic progressed, the MSE of MD2 method became larger and the point estimate have an far departure from observed timely CFR reflecting that it is not appropriate for the median and later stage of an epidemic. The MD1 method gave an underestimate of CFR during the outbreak with a little underestimation in the later stage of an epidemic. The accuracy of the MD3, MD4, MD5 and MD6 methods was going upward as epidemic progress in terms of bias and MSE. Among which MD5 and MD6 performed relatively well in the median and later stage. In the early stage of an epidemic, MD3 and MD4 gave a steady and reasonable estimate. However the accuracy of MD5 and MD6 were affected by the means of illness onset to cure. They gave an underestimate when the mean for cure is less than the mean for death. On the contrary, it make an overestimate. They make a reasonable estimate when they are equal. Notably, the effects are insensitive to the later stage of an epidemic for the MD5 and MD6 methods.Among methods based on individual data, the bias of estimate is worst in the early stage of disease with heavy censoring rate, e.g.,20days. It becomes close to the observed timely CFR due to the decreasing censoring rate. In the median and later stage of an epidemic, these methods provided reasonable close estimates. Obviously, the MI7 method gave an underestimation in the early stage of a pandemic. The MI8 method performs lightly better than the MI9 method in in terms of the criterion mentioned above, i.e., the point estimate and the width of 95% confidence interval, combined with MSE, particularly in the early stage. The M110 method performs not very well in the median and later stage. The affection of illness onset to cure for MI8 is same as MD5 and MD6.In scenario II where the observed timely CFR decreased from 0.8 to 0. The crude method have a biased estimate of CFR underestimating observed timely CFR until the later stage of an epidemic, i.e., day 80 and day 100. In this Scenario, methods based on individual data performed worse in the early stage of an epidemic, i.e., on day 20 and 40. It becomes milder as epidemic progressed with the small width of 95% confidence interval, and has an overestimate even though at the later stage of an epidemic. Among methods based on daily case notification data, the MD2 method gave a reasonable estimate in the early stage of an epidemic, but the amount of bias for the MD2 method was going upwards as epidemic progress. The MD1 method fluctuated during the overall epidemic with an overestimate in the early stage and an underestimate in the early stage of an epidemic. The accuracy of the others except for MD6 methods was going upwards to the observed timely CFR as disease progress. Among them MD5 performs best. The MD6 method is sensitive to the changes during the ongoing epidemic. There was no criterion to evaluate the effects of the cure means’performance due to the decreasing CFR, the effects for MD5, MD6 and MI8 were not discussed, however the effect for MD5 was still obvious in the early stage of an epidemic.ConclusionIn conclusion, methods based on individual data are more accurate for estimating CFR then those based on daily case notification data. The MI8 method performs best among all methods if individual data is available. The MD5 and MD4 methods also give a reasonable estimate during the course of an epidemic. In order to detect the changes of the epidemic, the MD6 method is used. Notably, Ghain suggested that it is more appropriate to use a rage rather than a point estimate in the early stage of an epidemic.Chapter 2 Methods and applications for estimation of reproduction number Section 1 SLICAR model considering the infectious of asymptomatic patients in latent and onset periodsBackgroudIt is inappropriate that Decisions only depends on the CFR, which measured the severity of an epidemic. The transmission potential is also a very important index for making public-health decisions. The transmission potential is quantified by basic reproduction number (R0), which is defined as the average number of secondary cases generated by a primary case in susceptible individuals. The critical value is set to 1. If Ro is less than 1, the epidemic will disappear naturally. Otherwise, it will spread and isolation policies should be taken to prevent the transmission. For transmission potential, Kermack, Aron, Schwartz and so on carried on the thorough research for common epidemiological model. But Aaron is pointed out that the asymptomatic cases of infectious for the spread of the disease cannot be ignored. Asymptomatic infection causes the body to produce specific immune response, only does not cause or causes only mild tissue damage, and so on do not show any clinical symptoms, signs, biochemical changes, even can only be found by immunological tests when the pathogens enter the body. Yang and so on pointed out that asymptomatic infection has certain infections. When the crowd for a larger proportion, its effect on the spread of disease cannot be ignored, Longini assumes that asymptomatic infection accounted for 33% in the study. But now there are no models which taking consideration of the asymptomatic infection and the latent periods.ObjectiveProviding a model including latent and onset periods of infectiousness in asymptomatic patients.MethodsSLICAR model was constructed by combining asymptomatic patients and SEIR model. We analyzed and checked the model using pandemic influenza A(H1N1) and Spanish flu data.ResultsSLICAR can fit the curves well for H1N1 and Spanish flu. The basic reproduction number (Ro) of H1N1 was 2.174 (coefficient of determination R2=0.802).2.636 and 3.675 represented 17-days and entire-epidemic Ro, respectively.ConclusionSLICAR model takes consideration of the infectious of asymptomatic patients in latent and onset periods. It is a comprehensive way for evaluating Ro, thus abundant information is available for taking measures.Section 2 Construction of SEIRD model and an application of EbolaBackgroudConstant case fatality risk (CFR) in simple SEIR-type model was inappropriate, because CFR during the entire epidemic was always changing over time. So daily CFR was adjusted in SEIRD (susceptible-exposed-infectious-recovered-dead) based on SEIRD-type model. We also applied the SEIRD model to Ebola in 1976, in 1995 and 2014. Meanwhile, we estimated the timely reproduction number (Rt) to capture the transmission potential and the effectiveness of interventions.ObjectiveProviding a epidemiologic model including daily case fatality risk for basic reproduction number, and conducting a comprehensive analysis to the Ebola outbreak.MethodsConstructing SEIRD model and applying it to the Ebola outbreak for basic reproduction number. Meanwhile, we estimate the timely reproduction number for Ebola outbreak.ResultsSEIRD model can be capable of fitting the epidemic curves. The different CFR methods does not much affect the estimation of basic reproduction number. The Ro is decreasing over time, which Ro for Ebola in 2014 is less than Ro for Ebola in 1976 and 1995.For Ebola in Congo in 1976, the Ro was 3.909 as of September 22 and 4.146 as of September 19 (Rt=1). The Rt was decreasing as the epidemic progressed. And Rt was below 1 after September 19, it means that the epidemic was controlled.For Ebola in 1995, the R0 was 1.887 by the end of April 28 and 1.853 as of May 21 (Rt=1). The trend for Rt in 1995 was similar with Rt in 1976. After May 21, the Rt value is below 1. In another words, the interventions for the outbreak had taken effect. For Ebola in 2014, the Ro of Sierra Leone and Guinea were 1.409 and 1.251 as of December 31, respectively. When the time point was Rt=1 (Sierra Leone and Guinea: November 8 and December 16), the R0 were 1.502 and 1.261, respectively. The Rt for Sierra Leone was decreasing over time. After November 8, Rt is below 1. In Guinea, the Rt fluctuated dramatically before August. Then it was decreasing and bellowed 1 after December 16.ConclusionIt is a comprehensive way that combined basic reproduction number and timely reproduction number to depict the spread of an epidemic and capture the transmission potential and the effectiveness of interventions. |
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