家畜伝染病発生数の時系列解析 Time series analysis of morbidities of animal infectious diseases
Time series analysis of morbidities of animal infectious diseases
数学的手法を用いた伝染病の発生予測は、理論疫学の分野で研究され、流行曲線の解析により各種の数式モデルが作られ、伝染病発生後の経過予測に貢献している。一方、官庁統計を用いた長期間にわたる疾病の発生推移についての研究は、記載疫学の分野で数多く行われている。しかし、この両者は、疫学の手法として最も遠い位置に置かれていることが多く、記載疫学的な材料を理論疫学的な手法で処理した研究は少ない。 記載疫学的な材料である統計資料を利用し、理論疫学的な手法により数式モデルを作り、そのモデルによる疾病の発生予測が可能になるとすれば、その価値は大きい。すなわち、数式モデルによる疾病の発生予測にもとづいて、効果的な疾病予防、防圧が可能になり、家畜衛生業務にきわめて大きな進歩をもたらすばかりでなく、その業務自体の経済的な価値を客観的に評価する基礎的な資料を提供することが可能になるからである。 上記の理由から、著者は、1947～1973年の「家畜衛生統計」を用いて、炭疽(牛)、気腫疽(牛)、ブルセラ病(牛)、トリコモナス病(牛)、結核(牛)、馬伝染性貧血、馬パラチフス、流行性脳炎(馬、疑似患畜数)、豚丹毒、豚コレラ、ひな白痢(鶏)、腐蛆病(単位は蜜蜂の群数)の12疾病の年間発生数について時系列解析を行なった。しかし、家畜伝染病発生数には、経済的な理由から、人為的影響が強くはいると思われるので、解析は、経済時系列解析における乗法モデルを用い、下記のように行った。 y_t=T + C + Iただし、 y_t=log(1+x_t)ただし、 x_t : 第t年度における年間発生数(t=1, …, N) T : y_tの傾向変動 C : y_tの循環変動 I : y_tの不規則変動 傾向変動は、重回帰分析により、年度tの1～4次の多項式として次のように求めた。 z^^^_pt=β_po + Σ^^p__i=1 β_pi t^i (p=1, …, 4; i=1, …, p; t=1, …, N)ただし、 p : 多項式tの次数 β_pi : 編回帰係数 また、y_tからz^^^_ptを差し引き、y_tの傾向変動を分離したw_ptについてペリオドグラム分析を行ない、次のw^^^_ptを求めた。 w^^^_pt=Σ^^n_p__j=1 a_pj sin(α_pj+2πt/μ_pj) (p=1, …, 4; j=1, …, n_p; t=1, …, N)ただし、 n_p : 統計的に優位な単振動の数 μ_pj : w_ptから求めた統計的に優位なj番目の周期 a_pj : 周期μ_pjのときの振幅 α_pj : 周期μ_pjのときの位相角 w^^^_pt : 統計的に優位な単振動を合成した第t年度の値 上記の手法で得た結果から、年度tに対するy_tの回帰方程式を次のように作成した。 y^^^_pt=z^^^_pt + w^^^_ptすなわち、 x^^^_pt=10^(β_po+Σ^^p__i=1 β_pi t^i+Σ^^np__j=1 a_pj sin(α_pj+2πt/μ_pj)) -1 (p=1, …, 4; i=1, …, p; j=1, …, n_p; t=1, …, N) このようにして12疾病それぞれについて得られたp=1, …, 4の回帰方程式y^^^_ptが、y_tに適合する程度を、次式の寄与率として求めた。 R_p=(S_Eq / S_yy)×100% (p=1, …, 4)ただし、 R_p : y^^^_pt のy_tに対する寄与率 S_Eq : y^^^_pt の回帰方程式の変動 S_yy : y_t の全変動 疾病ごとにy^^^_1t～y^^^_4tのいずれかの寄与率が95%以上、すなわち非常に良い適合を示したのは、トリコモナス病、結核病、馬伝染性貧血、ひな白痢であり、90～95%で良い適合を示したのは、ブルセラ病、馬パラチフス、腐蛆病であった。また、寄与率が80%台、すなわちやや低い適合を示したのは、流行性脳炎、豚丹毒、豚コレラで、炭疽、気腫疽の寄与率の最大値はそれぞれ42.05、50.70で、適合が悪かった。なお、多項式の次数による寄与率の差は、非常に良い適合を示した4疾病では小さく、気腫疽、豚コレラでは非常に大きかった。 上記の研究を開始したのちに、1974、1975年の発生数が公表されたので、そのそれぞれをx_N+1、x_N+2とし、これを変換したy_N+1、y_N+2に対するy^^^_p(N+1)、y^^^_p(N+2) の予測誤差を、次式のように求めてみた。 d_pk=y^^^_p(N+k) - y_N+k (p=1, …, 4; k=1, 2) 疾病ごとに予測誤差の絶対値|d_1k|～|d_4k|が最も小さいものを選び出すと、1974年については豚コレラ、気腫疽を除き、1975年についてはトリコモナス病、馬パラチフスを除き、他の予測誤差はすべて0.25以下であった。また|d_1k|～|d_4k|それぞれの平均値|d^^-_p|のうち最小のものについてみると、炭疽、気腫疽、トリコモナス病、流行性脳炎、豚コレラを除き、他の7疾病では0.25以下であった。この値は真数にして1.78倍以下である。したがって、予測精度はかなり良好なものといえる。ただし、寄与率と予測誤差の間の相関関係は統計的に有意でなかった。 つぎに、過去の数値に対する適合性と、予測精度の両面からy^^^_pt を評価するために、疾病およびy^^^_ptごとに、次のように有用性係数C_pを求めた。 C_p=W_R (1-R_p/100)+Σ^^2__k=1 W_d |d_pk|/3 (p=1, …, 4; k=1, 2)ただし、 W_R : 寄与率に対する重み係数(0.5とした) W_d : 予測誤差に対する重み係数(0.25とした) ただし、この重みづけは全く恣意的なものであり、予測誤差を3で除したことも、とくに明確な基準があって行なったわけではない。 疾病ごとに最も小さい有用性係数を選び出すと、結核病、馬伝染性貧血、ひな白痢で0.05より小、ブルセラ病、トリコモナス病、腐蛆病で0.1より小、馬パラチフス、流行性脳炎、豚丹毒で0.16より小であり、これら9疾病についての回帰方程式の有用性が高いことが示された。一方、炭疽、気腫疽の有用性係数は0.3台、豚コレラは0.4501と、前記9疾病の有用性係数から相当かけはなれた値を示した。 これらの成績を総合してみると、結核病、馬伝染性貧血、ひな白痢の3疾病については、適合性、予測精度、有用性いずれの点においても非常にすぐれており、つづいて、ブルセラ病、トリコモナス病、腐蛆病も良い成績をあげでいる。馬パラチフス、流行性脳炎、豚丹毒については前記6疾病よりやや劣るが、その有用性は十分に認められる。のこる炭疽、気腫疽、豚コレラの3疾病については、他の疾病に比し一段と劣った成績しか得られなかったが、従来しばしば行われてきた移動平均や単純な最小2乗法に比すれば、この方法のすぐれていることは明らかである。 このように仕分けされた疾病の特徴をみると、上位6疾病中からトリコモナス病を除いた、結核病、馬伝染性貧血、ひな白痢、ブルセラ病、腐蛆病はいずれも、定期検査によって患畜を摘発し、殺処分に付すことを主体とした防疫措置のとられている疾病であり、トリコモナス病も、つぎに位する馬パラチフスとともに種畜検査の対象となっている疾病である。これにたいし、下位の炭疽、気腫疽では寄与率がかけはなれて小さいが、両者とも芽胞形成菌であるBacillaceaeに属する細菌による疾病であることに注目すべきであろう。豚関係の2疾病のうち、豚丹毒では、その成績はやや劣る程度であったが、豚コレラでは、寄与率は80%以上を示しはしたものの、その成績は豚丹毒に比し劣り、その有用性係数は12疾病中最下位となった。また、流行性脳炎については、疑似患畜数を扱っているため、その疾病の内容が定かでない点もあるが、結果的に、豚コレラを上回り、豚丹毒を下回る成績を示した。 家畜疾病の生物学的ないしは実験室的な検討は、従来から微に入り細にわたって行なわれているのに反し経済活動としての畜産のなかでの家畜衛生、そのなかでの疾病といった考え方は、ことにわが国の試験研究機関のなかでは、ややもすると薄れがちである。 著者の当面の試みは、家畜伝染病の発生推移をモデル化することにあったが、上記のようにその目的は十分に達成され、しかもそのモデルを用いた発生予測の精度もかなり高いことが明らかになった。したがって著者が作成した数式モデルによる疾病の発生予測は、効果的な疾病予防、防圧を可能とし、家畜衛生業務の新披術として大いに貢献するばかりでなく、その業務自体の経済的評価のための客観的な基礎となるものと思われる。 今後、著者が試みた方法をさらに広い範囲に応用し、疾病の発生推移を左右する原因の解析を進めれば、疾病の生物学的な面での性格とともに、その経済ないしは社会的な面での特徴をも明らかにすることが可能であると思われる。このような意味で、著者の試みは、疫学的研究上のひとつの進路に先鞭をつけると同時に、コンピュータ利用の必要性を痛感させる研究であると信ずる。
In 1867, William Farr (1807-83), the founder of epidemiology in its modern form, applied his mathematical method on the rinderpest epizootic in Britain, and predicted that it would terminate within a certain stated time. It is said that his prediction proved to approximately accurate, while the contradictory remarks to the fact are found elsewhere. Afterward medical microbiology made remarkable progress, elucidating numbers of pathogenic agents of infectious diseases, and various new techniques were developed for the experimentation. Traditional epidemiology which had pursued the study of contagious diseases in populations received enough the benefit of the progress. With the decrease in acute infectious diseases and the growth of ecology, the scope and method of epidemiology have been gradually changed. There are little doubt that the epidemiology has been greatly encouraged by the modern statistical or stochastic methods introduced by R.A.Fisher in 1920's, and the rapid vulgarization of the computer techniques particularly since the latter period of 1950's. The mathematical forecasting of epidemics, however, remains one of the important subjects of theoretical epidemiology. On the basis of studies on epidemic curves, various mathematical models have been formulated and they have contributed to predicting the progress of epidemics. On the other hand, the long-term transition of diseases has been largely studied in the field of descriptive epidemiology. Any government health official should take advantage of the sequence of such studies for getting prospective views on ill health. However, the descriptive and theoretical studies are usually apart from each other, and the procedures of theoretical epidemiology have scarcely been applied on materials ordinarily used in descriptive studies. Therefore it will be of great value if transition of diseases may be predicted by mathematical models derived from vital statistics, viz. materials of descriptive epidemiology. Without proper perspectives, neither effective prophylaxis of diseases nor impartial appraisal of animal health services will be ultimately possible. Fostering such an idea in mind, the author attempted time series analysis of annual morbidities of the following twelve animal infectious diseases officially recorded on the Statistics of Animal Hygiene for the period from 1949 to 1973, published by the Bureau of Animal Industry, Ministry of Agriculture and Forestry; Anthrax in cattle, blackleg in cattle, bovine brucellosis, bovine trichomoniasis, bovine tuberculosis, equine infectious anemia, equine paratyphoid, equine infectious encephalitis (suspected cases), swine erysipelas, hog cholera, pullorum disease in chickens, and foulbrood. The morbidities were expressed in number of diseased cases with an exception of foulbrood which were in number of bee hives. The statistics of morbidity, so far as concerned to domestic animals, are susceptible to deliberate or economical influences, because the domestic animals are raised for economic purposes. Taking the fact into considerations, it was supposed that the methods often used in econometrics might be successfully applied on morbidity curve for a long time. This is the reason why the following multiplicative model was adopted for the present analysis. x_t = T×C×I (t= 1,...,N)where x_t: annual morbidity in the year t, T : secular trend of x_t, C : cyclic fluctuation of x_t, and I : irregular fluctuation of x_t. In order to avoid the impossibility of logarithmic calculation where x_t=0, the actual analysis was made on the following y_t; y_t = log(1+x_t),accordingly, y_t = T+C+I,namely, x_t = 10^T+C+I -1,where T : secular trend of y_t, C : cyclic fluctuation of y_t, and I : irregular fluctuation of y_t. By means of the multiple regression analysis, z^^^_pt, the estimates of secular trend of y_t, were obtained for each disease, in accordance with the following polynomial equations of the first to fourth degrees, for the year t; z^^^_pt = β_p0+Σ^^p__i=1 β_pi t^i (p=1,..., 4; i=1,..., p),where t : the year, p : degree of t in the polynomial equation, β_pi : partial regression coefficient of the term of the i-th degree in the polynomial equation of the p-th degree, and z^^^_pt : value of the year t obtained from the polynomial equation of the p-th degree. Partial regression coefficients were obtained by the least squares procedure, then z^^^_pt were formulated for each value of t. Subtracting z^^^_pt from y_t, trend-free w_pt was obtained as follows; w_pt = y_t - z^^^_pt. By application of the periodogram analysis to w_pt, w^^^_pt, the estimates of cyclic fluctuation of w_pt, were obtained as follows; w^^^_pt = Σ^^(n_p)__j=1 a_pj sin(α_pj+2πt/μ_pj) (p=1,..., 4; j=1,..., n_p; t=1,..., N),where n_p : number of sine curves with statistical significance, μ_pj : the statistically significant j-th cycle obtained from w_pt, a_pj : amplitude of the cycle μ_pj, α_pj : phase angle of the cycle μ_pj, and w^^^_pt : value of the year t obtained by cumulating sine curves with statistical significance. Finally the regression equation for the year t was formulated as follows; y^^^_pt = z^^^_pt+w^^^_pt,namely y^^^_pt = β_p0+Σ^^p__i=1 β_pi t^i+Σ^^(n_p)__j=1 a_pj sin(α_pj+2πt/μ_pj),namely x^^^_pt = 10^(β_p0+Σ^^p__i=1 β_pi t^i+Σ^^(n_p)__j=1 a_pj sin(α_pj+2πt/μ_pj)) - 1 (p=1,..., 4; i=1,..., p; j=1,..., n; t=1,..., N). Table 1 shows the regression equations of the first to fourth degrees obtained for each of the twelve diseases.[table 1] To evaluate those equations, the fitness of y^^^_pt to y_t was examined by the value of proportion defined as follows; R_P = (S_Eq/S_yy) × 100% (p = 1,..., 4)where S_yy = Σ^^N__t=1 (y_t - y^^-_t)^2, and S_Eq = Σ^^N__t=1 (y^^^_pt - y^^^-_pt)^2. The maximum value of the proportions of the four regression equations for each disease (Column 1 of Table 2) showed the excellent fitness in trichomoniasis, tuberculosis, equine infectious anemia, and pullorum disease with the value higher than 95%, the good fitness in brucellosis, equine paratyphoid, and foulbrood between 90 and 95%, and fairly good in equine encephalitis, swine erysipelas, and hog cholera above 80% level. In anthrax and blackleg, the fitness was inferior, with the maximum proportion of 42.05 and 50.70% respectively. The difference of proportions due to the degree of polynomial equations was slight as to the four diseases which showed the excellent fitness, but it was very large as to blackleg and hog cholera. During the above-mentioned analysis was under way, the morbidities of the diseases concerned for 1974 and 1975 were officially announced, the former in the Statistics of Animal Hygiene for 1974, and the latter in the Animal Hygiene Weekly, No.1390, 1976, both published by the Ministry of Agriculture and Forestry. The data for 1974 and 1975 were expressed as x_N+1 and x_N+2 respectively, which were transformed into y_N+l and y_N+2. The errors of prediction, viz. the deviations of y^^^_p(N+1) and y^^^_p(N+2) from y_N+1 and y_N+2 were calculated as follows; d_pk = y^^^_p(N+k) - y_N+k (p=1, ,4; k=1, 2). Among the absolute values of error of prediction, |d_1k| ～ |d_4k|, the smallest one (Columns 2 and 3 of Table 2) was picked out for each disease. As a result, the ten diseases other than hog cholera and blackleg in 1974, and the same number of diseases other than trichomoniasis and equine paratyphoid in 1975 gave the errors less than 0.25. The minimum values obtained from among the means of the errors for 1974 and 1975, |d^^-_p|, were less than 0.25 as to the seven diseases other than anthrax, blackleg, trichomoniasis, equine encephalitis and hog cholera (Column 4 of Table 2). As the antilogarithmic value of 0.25 is less than 1.78, it is believed that the morbidities were fairly precisely predicted. However, the correlation between the proportion and the error of prediction were not statistically significant.[table 2] In order to evaluate y^^^_pt, from both of the fitness for the past data and the preciseness of prediction, the utility coefficient, C_p, was defined as follows; C_p = W_R (1-R_P / 100)+Σ^^2__k=1 W_d |d_pk|/3where R_p : proportion of the regression equation, |d_pk| : error of prediction (absolute value) obtained by the extrapolation of regression equation, W_R : weight to proportion (fixed as 0.5), and W_d : weight to error of prediction (fixed as 0.25). Those weightings as well as the reduction of error of prediction to a one-third were made discretionally without a concrete criterion. According to this definition, the smaller the coefficient, the greater the utility of the regression equation. The smallest coefficient for each disease (Column 5 of Table 2) showed that it was less than 0.05 as to tuberculosis, equine infectious anemia and pullorum disease, less than 0.1 as to brucellosis, trichomoniasis, and foulbrood, and less than 0.16 as to equine paratyphoid, equine encephalitis and swine erysipelas. Accordingly, the utility of the regression equations for these nine diseases were incontestable. On the contrary, the coefficients exceeded 0.3 as to anthrax and blackleg, and 0.4501 as to hog cholera, showing that the regression equations for these three diseases were of doubtful utility. Looking through all the data mentioned above, the findings on the regression equations were summarized as follows; The excellent results were obtained as to tuberculosis, equine infectious anemia and pullorum disease, and secondly the good results as to brucellosis, trichomoniasis and foulbrood. The equations for equine paratyphoid, equine encephalitis and swine erysipelas took the third place, but it was believed that they held practical value. The remaining three diseases, anthrax, blackleg and hog cholera, showed the results inferior to the above-mentioned nine diseases, but there was little doubt that the present method was superior to moving average and simple least squares method often applied for this kind of analysis. The examination of the characteristics of the diseases categorized in this manner showed that tuberculosis, equine infectious anemia, pullorum disease, brucellosis and foulbrood, out of the six diseases of the higher rank, were all subject to regulatory inspection and to the so-called "stamping out." Trichomoniasis, the remainder of the higher rank, together with equine paratyphoid which took the next place, were both controlled by the Breeding Stock Law, besides the Animal Infectious Diseases Control Law. On the other hand, anthrax and blackleg which gave only a small value of proportion, are those caused by the organisms which belong to Bacillaceae, a family of endospore-forming bacteria. As for the two diseases of pigs, swine erysipelas took the third place, and hog cholera gave the results inferior to the former, showing the worst utility coefficient among the twelve diseases analyzed, although the value of proportion was more than 80%. Regarding equine encephalitis, the nature of the disease was ambiguous as the analysis was made on suspected cases, but it gave a result in between those of swine erysipelas and hog cholera. From biological and experimental points of view, diseases of domestic animals have been studied considerably in detail, but the research workers concerned sometimes make little account of the fact that the disease is one of the most important problems of the animal health services which should support animal husbandry in the framework of economical activities. In the field of medicine, the disease itself is deemed to be an evil necessarily subject to remedy. On the analogy of this, the veterinarians are apt to think the same as to animal diseases. However, domestic animals are primarily raised for economic purposes, and from an ethological point of view, they are possibly forced unnatural circumstances and often ill health, even though they are kept healthy. Therefore, with regard to animal diseases, it is most important to take account of their economic implications so far as they do not affect human health. The theoretical models of epidemics hitherto formulated are mainly concerned with those among the human beings, consequently they are scarcely relevant to economic idea. In choosing the materials and methods used for the present study, the author was always conscious of this point and after all adopted one of the econometric methods for the initial attempt to analyze the long-term transition of morbidity in domestic animals. Statistics used here were limited to those published by the government and also to the annual morbidities in real number of animals or bee hives. Analysis of seasonal fluctuations of morbidities was abandoned because it was supposed that the monthly morbidities were complicated with the seasonal variation of animal populations particularly in pigs, and the number of animals undergone any periodical inspection. Under the condition that the statistics of animal populations only as of February 1 were available, the decision was justifiable especially at the beginning of the trial. For the same reason, the morbidities were not transformed into ratio. For the time series analysis of economical fluctuation, either of additive or multiplicative model is assumed at first. In the present analysis, the author adopted the multiplicative model to avoid the inconvenience of additive model as stated by Takizawa. Serfling, and Housworth et al., employed the similar methods in their studies on excess mortality of influenza. As the transition of deaths due to influenza was much steadier or simpler in its character than those treated here, their regression equations were additive and of the first or second degree. In the field of veterinary medicine, no attempt had been made to carry on investigations of this genre, and the present analysis aimed, first of all, at constructing models of long-term transition of morbidities. As a result, the four regression equations were composed for each of the twelve diseases, and the most of them showed the satisfactory fitness to the original morbidity data. Furthermore, it was proved that by the extrapolation of the model, the morbidity could be predicted with fairly good preciseness. Consequently, the mathematical models formulated by the author will contribute to the betterment of the disease control, and also afford the sound basis for the impartial evaluation of animal health services. By the further application of the method introduced here, various factors which exert influences upon the transition of disease will be elucidated in regard with their biological nature as well as their economical and social aspects. In this sense, it may be said that the author's trial opened up a new field in epizootiology, and that it demonstrated a positive necessity of computer techniques in such a research.