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1 Tex S: odel equaions or edusa sochasic discree-ime ramework: In he main ex we described a sochasic ramework or modeling he spread o edusa hrough a randomly maing populaion; however we le ou several equaions or breviy. These are included here or compleeness. Equaion 5 in he main ex describes he number o larvae o genoype X A Y B over ime. Analogous equaions or he oher larval genoypes are provided here: Aa Aa AB AB L L ( L F( L T ( E E TE T L EL F L i i (S L L ( L F( L T ( E E TE T L EL F L i i (S L L ( L F( L T ( E E TE T L EL F L i i. (S3 In each case he irs erm accouns or survival o larvae having he given genoype rom one day o he nex he second erm accouns or newly haching eggs o his genoype and he hird erm accouns or ransormaion o larvae o he same genoype ino pupae. All symbols are deined in Tle S and he ehods secion o he manuscrip. X A X a eggs are produced by X a X a emales ha have maed wih X A Y B males (hal o he embryos rom which have genoype X A X a while X a X a emales ha have maed wih X a Y b males produce eggs ha are hal X a X a and hal X a Y b. Equaion 6 in he main ex describes he number o adul males o genoype X A Y B over ime. The oher adul male genoype X a Y b is described by he ollowing equaion: ( T ( ( E T EL F L i it. (S4
2 Here he irs erm accouns or survival o X a Y b aduls (denoed a ime by rom one day o he nex and he second erm accouns or ransormaion o X a Y b pupae ino aduls where hese pupae resul rom crosses beween X a X a emales and X a Y b males. As described in he main ex emales are assumed o mae only once and on he same day ha hey emerge so can hereore be described by boh heir genoype and he genoype o he male wih whom hey maed. Equaion 7 describes he number o emale aduls o genoype X A X a ha have maed wih X A Y B males over ime and he oher maed emale genoypes are described by he ollowing equaions: ( Aa Aa Aa AB Aa T ( ( E T EL F L i it AB ( AB AB AB T ( ( E T EL F L i it AB ( T ( ( E T EL F L i it AB (S5 (S6. (S7 For each o hese equaions he irs erm accouns or survival o adul emales having he given maed genoype rom one day o he nex and he second erm accouns or ransormaion o pupae o he given emale genoype ino aduls. The second erm is hen muliplied by he racion o he adul male populaion having eiher genoype X A Y B or X a Y b depending on he emale maed genoype. X A X a pupae emerge rom eggs produced by adul X a X a emales ha have maed wih X A Y B males (hal o he embryos rom which have genoype X A X a and X a X a pupae emerge rom eggs produced by crosses beween X a X a emales and X a Y b males (hal o he embryos rom which have he genoype X a X a. Using hese equaions we can derive several basic properies o he populaion such as he nonzero equilibrium densiies o larvae and aduls and he basic reproducive number i.e. he
3 average number o emale ospring produced by a single emale ha survive o adulhood a low populaion densiies in he sence o geneic conrol. The basic reproducive number can be deined inuiively as he rae o emale egg producion muliplied by he lie expecancy o an adul mosquio muliplied by he proporion o eggs ha will survive hrough all o he juvenile lie sages in he sence o densiy-dependence. This is given by R 0 EL (. (S8 The equilibrium populaion densiies can hen be calculaed by seing he populaion densiies o be equal across generaions in Equaions S-S4 and S7. This leads o he ollowing non-zero equilibria: Leq ( R (S9 0 eq T L L / R 0 ( 0 R E ( L / R0 (S0 where L eq and eq eq eq eq represen he oal populaion equilibria (i.e. L L L and eq eq eq. These ormulaions guide he parameer choices aken rom Deredec e al. [] as shown in Tle S: 3
4 Tle S: arameer values or sochasic discree-ime model Symbol: arameer: Value: Reerences: rimary parameers: T Duraion o egg sage day Depinay e al. [] E T Duraion o larval sage 4 days Depinay e al. [] L T Duraion o pupal sage day Depinay e al. [] E L oraliy rae o juvenile sages 0.68 /day olineaux & Gramiccia [3] Depinay e al. [] oraliy rae o wild-ype adul sage 0.3 /day olineaux & Gramiccia Daily egg producion per emale 3 /day Depinay e al. [] Equilibrium adul mosquio densiy 0000 This paper eq Derived parameers: g Average generaion ime 7. days T T T / [3] E L Survival probiliy or egg sage 0.83 T ( E E E Survival probiliy or larval sage T ( L L L Survival probiliy or pupal sage 0.83 T ( R Generaional populaion grown rae 9.3 Eq. S8 0 Srengh o densiy-dependen componen o larval survival 6300 Eq. S9-S0 4
5 odel equaions or emale-speciic RIDL sochasic discree-ime ramework: The dynamics o emale-speciic RIDL and auosomal X-shredders are described in he Resuls secion o he main ex; however he model ormulaion was le ou or breviy. They are boh included here or compleeness. For emale-speciic RIDL we represen he RIDL consruc as a single auosomal allele R wih a coesponding wild-ype allele r. We consider he case in which he lehaliy rai is emale-speciic lighlessness which allows larval developmen (and hence larval densiy-dependen compeiion o coninue unhindered while suppressing he adul emale populaion and allowing he ransgene o persis or a ew generaions hrough he male line. In his case he larval populaion can be described by a single varile L represening he oal larval populaion a ime : L T E E TE T L EL i i L L ( F ( L ( F ( L. (S The irs erm o his equaion accouns or survival o larvae rom one day o he nex he second erm accouns or newly haching eggs and he hird erm accouns or ransormaion o larvae ino pupae. Here represens he oal adul emale populaion size a ime and is given by where and represen he oal RR Rr number o wild-ype adul emales ha have maed wih males having genoypes RR Rr and respecively. RR Rr Adul males having genoypes RR Rr and are denoed by he variles respecively and are described by he ollowing equaions: RR Rr and RR RR ( RR (S RR TE T Rr Rr Rr Rr ( EL F( L ( i it Rr TE T 4 (S3 5
6 TE T ( EL F( L ( i it. (S4 Rr TE T 4 The irs erm o each o hese equaions accouns or survival o adul males having he given genoype rom one day o he nex and he second erm o equaions S3 and S4 accouns or ransormaion o pupae o he given genoype ino aduls. RR males canno be generaed hrough maing since only wild-ype emales are vile and hence a leas one wild-ype allele is always inheried among he nex generaion. ale Rr pupae emerge rom eggs produced by wild-ype adul emales ha have maed wih RR males (hal o he embryos rom which have he male Rr genoype and rom eggs produced by wild-ype adul emales ha have maed wih Rr males (a quarer o he embryos rom which have he male Rr genoype. Wild-ype male pupae emerge rom eggs produced by wild-ype adul emales ha have maed wih wild-ype males (hal o he embryos rom which have he wild-ype male genoype and rom eggs produced by wild-ype adul emales ha have maed wih Rr males (a quarer o he embryos rom which have he wild-ype male genoype. As described earlier emales are assumed o mae only once and on he same day ha hey emerge so can hereore be described by boh heir genoype and he genoype o he male wih whom hey maed. Since only wild-ype emales are vile here are hree resuling equaions represening emales ha have maed wih males having he genoypes RR Rr or. These are described by he ollowing equaions: ( RR RR TE T RR (S5 EL FL ( ( i i T RR Rr Rr TE T 4 6
7 ( Rr Rr TE T Rr EL FL ( ( i i T RR Rr Rr TE T 4 ( TE T EL FL ( ( i i T RR Rr Rr TE T 4 (S6. (S7 For each o hese equaions he irs erm accouns or survival o adul emales having he given maed genoype rom one day o he nex and he second erm accouns or ransormaion o wild-ype emale pupae ino aduls. The second erm is hen muliplied by he racion o he adul male populaion having eiher genoype RR Rr or depending on he emale maed genoype. As or males Rr pupae emerge rom eggs produced by wild-ype adul emales ha have maed wih RR males (hal o he embryos rom which have he emale Rr genoype and rom eggs produced by wild-ype adul emales ha have maed wih Rr males (a quarer o he embryos rom which have he emale Rr genoype. Wild-ype emale pupae emerge rom eggs produced by wild-ype adul emales ha have maed wih wild-ype males (hal o he embryos rom which have he wild-ype emale genoype and rom eggs produced by wild-ype adul emales ha have maed wih Rr males (a quarer o he embryos rom which have he wild-ype emale genoype. Sochasic and migraory consideraions are idenical o hose or he edusa sysem. odel equaions or auosomal X-shredder sochasic discree-ime ramework: For auosomal X-shredders we represen he X-shredding homing endonuclease gene (HEG as a single auosomal allele H wih a coesponding wild-ype allele h. The HEG creaes a bias among male gamees owards Y-bearing spermaozoa (we denoe he proporion o Y-beraing spermaozoa among male gamees by he symbol c. This leads o an excess o male ospring hence decreasing he emales populaion size and he oal populaion size since here are less emales o produce eggs. Since his gender disorion is ielevan a he larval sage we can use 7
8 equaion S8 again o describe he oal larval populaion size a ime ; however his ime he oal adul emale populaion size is given by hh hh hh hh hh hh (S8 where he irs genoype in he superscrip represens he emale genoype and he second genoype represens he genoype o he male wih whom hey maed. Adul males having genoypes and hh are denoed by he variles hh respecively and are described by he ollowing equaions: and m m E E ( ( ( m m EL F L i it E E (S9 m m m E hh E E m m ( E E hh EL F( L ( i it (S0 m m Ehh E hh hh m hh m Ehh hh E hh hh hh hh hh ( ( ( hh m hh m EL F L i it E hh E. (S The irs erm o each o hese equaions accouns or survival o adul males having he given genoype rom one day o he nex and he second erm accouns or ransormaion o pupae o he given genoype ino aduls. Since all crosses are vile only he raio o male o emale ospring is alered by he HEG here are up o seven crosses ha can generae a given genoype and so we denoe he number o male and emale eggs o genoype x produced by adul emales o genoype y ha have maed wih a male o genoype z by E and xm yz E x yz respecively. These quaniies are ime-dependen and he produc o he ecundiy o he emale genoype y he number o emales having he given maed genoype yz and he proporion o ospring o his maed genoype having he male or emale genoype z. I he paernal genoype 8
9 is ransgenic hen he proporion o male ospring is c; however i he aher is wild-ype boh genders are produced in equal numbers. The numbers o eggs rom each cross are given by he ollowing equaions: m TE T ( E E c c (S m m c c c c ( E E E E T E T (S3 m hh ( E hh E hh T E T (S4 m m c c c c ( E E E E T E T (S5 m m E E E c c c c c c hh m hh TE T E E E (S6 m hh m hh hh ( E hh E hh E hh E hh T E T (S7 m ( E E hh hh c c (S8 hh hh TE T m hh m hh hh hh c c c c ( Ehh Ehh Ehh Ehh T E T (S9 hh m hh hh hh hh ( Ehh hh Ehh hh T E T (S30 Adul emales are described according o heir maed genoype by he ollowing equaions: ( ( ( hh hh E E E hh ( ( ( E L FL i i T hh E (S3 ( ( ( hh hh E E E hh hh ( E E hh EL F( L ( i i T hh Ehh E hh (S3 9
10 ( ( ( hh hh hh hh hh hh hh hh hh E E hh hh hh hh hh hh hh ( ( ( E hh hh L FL i i T hh E hh E. (S33 For each o hese equaions he irs erm accouns or survival o adul emales having he given maed genoype rom one day o he nex and he second erm accouns or ransormaion o pupae o he given emale genoype ino aduls. The second erm is hen muliplied by he racion o he adul male populaion having eiher genoype or hh depending on he emale maed genoype. Sochasic and migraory consideraions are again idenical o hose or he edusa sysem. Reerences:. Deredec A Godray HCJ Bur A (0 Requiremens or eecive malaria conrol wih homing endonuclease genes. roc. Nal. Acad. Sci. USA 08: Depinay JO bogo C Killeen G Knols B Beier J e al. (004 A simulaion model o Arican Anopheles ecology and populaion dynamics or he analysis o malaria ransmission. alar. J. 3: olineaux L Gramiccia G (980 The Garki rojec: Research on he Epidemiology and Conrol o alaria in he Sudan Savanna o Wes Arica. Geneva: World Healh Organizaion ress. 3 p. 0
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