The Flu Seasons and the Missing Data: A Matched-Pair Analysis Northern and Southern Hemispheres and Hong Kong, China

Transcription

1 Journal of Human Vrology & Retrovrology The Flu Seasons and the Mssng Data: A Matched-Par Analyss Northern and Southern Hemspheres and Hong Kong, Chna Abstract Background: The matched-par analyss does not compare between vruses the magntude of postvty rates or the number of postve specmens from dfferent regons. Each vrus among n s match-pared wth tself n the two responses of ts own partal table. The Cochran-Mantel-Haenszel Test collapses all these partal tables wth all the 2n observatons n a 2x2 x n contngency table to yeld the margnal counts of the McNemar s Test. We want to know f, when and how t holds for our lve presentatons of the Laboratores real-tme observatons. Methods and Results: We used col (1,2) 1wk for North Amerca, Europe/Asa and South Amerca/Afrca/Australa/New Zealand. For Canada Row1/Row2 was (BC- Mantoba)/ (Ontaro-Atlantc). For the US Row1/Row2 was Regons (7-10)/ (1-6). In addton we performed smultaneous Proportonal Odds Comparson of Margns (4x4 Table). And we sequentally deleted Regons 1, 10, (9-10), (8-10), (1-4) and (1-5) to defne the effects of the mssng data. And we surveyed for ILI pneumonas n Hong Kong for matched-par regresson. A (H1) and A (H3) surged/resurged wth condton numbers (mult co lnearty) < (φ) (egen value / egen value mn) ( / mn) At above 2,962 the regresson coeffcents dverged n opposte drectons. Concluson: We defne the Influenza Season mathematcally wth the McNemar s Test usng the Laboratores real-tme observatons from the Amercas, Europe/Asa, Afrca and Australa/New Zealand. These real-tme sequental frames ( 12 21) from the weekly updated data show that Z n n 0.5 holds both for ( n + 12 n 21 ) the normal and for the approxmate standardzed test statstcs. We report lve how ths matched-par model performs wth the values of the nterm mssng data set to be zero, as these were the nterm observatons. Volume 1 Issue Vncent Kay Lo Ip* Research Artcle Department of Famly Practce, Rchmond Hosptal, Canada *Correspondng author: Vncent Kay Lo Ip, Department of Famly Practce, Rchmond Hosptal, 7000 Westmnster Hghway, Rchmond, BC, V6X 1A2, Canada, Tel: ; Emal: Receved: October 27, 2014 Publshed: November 08, 2014 Introducton The WHO Consensus on SARS n 2003 states, centres were to relate clncal data on the onset n the symptoms and sgns of SARS to vral shreddng studes both retrospectvely and prospectvely [1,2]. A textbook on mathematcal statstcs defnes Complete Suffcent Statstcs and the search for mnmum varance estmators [3]. The Current Stuaton The opportunstc [4] arborne transmsson of SARS could not get past the drect person-to-person small-partcle aerosols transmsson. The Avan Influenza (H5N1) cannot as yet spread by small-partcle aerosols [5] n human-to-human transmsson. Standard survellance for ILI s [6], excess hosptalzaton [7], outpatent Vsts [8], and travelng waves [9] do not explan the dscrepancy between the smultaneous onset spread of geographcally wdespread outbreaks [10-12] and the neffcent transmsson between person-to-person [10,13-15]. Worldwde stockplng strateges are stll beng formulated [14,16] on how to mprove the current strategy on vaccnaton and prophylactc ant-vral medcatons. The recent A (H1N1) stuaton n 2009_2010 remnds of the 1998 study [17]. Methods The MATCHED-PAIR Relatonshp - the Null Hypothess (Table 1) s that the moment to moment (.e., real-tme) changng or non-changng westerly (or northerly) wnds have no relaton wth the dstrbuton of the vrus solatons wthn the geographc areas. Ths becomes a test for the matched-par relatonshp [3]. The streamlne flow or the turbulence [18] has no relaton to the tmng and the geographc locaton of the actual vral landngs. From 2001 we used the log lnear model ML estmate b to show the estmated probablty that response on geographc locaton s x categores hgher than the response on tme equals exp b x tmes the reverse probablty. From 2010_2011 we used the logt model b to denote a parameter for each subject the odds that the row response falls n category j or below (Instead of above category j) are exp (β) tmes the odds for the column response. The cross-classfcaton n the above 2x2 Table (McNemar s Test) where n3638 actually presents 2n responses for the results Submt Manuscrpt J Hum Vrol Retrovrol 2014, 1(4): 00023

2 2/10 Table 1: McNemar s Test and Cochran-Mantel-Haenszel Test collapsng the 2x2x n contngency table wth n partal tables yelds the margnal counts of McNemar s Tests and each vrus among n s match-pared wth tself n ts own partal table. of two surveys, those of the frst survey n the horzontal row margnal counts and those of the second survey n the vertcal column margnal counts. The Cochran-Mantel-Haenszel Tests on the above rght present these same data dfferently as n3638 separate 2x2 partal tables, one partal table for the two matched responses from each vrus. Each 2x2 partal table has one column for each possble outcome, and the results of the frst survey n row 1, and the results of the second survey n row 2 and collapsng the 2x2 x n contngency table for USA and Canada 11 yelds the margnal counts of the McNemar s tests. The Mssng Data The 10-Regon who/nrevss regonal map contaned the full verson of the offcal data Regons. The 9-Regon who/nrevss regonal map showed how the observatons could be mssng or msplaced. The offcal 10-Regon map carred the numbers (1-10) of the data Regons n bold and these numbers (together wth ther data) corresponded to the respectve Regons n the 9-Regon map, where Regon 10 and ts data were mssng. We sequentally defned Regons 1, 10, (9-10), (8-10), (1-4) and (1-5) as further mssng data and we presented these comparsons n the extreme condtons of. For weekly real-tme computaton we needed to defne the entty of the nterm mssng data and whch would be updated n the subsequent weeks. Results The McNemar s Test and the Wald test (Proportonal Odds Model) On Fgure 1-3 for Canada, Fgure 4 for the US, Fgure 5 for North Amerca, Fgure 6 for Europe/Asa, and Fgure 7 for the Southern Hemsphere, we tested the matched-par model generally wth the standardzed normal test statstc z of the McNemar s Test on the upper left and compared ths wth the approxmate z based on the Proportonal Odds Model on the upper rght. On the lower row wthn each graph we supermposed the 2x2 (2wk) and 4x4 (4wks) Tables to compare ther real-tme dfferences, A(H3) on the lower left and A(H1) on the lower rght. For Fgure 2, the left upper dagram showed the full verson of the offcal data (2x2 4wks) Row1/Row2(7-10)/ (1-6)10R7 and (4x4 4wks) Row1/Row2/Row3/Row4 Regons(910/78/56/1-4). The rght upper (AH1), left lower (AH3) and rght lower (B) dagrams showed the effects of the mssng data (mn and ) from both sdes of Row1 or Row2 n the 2x2 Table Column(1, 2)2w-wks. The three centre-most curves n each of these three dagrams were represented by 10R7Row1/Row2(7-10)/(1-6), 9R7 wth out10row1/row2 (7-9)/(1-6) and 9R7 wthout1row1/row2(7-10)/(2-6). The upper boundares were represented by 8R7Row1/Row2 (7-8)/(1-6) and 7R7Row1/Row2(7)/(1-6. And the lower boundares were by 6R7(7-10)/(5-6) wthout Regons (1-4) and 5R7(7-10)/(6) wthout Regons (1-5). Wth these mssng data defned for Row1 the entre curves for 8R7Row1/Row2(7-8)/ (1-6) and for 7R7Row1/Row2(7)/(1-6) moved upwards along the y-axs. The opposte occurred for Row1/Row2 6R7(7-10)/ (5-6) wthout Regons (1-4) and for Row1/Row25R7(7-10)/ (6) wthout Regons (1-5). These moved downwards along the y-axs. In Fgure 3 we computed the full data from Canada, USA, as Row1/Row2North/South Canada/USA BC-Atlantc/ Regons (1-10) and the full data from Mexco, Guatemala and USA as Row1/Row2 South/North Mexco-Guatemala/USA. For Canada/USA we computed these to be wthn the boundares formed wth Row1BC- Atlantc Provnces wthout the Thunder bay data and wth Row2Regons (1-9) wthout Regon 10. On the rght upper graph the data were Row1/Row2/Row3/Row4 Canada woont/ Ontaro/ Regon (125810)/Regon (34679). In comparng the 2x2 Table wth the 4x4 Table t was found that the orgnal Row1/Row2 boundary should be kept ntact as the Row2/Row3 boundary. For the Same reason the 2x2 Table for Mexco-Guatemala/USA was compared wth Row1/Row2/ Row3/Row4 Guatemala/ Mexco/ Regons (34679) / Regons (125810). Fgures 4 and 5 The McNemar s Test and the Proportonal Odds Model The full verson of the data represented by 10R7Row1/ Row2(7-10)/(1-6) were computed to be wthn the two boundares set between 9R7(wthout 10) Row1/Row2 (7-9)/(1-6) and 9R7 (wthout 1) Row1/Row2 (7-10)/(2-6).

3 3/10 Fgure 1: The McNemar s Test and the Wald Test for Vrus Isolaton Data wk 39 Canada. Left upper, Col (1,2) 1wk, and rght upper, Col(1,2,3,4)1wk. Z on left s standardzed normal statstc. Zpropodd on rght s approxmate z from proportonal odds model. On left(for upper) are for full verson Canada data computed wthn the boundares set by the mssng data from row1 and row2 of the 2x2 Table. Rght(for upper) 4x4 Table showed the computatons for all Canada data wthn the boundares set wthout the Saskatchewan data n Row3.In the lower left, data for A(H3) (2x2) wth and wthout sask data n Row1 compares wth A (H3)(4x4) wth and wthout sask data n Row2. On rght lower, A(H1) both 2x2 and 4x4 supermposed. Fgure 2: The Mssng Data (mnmzed and mzed). USA Season 2013_14 year to wk39.left upper, z(2x2) for Col(1,2)2wks and zodds(4x4) for Col(1,2,3,4)1wk, and zorzodds s ether z or zodds (approxmate z). z s normal test statstc row1/row2(7-10)/(1-6)10r7 and approxmate z (zodds) s for row1/row2/row3/ row4regons(910/78/56/1234). For rght Upper (H1), left lower (H3), and rght lower(b), Col(1,2)2wks, and 10R7Regons(7-10)/(1-6), 9R7wo10 (7-9)/(2-6), 8R7(7)/(1-6), 6R7(1-4) (7-10)/(5-6) 6R7 wthout(1-4), 5R7(7-10)/(6) wthout (1-5). For these three fgures, mssng data n row 1 elevate z, and n row 2 they lower z.

4 4/10 Fgure 3: Row1/Row2Canada/USA and Row1/Row2Mexco-Guatemala/USA. North Amercan Season year to wk 39 Matched-par behavour. Upper. Row1/Row2Canada/USA on left. Row1 from the left become on the rght Row1/Row2Canada woot/ontaro. Row2 From the left becomes on the rght Row3/Row4 Regons(125810)/Regons(34679). Lower Left. Row1/Row2Guatemala. Mexco/USA. Ths Row1 become on the rght lower Row1/Row2Guatemala/Mexco. Row2 on the left lower becomes Row3/Row4Regons (34679)/ Regons(125810) on the rght lower z s normal test statstc of McNemar s Test. Zpropodd s approxmate z of the proportonal odds comparson of margns (4*4) Fgure 4: The McNemar s Test and the Proportonal Odds Model US wk39. Upper Row compares the general matched-pared between the 2*2 and the 4*4 Tables. On left column (1,2)1.week compares wth rght column (1,2,3,4)1.week. z on left s standard normal statc. Zpropodd on rght s approxmate z from proportonal odds model. The upper left and the rght are for full verson 10 Regons computed wthn boundares set by the mssng data as defned. For the lower row each graph compares the specfc matched-par behavour between 2*2 (2wks) and the 4*4 (4wks) Tables. Left s A(H3) and rght s A(H1).

5 5/10 Fgure 5: The McNemar s Test and the Proportonal Odds Model. N Amerca 2014 wk 39. Upper Row shows the general matched-par behavour between the 2*2 and the 4*4 Tables. Left Row1/Row2Canada. Regons (125810)/Regons (34679). Mexco-Guatemala Row1 from the left becomes on the rght Row1/Row2Canada/Regons (125810). Row2 from the left s now on the rght Row3/Row4Regons (34679)/Mexco-Guatemala. Left upper, z s normal test statstc of the 2*2 Table. Rght upper, zpropodd s approxmate z of the 4*4 Table. Lower Row, on each graph the specfc matched-par behavour between the 2*2(2wks) and 4*4(4wks): A (H3) on the left and A(H1) on the rght. Fgure 6: WHO data for Europe/Asa. Europe/Asa 2014 wk39 Upper Row shows the general matched.par behavour between the 2*2 and 4*4 Tables. Left Row1/Row2Europr/Asa. Row1 from the left becomes on the rght Row1/Row2Iceland_Brtsh_Isles/Contonental_Europe. Row2 from the left s now on the rght Row3/ Row4Contnental_Asa/Japan_Phlppnes_Indonesa. Left upper, z s normal test statstc of the 2*2 Table. Rght upper, zpropdd s approxmate z of the 4*4 Table. Lower Row, on each graph the specfc matched.par behavour between the 2*2(2wks) and 4*4 (4wks): A (H3) on the lower rght.

6 6/10 Fgure 7: WHO data for the Southern Hemsphere. Southern Hemsphere 2014 wk39. Upper Row shows the general matched-par behavour between the 2*2 and the 4*4 Tables. Left Row1/ Row2South_Amerca_Afrca/Australa-New Zealand. Row1 from the Left s on the rght Row1/Row2South Amerca/Afrca. Row2 from the left s now on the rght Row2/Row4Australa/New Zealand. Left upper, z s normal test statstc of the 2*2 Table. Rght upper, zpropodd s approxmate z of the 4*4 Table. Lower Row, on each graph the specfc matched. Par behavour between the 2*2(2wks) and 4*4(4wks): A(H3) on the lower left and A(H1) on the lower rght. From a hgher vantage pont we computed the whole of North Amerca wth row1/row2 Canada-Regons (125810)/Regons (34679)-Mexco-Guatemala and wth row1/ row2/ row3/ row4 Canada/Regons (125810)/Regons (34679)/Mexco- Guatemala n (Fgure 5). The Geographc Asymmetry by 4x4 Squared Tables We examned the west-to-east symmetry coeffcents from week 40 onwards from 2001_2002. The 4x4 Squared Tables was appled weekly to examne the symmetry of the floatng 12_week average. For convenence we dvded the Seasons nto those domnated by A(H1N1) and those domnated by A(H3N2). The Seasons domnated by Influenza A(H1N1) The ntal analyss wth the 4x4 table between 2001 and 2008 showed that the symmetry coeffcents for A (H1N1) were dfferent from those for A(H3N2). There were three A(H1N1)- domnated Seasons and ther mum symmetry coeffcents ranged from (2002_2003), to (2008_2009) and to (2006_2007). The average was 1.06, and ths was much lower than 2.18 whch was the average for the fve A (H3N2) Seasons. Season For the major part of the Season A (H1N1) appeared to have postve symmetry coeffcents. The symmetry coeffcents for A (H3N2) appeared less postve and they seemed to be catchng up wth a steeper slope. At the end of the Season all the coeffcents were converged together. Ths was the year when the hybrd A (H1N2) was reported. Season In the US we found that A(H3N2) behaved dfferently from A(H1N1). For Influenza A(H3N2), the symmetry for weeks 01 to 12 OR0.62(0.57, 0.68) showed no change from OR0.39(0.23, 0.67) for weeks 41_52. Ths dfference can be examned more closely by the weekly computatons. The scatter dagram of the weekly symmetry coeffcents for A(H1N1) closely resembled that for A(H3N2) but Influenza A(H1N1) had a 6-8 week lead ahead of A(H3N2). Ths 6-8 week lead was reflected at the weekly ratos of H1 to H3 antgens. The cdc ratos for H1 to H3 antgens were decdedly elevated at 10/1 for week 49, at 99/7 for week 6 and 296/40 for week 12. On the background of ths ncreased H1/H3 rato only n the US, we found that t was not just smple concdence that the west-to-east trajectory of Influenza A(H1N1) was ncreasng n the US from OR0.76(0.65, 0.91) for week 41_52 to OR1.95(1.82, 2.1) for weeks 01_12. We summarzed the evdence for ths dfference wth the McNemar s Test for Canada/USA between Influenza A(H1N1) and Influenza A(H3N2). A(H1N1) had counts z(180/730) /( )^ The same test for A (H3N2) had counts z(467-49)/ (467+49)^ There was very strong evdence and probablty for the A (H1N1) solatons to ncrease towards the US. The Influenza A(H3N2) Seasons A (H3N2) was domnant for 2001_2002, 2003_2004, 2004_2005, 2005_2006 and 2007_2008. Except for 2007_2008 the symmetry coeffcents for A(H1N1) were dffcult to compute. For Season 2007_2008, A (H1N1) showed a pattern dentcal to that for A(H3N2). These fve Seasons had ther mum symmetres rangng from 1.62 to 2.98 wth an average of Except for Season 2005_2006, whch had the lowest symmetry coeffcent at , all had ncreased excess mortalty.

7 7/10 Season 2001_2002: The symmetry coeffcents for A (H3N2) remaned close to the mum level around from weeks 05_16 to 08_19. The symmetry coeffcents for Influenza A (H1N1) could only be brefly computed at at week 04_15. The coeffcents for Influenza B remaned negatve throughout from week 40_51 to week 06_17. Season 2003_2004: Ths was the year that was noted for the early and prolonged respratory complcatons. The negatve symmetry coeffcents for A (H3N2) were converted postve very early at week 43_01 to These remaned at 2 or above 2 from week 44_02 onwards to week 06_17 and peaked at Durng ths perod from week 43 to week 46 the drft varant A/ Fujan/411/2002 arrved at the US and Canada. Season 2004_2005: For ths Season the symmetry coeffcents converted from negatve to between weeks 50_09. Ths peaked at week 05_16 at The A/Calforna/7/04 arrved the US durng week 2 and to Canada and Europe n week 07. By week 16 the mass mgraton was complete and A/Calforna/7/04 had all but dsplaced the A/Fujan/411/2002. Season : Among the fve A (H3N2)-domnated seasons ths Season had the lowest symmetry coeffcent, Ths was the season wthout ncreased excess mortalty. Season : A(H3N2) showed a massve ncrease n ther west-to-east asymmetry OR14.1(12, 16.6) from OR0.1(0.08, 0.14). Compared wth 2003_2004 where the mum coeffcent was 2.985, the symmetry coeffcents for Season 2007_2008 were converted to postve values late n the Season. Once converted onto postve terrtory however, these symmetry coeffcents of the 2007_2008 Season ncreased very quckly to at week 09_20. Increased excess mortalty was reported and ths occurred late. Vral actons n Hong Kong by Matched-Par Regresson (Fgure 8): We also performed matched-par regresson based on the surveys from authentcated but anonymous questonnares from well-defned responders, whch were found to be relable and accurate [19-22]. Randomzaton and Samplng Smple Random Samplng [22] to Hong Kong s actvely practcng N3,000 doctor populaton, n100 to n200 questonnares were sent every week to surveys on the symptoms and complcatons of Influenza-lke-Illnesses,.e., fever [yes/ no], cough [yes/no], dyspnoea [yes/no], pneumona [yes/no], darrhea [yes/no] and patent s hstory of contact wth ILI s [yes/ no]. Insttutonal communty clncs were ncluded. Overseas addresses were excluded. Mathematcal Model for Regresson Analyss The model s an example of mxed model, contanng the random effect (ntercept) and the fxed effect ( b ). For PROC GLIMMIX, the outcome 1 s success and outcome 0 s falure for the response of pneumona n the unvarate 2-stacked data format. Let ψ denote the probablty of a success for subject s response: ( ψ ) + 1 ( 1 1) + 2 ( 3 2) + 3[ ] + 4[ ] + 5[ ] [ MYCO] [ RSV ] [ Contact estmator] Logt A H N A H N B C PFLU Equaton 2 a n the ntercept representng an unobserved sample from a probablty dstrbuton, presumed to be normal dstrbuton wth unknown mean and standard devaton. Response (event 1 ) denotes the probablty of a success for subject s response; b, b, b and b are coeffcents for weekly respratory vrus solatons; Estmator H1N1, H3N2, B, C, PFLU, MYCO, rsv, vrus*vrus. fh1, fh3 and fb are H1N1, H3N2 or b expressed as fracton of samples; contact number of patents per week who develop ILI s after contact wth other patents who had ILI s. We re-computed our data wth PROC GLIMMIX from December 2004 onwards. We performed unvarate regresson analyss for pneumona usng Response (event1), 1 beng success and 0 beng falure for pneumona. The two response parameters of pneumona (bnary) and contact (Posson) were stacked n the 2-tered unvarate format [23]. We computed the random (a ) effects and the fxed (β) effects for the dfferent components of Vrus Isolaton Data n accordance wth our ( ) model, and Probablty p exp a + b for 1+ exp( a + b ) Response(event1), ths beng the subject-specfc effects from the th questonnare and th partal table of the McNemar s Test. Note: Model s modfed from PROC GLIMMIX documentaton [23,24]. The McNemar s Test and matched-par data provded the model for ths survey for matched-par data, 1 success and 0 falure for ILI pneumona. PROC GLIMMIX computed the random (a ) and fxed (β) effects for A(H3N2) and A(H1N1). We p ( event ) Probablty '1' exp ( a + b ) ( a b ) 1+ exp + defne as the subject-specfc-effects from the th questonnare and partal table. We defned the fve core Seasons from 2005 to 2009 to be: February_May 2005 Influenza A (H1N1) wth the probablty for success 1 for pneumona wth the th questonnare p0.0203, φ 533; December_March 2006 wth p for A(H3N2) and φ 261, January_Aprl 2007 wth p for A(H1N1) and φ 2,202, February_Aprl 2008 wth p for A(H3N2) and φ 38 and the current Season February to Aprl 2009 p for A(H1N1) and φ 2,962. Multcollneartes We defned that multcollneartes exsted when the condton numbers were above φ 2,962. As we plotted

8 8/10 Fgure 8: The Core seasons on the x-axs below <log (Condton Number) the log of the condton numbers between the subtypes A(H3N2) and A(H1N1) on the x-axs aganst ther coeffcents on the y-axs we found that the coeffcents for the fve core Seasons had formed a horzontal lne ndstngushable from the x-axs. On the rght beyond the log for the condton number φ 2,962, we found the coeffcents for the two subtypes A(H3N2) and A(H1N1) to be vsbly dverged from each other. These dverged coeffcents were from the two pre-season perods, the frst one beng January to March 2005 for A(H3N2) φ 51,121 and second one from January to March 2008 wth the condton number φ / The dvergng coeffcents also came from the three post-season perods Aprl to June 2006 for A (H3N2) φ 6,776.93/ ,668, February to May 2007 for A(H3N2) φ 4,801.89/ ,613 and May to July 2008 for A(H1N1) φ 3321/ ,439. Co-crculatons We defned co-crculatons to nclude our computed conjont regressor A (H1N1)_A (H3N2) as per our mathematcal model. In ths format the two sub-types were truly movng wth each other. The coeffcents for these conjont regressor A (H3N2)_A(H1N1) dd not seem to be nflated by the multcollneartes ( movng wth each other ) and they seemed to lnk up the dfferent components of the Season. The Baselne n 2005 For 2005 we dd not compute any coeffcent for our conjont regressor. In addton the p s for pneumona for both A(H3N2) and A(H1N1) durng the pre-season part (January_March 2005) and durng the core part (February to May 2005) of the Season were zero or very close to zero OR1.82(1.14, 2.90) p For the rest of 2005 the random (a ) effects were very low, from to We therefore defned ths perod to be our baselne OR0.73(0.61, 0.87) p for A (H3N2) and OR1.64(1.11, 2.42) p for A (H1N1). The entre Season (the expanded outbreak) By an expanded outbreak we referred to the entre Flu Season. For 2005, 2006 and 2007 we found the expanded outbreak to nclude the core Season plus ether one of the two other components of the Season,.e., the pre-season plus the core Season or the core Season plus the post-season. For 2008 the entre Flu Season conssted all three components, the pre- Season, the core Season and the post-season. The tug-of-war/stalemate between 2006 and 2007PROC GLIMMIX provded a method for us to examne n detal the perod between the 2007 and 2008 outbreaks. Usng the weekly data for July_October 2006 the th probablty for success 1 for pneumona for A(H3N2) was p Low 95% was p These numbers for A(H1N1) was p Hgh 95% was p Usng the monthly data from ths same perod (July_Oct) the th probablty for success 1 for pneumona for A(H3N2) was p Low 95% was p These numbers for pneumona for A (H1N1) was p Hgh 95% was p The weekly data for Sept_Dec 2006 showed that ths stalemate contnued tll the end of December 2006 when the th probablty for success 1 for pneumona for A(H3N2) ended at p Low 95% was p The same numbers for A (H1N1) was p Its hgh 95% range was p Season 2006_2007 the surge/resurge of A(H1N1) Durng the Flu Season of January_Aprl 2007 we saw the

9 9/10 p probablty for success for pneumona for A(H3N2) reduced from p to p and the p for A(H1N1) ncreased from p to p For May_August 2007 the p for A (H1N1) contnued to be hgher than that for A(H3N2), whether based on the weekly data (p0.0114) or based on the monthly data (p0.1945). For May_Dec 2007 the p could be computed for A(H1N1) p but not for A(H3N2). Season 2007_2008 the late return of A(H3N2) The conjont co-crculatons for A(H3N2)_A(H1N1) appeared n December 2007 and ths contnued nto the monthly data from January to March as OR1.19(1.01, 1.4) and from Aprl to July 2008 as OR1.02 (1.001, 1.04) wthout any dvergence for A(H3N2) and A(H1N1) n data 1, 2, 3, 6, 9 and 14. For January_ Mar 2008, we computed the condton number φ (egen value /egen value mn) / At these astronomcal levels for the condton numbers, Proc Glm mx gave the random (a ) effects of The fxed (β) effects for A (H3N2) were ( ) and for A(H1N1) We could not nterpret these except to say that A (H3N2) and A(H1N1) had moved wth each other. The 2007_2008 Flu Season had three components, the January to March pre-season wth hgh condton numbers and marked dvergence, the February & Aprl 2008 core Season wth low condton number φ (1,618.95/42.47)38.12 wthout dvergence. The thrd component of the Season from May to July 2008 had the condton number φ (3,772.26/ )90,095. The β coeffcents for the conjont regress or A (H3N2)_A(H1N1) seemed to lnk the three components to form the same Season. Pandemc Season 2008_2009 Ths Season was n fact an ant-cl when compared wth 2007_2008. The perod Nov_Feb brought n the ncreased condton number of φ 6,024. Only the coeffcents for the conjont co-crculaton A (H3N2)_A(H1N1) could be calculated. The three perods that followed,.e., Dec_Feb, Feb_Aprl and for Feb_May had mnmal coeffcents respectvely for A (H1N1) and for A(H3N2). From these we could defne Feb_Aprl as the core Season and that the condton number log φ log(2,962)7.99 could now be defned as the new upper lmt wthout dvergence of the regresson coeffcents. The respectve probabltes for success 1 for pneumona wth the th questonnare could then be represented by these perods Dec_Feb, Feb_Aprl and Feb_ May. These were for A(H1N1) and for A(H3N2). These showed a postve trend as well as an mprovng statstcs of ft (Fgure 8). Fgure 9: General Flow Dagram wthn the context of the matched-par analyss, when the S, I and R compartments became ndstngushable from each other as β, the parameter for drect person-to-person contact transmssons, s n the lmt of β -0. Dscusson In the face of the explosveness of the 1918 nfluenza pandemc [24], the nvestgators were surprsed to fnd that ther estmates of the Basc Reproducton Number (R 0 ) were much lower than expected. The nvestgators of the SIRC model defned the epdemologcal parameters of (δ ) and (γ ) as the nverses of the average tme spent by ndvduals n each of the compartments of R and C respectvely [25]. Wthn the context of the 4x4 Squared Table, we thnk that the lower than expected R0 found for the 1918 pandemc should lead to the drecton when (b) the parameter for drect person-to-person contact transmsson was n the lmt of : Equaton ( ) S+ I + R µ 1 S + I + R + γ C δ R Equaton 3 shows that when the epdemologcal parameter for drect person-to-person contact transmsson (b) s n the lmt of b-0 the compartments of S, I and R become ndstngushable from each other. The graphcal form (Fgure 9) resembles our everyday physcal realtes of the 4x4 and 2x2 Squared Tables. The SIRC model defned (γ -1) and (δ -1) as the average tme spent by ndvduals n each of the compartments of C and 1 1 R respectvely, and that the functon δ + γ represented the average age of a cross-mmune ndvdual, 25 and that they rδ were nter-related γ [25,26]. Among the three levels σδ r ( a δ,andγ ) of the tme span factor n the orgnal general flow dagram [25], we found one non-redundant dummy [19] varable besde (a). We propose that one epdemologcal parameter asde from the two epdemologcal parameters ( γ, δ ) be needed for (Fgure 9). At least one non-redundant epdemologcal parameter should exst between the cross mmune compartment and the sngle combned SIR compartment. Acknowledgment To the Department of Mathematcs, Hong Kong Baptst Unversty, Professor KT Fang, Professor Fred Hckernell and Professor SN Chu.

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