▼♦❞❡❧❛çã♦ ▼❛t❡♠át✐❝❛ ❞❡ ❊♣✐❞❡♠✐❛s
❆♥❛ ❈❛t❛r✐♥❛ ❈❛r♥❡✐r♦ ❆r❛ú❥♦
❘❡❧❛tór✐♦ ❋✐♥❛❧ ❞♦ ❚r❛❜❛❧❤♦ ❞❡ Pr♦❥❡t♦ ❛♣r❡s❡♥t❛❞♦ à ❊s❝♦❧❛ ❙✉♣❡r✐♦r ❞❡ ❚❡❝♥♦❧♦❣✐❛ ❡ ●❡stã♦
■♥st✐t✉t♦ P♦❧✐té❝♥✐❝♦ ❞❡ ❇r❛❣❛♥ç❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ●r❛✉ ❞❡ ▼❡str❡ ❡♠
❚❡❝♥♦❧♦❣✐❛ ❇✐♦♠é❞✐❝❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦ã♦ P❛✉❧♦ ❆❧♠❡✐❞❛ ❈♦✲♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇❛❧s❛
▼♦❞❡❧❛çã♦ ▼❛t❡♠át✐❝❛ ❞❡ ❊♣✐❞❡♠✐❛s
❆♥❛ ❈❛t❛r✐♥❛ ❈❛r♥❡✐r♦ ❆r❛ú❥♦
❘❡❧❛tór✐♦ ❋✐♥❛❧ ❞♦ ❚r❛❜❛❧❤♦ ❞❡ Pr♦❥❡t♦ ❛♣r❡s❡♥t❛❞♦ à ❊s❝♦❧❛ ❙✉♣❡r✐♦r ❞❡ ❚❡❝♥♦❧♦❣✐❛ ❡ ●❡stã♦
■♥st✐t✉t♦ P♦❧✐té❝♥✐❝♦ ❞❡ ❇r❛❣❛♥ç❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ●r❛✉ ❞❡ ▼❡str❡ ❡♠
❚❡❝♥♦❧♦❣✐❛ ❇✐♦♠é❞✐❝❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦ã♦ P❛✉❧♦ ❆❧♠❡✐❞❛ ❈♦✲♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇❛❧s❛
✏❊st❡ tr❛❜❛❧❤♦ ❞❡ ♣r♦❥❡t♦ ✐♥❝❧✉✐ ❛s ❝rít✐❝❛s ❡ s✉❣❡stõ❡s ❢❡✐t❛s ♣❡❧♦ ❏úr✐✑
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦✱
❆♦s ♠❡✉s ♣❛✐s ▼ár✐♦ ❡ ▼❛r✐❛✱ ♦s ♣r✐♠❡✐r♦s q✉❡ ✐♥❝❡♥t✐✈❛r❛♠ ♦s ♠❡✉s ❡st✉❞♦s✱ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠✐♠ ❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❡♠ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳
➪ ♠✐♥❤❛ ✐r♠ã✱ ❝✉♥❤❛❞♦ ❡ s♦❜r✐♥❤♦ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳
❆♦s ♣r♦❢❡ss♦r❡s ❈❛r❧♦s ❇❛❧s❛ ❡ ❏♦ã♦ P❛✉❧♦ ❆❧♠❡✐❞❛ ♣♦r t♦❞♦ ♦ ✐♥❝❡♥t✐✈♦✱ t❡♠♣♦✱ tr❛❜❛❧❤♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ♣❛rt✐❧❤❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ❞❡❞✐❝❛r❛♠ ♥❛ ♦r✐❡♥t❛çã♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳
P♦r ú❧t✐♠♦✱ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ♥❛♠♦r❛❞♦ ♣❡❧♦ s❡✉ ❛♣♦✐♦✱ ❝❛r✐♥❤♦ ❡ ❝♦♠♣r❡❡♥sã♦ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❡t❛♣❛✳
❘❡s✉♠♦
◆❡st❛s ú❧t✐♠❛s ❞é❝❛❞❛s ❛❧❣✉♠❛s ❞♦❡♥ç❛s ❝♦♠♦ ❛ ✈❛rí♦❧❛✱ ♦ s❛r❛♠♣♦✱ ❛ ❙❆❘❙ ✭sí♥❞r♦♠❡ r❡s♣✐r❛tór✐❛ ❛❣✉❞❛✮ ❡ ❛ ❣r✐♣❡ tê♠ r❡❝❡❜✐❞♦ ♠✉✐t❛ ❛t❡♥çã♦ ♣♦r ♣❛rt❡ ❞♦s ✐♥✈❡st✐❣❛❞♦r❡s✳ ❚r❛t❛♥❞♦✲s❡ ❛ ❡♣✐❞❡♠✐❛ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ s❛ú❞❡ ♣ú❜❧✐❝❛ é ✐♠♣♦rt❛♥t❡ ❢❛③❡r ❛ ♠♦✲ ❞❡❧❛çã♦ ❞❛ s✉❛ ♣r♦♣❛❣❛çã♦✱ ❞❡ ♠♦❞♦ ❛ q✉❡ s❡ ♣♦ss❛ ❛t✉❛r s♦❜r❡ ❡❧❛ ❡ ❛ss✐♠ ♣r❡✈❡♥✐r ✉♠ s✉rt♦✳ ❉❡st❡ ♠♦❞♦✱ ❡①✐st❡♠ ❞✉❛s ❣r❛♥❞❡s ❝❧❛ss❡s ❞❡ ♠♦❞❡❧♦s ❡♣✐❞❡♠✐♦❧ó❣✐❝♦s✱ ♦s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡ ♦s ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s✳
◗✉❛♥t♦ ❛♦s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡st❡s ✉t✐❧✐③❛♠ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❛ ♠♦✲ ❞❡❧❛r ♦ ❝r❡s❝✐♠❡♥t♦ ❡ ♦ ❞❡❝r❡s❝✐♠❡♥t♦ ❞❛s ♣♦♣✉❧❛çõ❡s ❡♥✈♦❧✈✐❞❛s✳ Pr❡ss✉♣õ❡♠ q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞✐ár✐♦ ❞❡ ✉♠ ✐♥❞✐✈í❞✉♦ é s❡♠♣r❡ ♦ ♠❡s♠♦✳ ❊st❡ ♠♦❞❡❧♦ ❝♦rr❡s♣♦♥❞❡ ❛ s✐t✉❛çõ❡s ♠❛✐s ♣r♦✈á✈❡✐s ❞❡ ♦❝♦rr❡r✳ ❖s ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s ✉t✐❧✐③❛♠ ✐♥❢♦r♠❛çã♦ ❡st❛✲ tíst✐❝❛ s♦❜r❡ ❛ ♣♦♣✉❧❛çã♦ ❛❢❡t❛❞❛ ❣❡r❛♥❞♦ ♣❛❞rõ❡s✳ ❊st❡s ♠♦❞❡❧♦s sã♦ ♠❛✐s r❡❛❧✐st❛s ❞♦ q✉❡ ♦s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ♣♦rq✉❡ ✐♥❝❧✉❡♠ ✈ár✐❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦✳ ❘♦❝❤❛ ❬✷✶❪ r❡s✉♠✐✉ ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡①✐st❡♥t❡s ♥❛ ❧✐t❡✲ r❛t✉r❛✳ Pr♦♣ôs ♠♦❞❡❧♦s ♠❛✐s ❝♦♠♣❧❡t♦s ♣❛r❛ ❡st✉❞❛r ❛ ♠❛❧ár✐❛ ❡ ❛ ❙■❉❆✳ ❚s✉❥✐❣✉❝❤✐ ❬✷✹❪ ❡st✉❞♦✉ ♠♦❞❡❧♦s ❛♣❧✐❝❛❞♦s à ❡♣✐❞❡♠✐♦❧♦❣✐❛✱ ❝♦♠♦ ♦s ♠♦❞❡❧♦s SIS✱ SIR ❡ SIRS✱ ❞❡
♠♦❞♦ ❛ ❞❡s❡♥✈♦❧✈❡r ✉♠ ♠♦❞❡❧♦ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ❞✐♥â♠✐❝❛ ❞❛ ❞✐s♣❡rsã♦ ❞❛ ■♥✢✉❡♥③❛ ❆ ✭❍✶◆✶✮ ♥✉♠❛ ♣♦♣✉❧❛çã♦✳
❆ss✐♠✱ ♥❡st❡ tr❛❜❛❧❤♦ ❢❛③✲s❡ ✉♠❛ s✐st❡♠❛t✐③❛çã♦ ❞♦s ♣r✐♥❝✐♣❛✐s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥ís✲ t✐❝♦s ❡①✐st❡♥t❡s✱ t❛❧ ❝♦♠♦ ❘♦❝❤❛ ❬✷✶❪ ❡ ❚s✉❥✐❣✉❝❤✐ ❬✷✹❪ ❡ ❛✐♥❞❛ ❢❛③✲s❡ ❛ s✉❛ ❝♦♥✈❡rsã♦ ❡♠ ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s✳ ❖s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡st✉❞❛❞♦s ❡ ✐♠♣❧❡♠❡♥t❛❞♦s ❡♠ ❝❛s♦s ❞❡ ❡st✉❞♦ ❢♦r❛♠ ♦s s❡❣✉✐♥t❡s✿ SIS✱SIR✱ SIRS✱ SIQS✱SIQR✱ M SEIRS✱ M SEIR ❡ SEIR✳ ❉❡ ♠♦❞♦ ❛ ❛♣❡r❢❡✐ç♦❛r ❛❧❣✉♥s ❞♦s ♠♦❞❡❧♦s ❡st✉❞❛❞♦s✱ ❢♦✐ ♥❡❝❡ssár✐♦ ❢❛③❡r ❛❧t❡✲
r❛çõ❡s✳ P♦rt❛♥t♦✱ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❢♦✐ ♥❡❝❡ssár✐♦ ❢❛③❡r ✉♠
❛❥✉st❛♠❡♥t♦ às ❡q✉❛çõ❡s ❡ ❞❡t❡r♠✐♥❛r ✉♠❛ ♥♦✈❛ ❝♦♥st❛♥t❡ q✉❡ s❡ ❞❡s✐❣♥♦✉ ❞❡ ❑✳ ❊st❛ ❝♦♥st❛♥t❡ ♣❡r♠✐t❡ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ✈❛r✐❛r ♦s ♣❛râ♠❡tr♦s ❞❡ ♥❛t❛❧✐❞❛❞❡ ❡ ♠♦rt❛❧✐❞❛❞❡✳ ◆❛ ❧✐t❡r❛t✉r❛✱ ♣♦r ❡①❡♠♣❧♦ ❡♠ ❬✶✹❪✱ ❝♦♥s✐❞❡r❛✲s❡ ♦s ♣❛râ♠❡tr♦s ❞❡ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦✲ ♥❛❧ ✐❣✉❛✐s ❞❡ ♠♦❞♦ ❛ q✉❡ ❤❛❥❛ s❡♠♣r❡ ✉♠ ❡q✉✐❧í❜r✐♦ ❡♥tr❡ ♥❛s❝✐♠❡♥t♦s ❡ ♠♦rt❡s✱ q✉❛♥❞♦ s❡ ♣r❡t❡♥❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ❞❡ t❛♠❛♥❤♦ ❝♦♥st❛♥t❡✳
❖s ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s ❢♦r❛♠ ❛♣❧✐❝❛❞♦s ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❡ ✉♠❛ ❡♣✐✲ ❞❡♠✐❛ ♥✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ✭❡①❡♠♣❧♦ ❜❛s❡❛❞♦ ❡♠ ❬✷✵❪✮ ❛♣❧✐❝❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡ ❛✐♥❞❛ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ ❝♦♥tá❣✐♦ ❞❛ ●r✐♣❡ ♣❡ss♦❛ ❛ ♣❡ss♦❛ ♥✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ❡st✉❞❛♥t✐❧✱ ❝♦♠♦ ❛ ❞❛ ❊❙❚✐● ✉s❛♥❞♦ ✉♠ ♠♦❞❡❧♦SIR ❡st♦❝ást✐❝♦✳ ❋♦r❛♠ ✐♥tr♦❞✉③✐❞♦s
❢❛t♦r❡s ❝♦♠♦ ❛ ✈❛❝✐♥❛çã♦✱ ❛ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧✱ ❛ ❤❡t❡r♦❣❡♥❡✐❞❛❞❡ ❞❛ ♣♦♣✉❧❛çã♦ ❡ ❛ ✈❛r✐❛çã♦ ❞♦ ♣❡rí♦❞♦ ❞❡ ❝♦♥tá❣✐♦ ♥✉♠❛ ♣♦♣✉❧❛çã♦ ❤❡t❡r♦❣é♥❡❛✳
P♦r ú❧t✐♠♦✱ t♦❞♦s ♦s ♠♦❞❡❧♦s ❡st✉❞❛❞♦s ❢♦r❛♠ ✐♠♣❧❡♠❡♥t❛❞♦s ❡♠ ▼❛t❧❛❜❘✳ P❛r❛
t♦❞♦s ♦s ♠♦❞❡❧♦s ❢❛③❡♠✲s❡ ❛♣❧✐❝❛çõ❡s ❞❡ ♣r♦❜❧❡♠❛s ♣rát✐❝♦s✱ ♣r♦♣♦st♦s ♣♦r ♥ós ♦✉ r❡❝♦✲ ❧❤✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛✳ ❆♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✱ ♣❛r❛ ❛❧❣✉♥s ♠♦❞❡❧♦s✱ sã♦ ❛✐♥❞❛ ❝♦♠♣❛r❛❞♦s ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❝♦♠ ♠♦❞❡❧♦s ❞✐❢❡r❡♥t❡s✳ ❖s r❡s✉❧t❛❞♦s ♣❡r♠✐t❡♠ ❝♦♠♣r❡❡♥❞❡r ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✉rt♦ ❡♠ ❢✉♥çã♦ ❞❡ ❞❡t❡r♠✐♥❛❞♦s ❢❛t♦r❡s ❝♦♠♦ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ♠❡❞✐❞❛s ❞❡ ♣r❡✈❡♥çã♦ ❡ ❞❡ ✐s♦❧❛♠❡♥t♦✱ ✈❛❝✐♥❛çã♦ ❡ q✉❛r❡♥t❡♥❛✱ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ♥❛t❛❧✐❞❛❞❡ ❡ ❞❡ ♠♦rt❛❧✐❞❛❞❡ ❡ ❛ ❝♦♥s✐❞❡r❛çã♦ ❞❡ ❤❡t❡r♦❣❡♥❡✐❞❛❞❡✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❊♣✐❞❡♠✐❛✱ ▼♦❞❡❧♦s ❉❡t❡r♠✐♥íst✐❝♦s✱ ▼♦❞❡❧♦s ❊♣✐❞❡♠✐♦❧ó❣✐❝♦s✱ ▼♦❞❡❧♦s ❊st♦❝ást✐❝♦s✳
❆❜str❛❝t
❚❤❡s❡ ❧❛st ❞❡❝❛❞❡s s♦♠❡ ❞✐s❡❛s❡s s✉❝❤ ❛s s♠❛❧❧♣♦①✱ ♠❡❛s❧❡s✱ ❙❆❘❙ ✭❙❡✈❡r❡ ❆❝✉t❡ ❘❡s♣✐✲ r❛t♦r② ❙②♥❞r♦♠❡✮ ❛♥❞ ✐♥✢✉❡♥③❛ ❤❛✈❡ r❡❝❡✐✈❡❞ ♠✉❝❤ ❛tt❡♥t✐♦♥ ❢r♦♠ r❡s❡❛r❝❤❡rs✳ ❇❡✐♥❣ t❤❡ ❡♣✐❞❡♠✐❝ ❛ ♣✉❜❧✐❝ ❤❡❛❧t❤ ♣r♦❜❧❡♠✱ ✐s ✐♠♣♦rt❛♥t t♦ ❞♦ t❤❡ ♠♦❞❡❧✐♥❣ ♦❢ ✐ts ♣r♦♣❛❣❛✲ t✐♦♥ s♦ t❤❛t ✇❡ ❝❛♥ ❛❝t ✉♣♦♥ ✐t ❛♥❞ t❤✉s ♣r❡✈❡♥t✐♥❣ ❛♥ ♦✉t❜r❡❛❦✳ ❚❤✉s✱ t❤❡r❡ ❛r❡ t✇♦ ♠❛❥♦r ❝❧❛ss❡s ♦❢ ❡♣✐❞❡♠✐♦❧♦❣✐❝❛❧ ♠♦❞❡❧s✱ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s ❛♥❞ t❤❡ st♦❝❤❛st✐❝ ♠♦❞❡❧s✳
❆s ❢♦r ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s✱ t❤❡② ✉s❡ ❞✐❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s t♦ ♠♦❞❡❧ t❤❡ ❣r♦✇t❤ ❛♥❞ ❞❡❣r♦✇t❤ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐♥✈♦❧✈❡❞✳ ■t ♣r❡s✉♣♣♦s❡s t❤❛t t❤❡ ❞❛✐❧② ❜❡❤❛✈✐♦r ♦❢ ❛♥ ✐♥✲ ❞✐✈✐❞✉❛❧ ✐s ❛❧✇❛②s t❤❡ s❛♠❡✳ ❚❤✐s ♠♦❞❡❧s ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♠♦st ♣r♦❜❛❜❧❡ s✐t✉❛t✐♦♥s t❤❛t ❝❛♥ ♦❝❝✉r✳ ❚❤❡ st♦❝❤❛st✐❝ ♠♦❞❡❧s ✉s❡ st❛t✐st✐❝ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ✐♥❢❡❝t❡❞ ♣♦✲ ♣✉❧❛t✐♦♥ ❛♥❞ ❣❡♥❡r❛t❡ ♣❛tt❡r♥s✳ ❚❤❡s❡ ♠♦❞❡❧s ❛r❡ ♠♦r❡ r❡❛❧✐st✐❝ t❤❛♥ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s ❜❡❝❛✉s❡ t❤❡② ✐♥❝❧✉❞❡ s❡✈❡r❛❧ ❜❡❤❛✈✐♦r ♣♦ss✐❜✐❧✐t✐❡s✳
❘♦❝❤❛ ❬✷✶❪ s✉♠♠❛r✐③❡s s♦♠❡ ♦❢ t❤❡ ♠❛✐♥ ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s ❡①✐st✐♥❣ ✐♥ t❤❡ ❧✐t❡✲ r❛t✉r❡✳ ❙❤❡ ♣r♦♣♦s❡❞ ♠♦r❡ ❝♦♠♣❧❡t❡ ♠♦❞❡❧s ❢♦r st✉❞②✐♥❣ ♠❛❧❛r✐❛ ❛♥❞ ❆■❉❙✳ ❚s✉❥✐❣✉✲ ❝❤✐ ❬✷✹❪ st✉❞✐❡❞ ♠♦❞❡❧s ❛♣♣❧✐❡❞ t♦ ❡♣✐❞❡♠✐♦❧♦❣②✱ ❛s ♠♦❞❡❧s ❙■❙✱ ❙■❘ ❛♥❞ ❙■❘❙ ✐♥ ♦r❞❡r t♦ ❞❡✈❡❧♦♣ ❛ ♠♦❞❡❧ t♦ ❞❡s❝r✐❜❡ t❤❡ ❞②♥❛♠✐❝s ♦❢ ❞✐s♣❡rs✐♦♥ ■♥✢✉❡♥③❛ ❆ ✭❍✶◆✶✮ ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✳
■♥ t❤✐s ✇♦r❦ ✇❡ ♠❛❦❡ ❛ s②st❡♠❛t✐③❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❡①✐st✐♥❣ ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s✱ s✉❝❤ ❛s ❘♦❝❤❛ ❬✷✶❪ ❛♥❞ ❚s✉❥✐❣✉❝❤✐ ❬✷✹❪ ❛♥❞ ❞♦ ✉♣ ✐ts ❝♦♥✈❡rs✐♦♥ ✐♥t♦ st♦❝❤❛st✐❝ ♠♦❞❡❧s✳ ❚❤❡ ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧s st✉❞✐❡❞ ❛♥❞ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❝❛s❡ st✉❞✐❡s ✇❡r❡ ❛s ❢♦❧❧♦✇s ❙■❙✱ ❙■❘✱ ❙■❘❙✱ ❙■◗❙✱ ❙■◗❘✱ ▼❙❊■❘❙✱ ▼❙❊■❘ ❛♥❞ ❙❊■❘✳ ■♥ ♦r❞❡r t♦ ✐♠♣r♦✈❡ s♦♠❡ ♦❢ t❤❡ ♠♦❞❡❧s st✉❞✐❡❞✱ ✐t ✇❛s ♥❡❝❡ss❛r② t♦ ♠❛❦❡ ❝❤❛♥❣❡s✳ ❋♦r ♠♦❞❡❧s ✇✐t❤ ❞②♥❛♠✐❝ ♣♦♣✉❧❛t✐♦♥ ✐t ✇❛s ♥❡❝❡ss❛r② t♦ ♠❛❦❡ ❛♥ ❛❞❥✉st♠❡♥t t♦ t❤❡ ❡q✉❛t✐♦♥s ❛♥❞ ❞❡t❡r♠✐♥❡ ❛ ♥❡✇ ❝♦♥st❛♥t
❞❡s✐❣♥❛t❡❞ ❑✳ ❚❤✐s ❝♦♥st❛♥t ❛❧❧♦✇s ✐t t♦ ❜❡ ♣♦ss✐❜❧❡ t♦ ✈❛r② t❤❡ ❜✐rt❤ ❛♥❞ ♠♦rt❛❧✐t② ♣❛r❛♠❡t❡rs✳ ■♥ ❧✐t❡r❛t✉r❡✱ ❢♦r ❡①❛♠♣❧❡ ✐♥ ❬✶✹❪✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❞②♥❛♠✐❝ ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡r ❡q✉❛❧ s♦ t❤❛t t❤❡r❡ ✐s ❛❧✇❛②s ❛ ❜❛❧❛♥❝❡ ❜❡t✇❡❡♥ ❜✐rt❤ ❛♥❞ ❞❡❛t❤✱ ✇❤❡♥ ✇❡ ✇❛♥t ❛ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❝♦♥st❛♥t s✐③❡✳
❚❤❡ st♦❝❤❛st✐❝ ♠♦❞❡❧s ✇❡r❡ ❛♣♣❧✐❡❞ t♦ ♠♦❞❡❧ t❤❡ ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❛♥ ❡♣✐❞❡♠✐❝ ✐♥ ❛ ❝❧❛ssr♦♦♠ ✭❡①❛♠♣❧❡ ❜❛s❡❞ ♦♥ ❬✷✵❪✮ ❜② ❛♣♣❧②✐♥❣ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❛♥❞ ②❡t t❤❡ st✉❞② ♦❢ ■♥✢✉❡♥③❛ ♣❡rs♦♥ t♦ ♣❡rs♦♥ ✐♥ ❛ st✉❞❡♥t ❝♦♠♠✉♥✐t②✱ ❛s ❊❙❚✐● ❜② ✉s✐♥❣ ❛ st♦❝❤❛st✐❝ ♠♦❞❡❧ ❙■❘✳ ❋❛❝t♦rs ✇❡r❡ ✐♥tr♦❞✉❝❡❞ s✉❝❤ ❛s ✈❛❝❝✐♥❛t✐♦♥✱ ❞②♥❛♠✐❝ ♣♦♣✉❧❛t✐♦♥✱ ❤❡t❡r♦❣❡♥❡✐t② ❛♥❞ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ✐♥❢❡❝t✐♦♥ ♣❡r✐♦❞ ✐♥ ❛ ❤❡t❡r♦❣❡♥❡♦✉s ♣♦♣✉❧❛t✐♦♥✳
❋✐♥❛❧❧②✱ ❛❧❧ ♠♦❞❡❧s st✉❞✐❡❞ ✇❡r❡ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ▼❛t❧❛❜❘✳ ❋♦r ❛❧❧ ♠♦❞❡❧s ❛r❡ ♠❛❞❡
♦❢ ♣r❛❝t✐❝❛❧ ♣r♦❜❧❡♠s ❛♣♣❧✐❝❛t✐♦♥s✱ ♣r♦♣♦s❡❞ ❜② ♦r t❛❦❡♥ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❉✉r✐♥❣ t❤❡ ✇♦r❦ ❢♦r s♦♠❡ ♠♦❞❡❧s ❛r❡ ❛❧s♦ ❝♦♠♣❛r❡❞ t❤❡ r❡s✉❧ts ✇✐t❤ ❞✐✛❡r❡♥t ♠♦❞❡❧s✳ ❚❤❡ r❡s✉❧ts ❛❧❧♦✇ ✉s t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛♥ ♦✉t❜r❡❛❦ ❛s ❛ r❡s✉❧t ♦❢ ❝❡rt❛✐♥ ❢❛❝t♦rs s✉❝❤ ❛s t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ♣r❡✈❡♥t✐✈❡ ♠❡❛s✉r❡s ❛♥❞ ✐s♦❧❛t✐♦♥✱ q✉❛r❛♥t✐♥❡ ❛♥❞ ✈❛❝❝✐♥❛t✐♦♥✱ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❜✐rt❤ ❛♥❞ ♠♦rt❛❧✐t② ❛♥❞ t❤❡ ❝♦♥s✐❞❡r❛t✐♦♥ ♦❢ ❤❡t❡r♦❣❡♥❡✐t②✳
❑❡②✇♦r❞s✿ ❊♣✐❞❡♠✐❝✱ ❉❡t❡r♠✐♥✐st✐❝ ▼♦❞❡❧s✱ ❊♣✐❞❡♠✐♦❧♦❣✐❝❛❧ ▼♦❞❡❧s✱ ❙t♦❝❤❛st✐❝ ▼♦❞❡❧s✳
❈♦♥t❡ú❞♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐
▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s ①✈
▲✐st❛ ❞❡ ❙✐♠❜♦❧♦❣✐❛ ①✈✐✐
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❖r❣❛♥✐③❛çã♦ ❞♦ ❘❡❧❛tór✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✷ ▼♦❞❡❧❛çã♦ ♠❛t❡♠át✐❝❛ ❞❡ ❡♣✐❞❡♠✐❛s ✺
✷✳✶ ▼♦❞❡❧❛çã♦ ❞❡t❡r♠✐♥íst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ▼♦❞❡❧❛çã♦ ❡st♦❝ást✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❚❡♦r✐❛ q✉❛❧✐t❛t✐✈❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸✳✶ ❖ ♣❧❛♥♦ ❞❡ ❢❛s❡ ❡ ♦ r❡tr❛t♦ ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸✳✷ P♦♥t♦s ❝rít✐❝♦s ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✸ ▼♦❞❡❧♦s ❉❡t❡r♠✐♥íst✐❝♦s ✶✸
✸✳✶ ▼♦❞❡❧♦ Pr❡❞❛❞♦r✲Pr❡s❛✿ ❙✐st❡♠❛ ❞❡ ▲♦t❦❛✲❱♦❧t❡rr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✶✳✶ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✷ ▼♦❞❡❧♦s ❈♦♠♣❛rt✐♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✸ ▼♦❞❡❧♦s SI ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸✳✸✳✶ ▼♦❞❡❧♦ SIR ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸✳✸✳✷ ▼♦❞❡❧♦ SIR ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸✳✸✳✸ ▼♦❞❡❧♦ SIS ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✸✳✹ ▼♦❞❡❧♦ SIRS ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✸✳✺ ▼♦❞❡❧♦ SIRS ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✸✳✻ ▼♦❞❡❧♦ SIRS ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ ✈❛❝✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✹ ▼♦❞❡❧♦s M SEIRS✱ M SEIR ❡ SEIR ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸✳✹✳✶ ▼♦❞❡❧♦ M SEIRS ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸✳✹✳✷ ▼♦❞❡❧♦ M SEIR ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✹✳✸ ▼♦❞❡❧♦ SEIR ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✹✳✹ ▼♦❞❡❧♦ SEIR ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ ✈❛❝✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✺ ▼♦❞❡❧♦s SIQS ❡ SIQR ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✸✳✺✳✶ ▼♦❞❡❧♦ SIQS ♣♦r ❛çã♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ s✐♠♣❧❡s ❡♠ ♠❛ss❛ ✳ ✳ ✳ ✳ ✳ ✹✹
✸✳✺✳✷ ▼♦❞❡❧♦ SIQR ♣♦r ❛çã♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ s✐♠♣❧❡s ❡♠ ♠❛ss❛ ✳ ✳ ✳ ✳ ✳ ✹✻
✹ ▼♦❞❡❧♦s ❊st♦❝ást✐❝♦s ✹✾
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✷ Pr♦♣❛❣❛çã♦ ❞❡ ✉♠❛ ■♥❢❡çã♦ ♥✉♠❛ ❙❛❧❛ ❞❡ ❆✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✷✳✶ ❙✐♠✉❧❛çã♦ s❡♠ ♠♦❜✐❧✐❞❛❞❡ ❡ ✈❛❝✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✷✳✷ ❙✐♠✉❧❛çã♦ ❝♦♠ ♠♦❜✐❧✐❞❛❞❡ ❡ s❡♠ ✈❛❝✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✷✳✸ ❙✐♠✉❧❛çã♦ ❝♦♠ ♠♦❜✐❧✐❞❛❞❡ ❡ ✈❛❝✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✷✳✹ ❊st✐♠❛t✐✈❛ ❞♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✐♥❢❡t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✷✳✺ ❊st✐♠❛t✐✈❛ ❞❛ t❛①❛ ❞❡ ✈❛❝✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✸ ▼♦❞❡❧♦ SIR ❊st♦❝ást✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✹✳✸✳✶ ❊①❡♠♣❧♦ ❞❡ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ▼♦❞❡❧♦ SIR ❊st♦❝ást✐❝♦ ✳ ✳ ✳ ✳ ✺✾
✹✳✹ ❙✐♠✉❧❛çã♦ ❞♦ ❝♦♥tá❣✐♦ ❞❛ ●r✐♣❡ ♥✉♠❛ ❈♦♠✉♥✐❞❛❞❡ ❊s❝♦❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✹✳✶ ▼♦❞❡❧♦s SIR ❡st♦❝ást✐❝♦ s✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✹✳✹✳✷ ▼♦❞❡❧♦s SIR ❡st♦❝ást✐❝♦ ❝♦♠ ✈❛❝✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✹✳✹✳✸ ▼♦❞❡❧♦ SIR ❡st♦❝ást✐❝♦ ❝♦♠ ✈❛❝✐♥❛çã♦ ❡ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ✻✸
✹✳✹✳✹ ▼♦❞❡❧♦ SIR ❡st♦❝ást✐❝♦ ❝♦♠ ❤❡t❡r♦❣❡♥❡✐❞❛❞❡ ❞❛ ♣♦♣✉❧❛çã♦ ✳ ✳ ✻✹
✹✳✹✳✺ ▼♦❞❡❧♦SIR ❡st♦❝ást✐❝♦ ❝♦♠ ❤❡t❡r♦❣❡♥❡✐❞❛❞❡ ❞❛ ♣♦♣✉❧❛çã♦ ❡ ✈❛✲
r✐❛çã♦ ❞♦ ♣❡rí♦❞♦ ❞❡ ❝♦♥tá❣✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✺ ❈♦♥❝❧✉sõ❡s ✻✼ ✺✳✶ Pr✐♥❝✐♣❛✐s ❖❜s❡r✈❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✺✳✷ Pr✐♥❝✐♣❛✐s ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✺✳✸ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
❇✐❜❧✐♦❣r❛✜❛ ✼✶
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ P❧❛♥♦ ❞❡ ❢❛s❡ x1x2 ❡ r❡tr❛t♦ ❞❡ ❢❛s❡ ❞❡ ❛❧❣✉♠❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛
❧✐♥❡❛r ✭✷✳✶✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷✳✷ ❊①❡♠♣❧♦s ❞❡ ♣♦♥t♦s s❡❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✸ ❊①❡♠♣❧♦ ❞❡ r❡tr❛t♦s ❞❡ ❢❛s❡ ❡①✐❜✐♥❞♦ ✭❛✮ ✉♠ ♣♦ç♦ ❡ ✭❜✮ ✉♠❛ ❢♦♥t❡✳ ✳ ✳ ✶✷
✸✳✶ ❊①❡♠♣❧♦ ❞❡ ♠♦❞❡❧♦ ❞❡ ▲♦t❦❛✲❱♦❧t❡rr❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✸✳✷ ❊①❡♠♣❧♦ ❞❡ ♠♦❞❡❧♦ ❞❡ ▲♦t❦❛✲❱♦❧t❡rr❛✱ ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♣ró①✐♠❛s ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✸✳✸ ▼♦❞❡❧♦ SIR✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸✳✹ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦ SIR✳ ❆♠❜❛s ❛s ✜❣✉r❛s tê♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐✲
❝✐❛❧ s0 = 0.99✱ i0 = 0.01❡ r0 = 0✳ ❆ ✜❣✉r❛ à ❡sq✉❡r❞❛✱ ✸✳✹❛ é ❢❡✐t❛ ♣❛r❛
♦ ❝❛s♦ ❞❡ β = 1.5❡ µ= 1 ❡ ❛ ✜❣✉r❛ à ❞✐r❡✐t❛✱ ✸✳✹❜ é ❢❡✐t❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡
β = 0.5 ❡ µ= 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸✳✺ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ R0 ❡♠ ❢✉♥çã♦ ❞♦ t♦t❛❧ ❞❡ ✐♥❢❡t❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸✳✻ ▼♦❞❡❧♦ SIR ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸✳✼ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦SIR ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧✳ ❆♠❜❛s ❛s ✜❣✉r❛s
tê♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ s0 = 0.99✱ i0 = 0.01 ❡ r0 = 0✳ ❆ ✜❣✉r❛
à ❡sq✉❡r❞❛✱ ✸✳✼❛ é ❢❡✐t❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ β = 1.5✱ µ = 1✱ δ = 0.001 ❡
γ = 0.0005❡ ❛ ✜❣✉r❛ à ❞✐r❡✐t❛✱ ✸✳✼❜ é ❢❡✐t❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡β = 0.5✱µ= 1✱ δ = 0.001 ❡ γ = 0.0005✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✽ ▼♦❞❡❧♦ SIS P❛❞rã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✸✳✾ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦SIS✳ ❆♠❜❛s ❛s ✜❣✉r❛s tê♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ s0 = 0.7 ❡ i0 = 0.3✳ ❆ ✜❣✉r❛ à ❡sq✉❡r❞❛ ✸✳✾❛ é ♦ ❝❛s♦ ❝♦♠ β = 1.4 ❡
α = 0.2❡ ❛ ✜❣✉r❛ à ❞✐r❡✐t❛ ✸✳✾❜ é ♦ ❝❛s♦ ❝♦♠ β = 0.6 ❡ α= 0.2✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✵ ▼♦❞❡❧♦ SIRS✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✶✶ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦ SIRS✳ ❆♠❜❛s ❛s ✜❣✉r❛s tê♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦
✐♥✐❝✐❛❧s0 = 0.9✱i0 = 0.1❡r0 = 0✳ ❆ ✜❣✉r❛ à ❡sq✉❡r❞❛✱ ✸✳✶✶❛ é ❢❡✐t❛ ♣❛r❛
♦ ❝❛s♦ ❞❡ β = 1✱ µ= 1
3 ❡ λ= 0.1✳ ◆❛ ✜❣✉r❛ à ❞✐r❡✐t❛✱ ✸✳✶✶❜ ♠❛♥t❡✈❡✲s❡
t♦❞♦s ♦s ♣❛râ♠❡tr♦s ✐❣✉❛✐s à ❡①❝❡çã♦ ❞❡ λ= 0.2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✶✷ ▼♦❞❡❧♦ SIRS ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸✳✶✸ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦SIRS ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧✳ ❆♠❜❛s ❛s ✜❣✉r❛s
tê♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ s0 = 0.9✱ i0 = 0.1 ❡ r0 = 0✳ ❆ ✜❣✉r❛ à
❡sq✉❡r❞❛✱ ✸✳✶✸❛ é ❢❡✐t❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ β = 1✱ µ = 13✱ λ = 0.1✱ δ = 0.001 ❡ γ = 0.0005✳ ◆❛ ✜❣✉r❛ à ❞✐r❡✐t❛✱ ✸✳✶✸❜ ♠❛♥t❡✈❡✲s❡ t♦❞♦s ♦s ♣❛râ♠❡tr♦s
✐❣✉❛✐s à ❡①❝❡çã♦ ❞❡ δ= 0.01❡ γ = 0.0001✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✶✹ ▼♦❞❡❧♦ SIRS ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ ✈❛❝✐♥❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✸✳✶✺ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦SIRS ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ ✈❛❝✐♥❛çã♦✳ ❆♠✲
❜❛s ❛s ✜❣✉r❛s tê♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ s0 = 0.9✱ i0 = 0.1 ❡ r0 = 0✳
❆ ✜❣✉r❛ à ❡sq✉❡r❞❛✱ ✸✳✶✺❛ é ❢❡✐t❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ β = 1✱ µ= 1
3✱ λ= 0.1✱
δ = 0.001✱ γ = 0.0005 ❡ ν = 0.1✳ ◆❛ ✜❣✉r❛ à ❞✐r❡✐t❛✱ ✸✳✶✺❜ ♠❛♥t❡✈❡✲s❡
t♦❞♦s ♦s ♣❛râ♠❡tr♦s ✐❣✉❛✐s à ❡①❝❡çã♦ ❞❡ ν = 0.3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✶✻ ▼♦❞❡❧♦ M SEIRS✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸✳✶✼ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦ M SEIRS✳ ❆ ✜❣✉r❛ t❡♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ m0 = 0✱ s0 = 0.85✱ e0 = 0.1✱ i0 = 0.05❡ r0 = 0✳ ❖s r❡st❛♥t❡s ♣❛râ♠❡tr♦s
sã♦ ρ= 0.01✱ β = 1.5✱ λ= 0.1✱ µ= 0.3✱ε= 0.2✱δ = 0.001 ❡ γ = 0.0005✳ ✸✻
✸✳✶✽ ▼♦❞❡❧♦ M SEIR✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✶✾ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦ M SEIR✳ ❆ ✜❣✉r❛ t❡♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ m0 = 0✱ s0 = 0.85✱ e0 = 0.1✱ i0 = 0.05❡ r0 = 0✳ ❖s r❡st❛♥t❡s ♣❛râ♠❡tr♦s
sã♦ γ = 0.0005✱ δ= 0.001✱ β= 1.5✱µ= 0.3✱ ε= 0.2 ❡ρ❂✵✳✵✶✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸✳✷✵ ▼♦❞❡❧♦ SEIR✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✷✶ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦SEIR✳ ❆ ✜❣✉r❛ t❡♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ s0 =
0.85✱ e0 = 0.1✱ i0 = 0.05 ❡ r0 = 0✳ ❖s r❡st❛♥t❡s ♣❛râ♠❡tr♦ sã♦ β = 1✱
µ= 0.3 ❡ε = 0.2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✷✷ ▼♦❞❡❧♦ SEIR ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ ✈❛❝✐♥❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✷✸ ❊①❡♠♣❧♦ ❞♦ ♠♦❞❡❧♦ SEIR ❝♦♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ ✈❛❝✐♥❛çã♦✳ ❆
✜❣✉r❛ t❡♠ ❝♦♠♦ ❝♦♥✜❣✉r❛çã♦ ✐♥✐❝✐❛❧ s0 = 0.85✱ e0 = 0.1✱ i0 = 0.05 ❡
r0 = 0✳ ❖s r❡st❛♥t❡s ♣❛râ♠❡tr♦s sã♦ β = 1✱ µ= 0.3✱ ε = 0.2✱ δ = 0.001✱
γ = 0.0005✱ ❞= 0.0001 ❡ν = 0.1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✸✳✷✹ ▼♦❞❡❧♦ SIQS ♣♦r ❛çã♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ s✐♠♣❧❡s ❡♠ ♠❛ss❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✸✳✷✺ ▼♦❞❡❧♦SIQS♣♦r ❛çã♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ s✐♠♣❧❡s ❡♠ ♠❛ss❛ ❝♦♠ ♦s s❡❣✉✐♥t❡s
✈❛❧♦r❡s ♣❛r❛ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✿ s0 = 0.9✱ i0 = 0.1 ❡ q0 = 0✳ ❖s
r❡st❛♥t❡s ♣❛râ♠❡tr♦s sã♦ θ = 0.3✱ δ = 0.001✱ γ = 0.0001✱ d = 0.0005✱
β = 1✱α = 0.1❡ ω = 0.1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✸✳✷✻ ▼♦❞❡❧♦ SIQR ♣♦r ❛çã♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ s✐♠♣❧❡s ❡♠ ♠❛ss❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✸✳✷✼ ▼♦❞❡❧♦ SIQR ♣♦r ❛çã♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ s✐♠♣❧❡s ❡♠ ♠❛ss❛ ❝♦♠ s0 = 0.9✱
i0 = 0.1✱q0 = 0❡r0 = 0✳ ❖s r❡st❛♥t❡s ♣❛râ♠❡tr♦s sã♦θ = 0.3✱δ= 0.001✱
γ1 = 0.0001✱ γ2 = 0.00005✱ d= 0.0005✱ β = 1✱ µ= 0.1❡ τ = 0.1✳ ✳ ✳ ✳ ✳ ✹✽
✹✳✶ ❉✉❛s s✐♠✉❧❛çã♦ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❛ ❣r✐♣❡ ♥✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ❝♦♠ ♥ = 6 ❡ ♠ = 5✳ ❆♠❜❛s ❛s ❋✐❣✉r❛s ✹✳✶❛ ❡ ✹✳✶❜ ❛♣r❡s❡♥t❛♠ ❛ ♣r♦♣♦rçã♦ ❞❡ ✐♥❞✐✈í❞✉♦s s✉s❝❡tí✈❡✐s✱ ✐♥❢❡t❛❞♦s ❡ r❡❝✉♣❡r❛✲ ❞♦s ♣♦r ❞✐❛ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ♠❡s♠♦s ♣❛râ♠❡tr♦s ❦= 4 ❡ τ = 0.2✳ ✳ ✳ ✳ ✳ ✺✷
✹✳✷ ❉✉❛s s✐♠✉❧❛çã♦ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❛ ❣r✐♣❡ ♥✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ❝♦♠ ❞✐♠❡♥sã♦ ♥= 6♣♦r ♠ = 5❡ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♠♦✲ ❜✐❧✐❞❛❞❡ ❞❡ ❧✉❣❛r✳ ❆♠❜❛s ❛s ❋✐❣✉r❛s ✹✳✷❛ ❡ ✹✳✷❜ ❛♣r❡s❡♥t❛♠ ❛ ♣r♦♣♦rçã♦ ❞❡ ✐♥❞✐✈í❞✉♦s s✉s❝❡tí✈❡✐s✱ ✐♥❢❡t❛❞♦s ❡ r❡❝✉♣❡r❛❞♦s ♣♦r ❞✐❛ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ♠❡s♠♦s ♣❛râ♠❡tr♦s ❦= 4✱τ = 0.2 ❡ δ= 0.2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✹✳✸ ❉✉❛s s✐♠✉❧❛çã♦ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❛ ❣r✐♣❡ ♥✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ❞❡ ❞✐♠❡♥sã♦ ♥ = 6 ♣♦r ♠ = 5 ❝♦♠ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♠♦❜✐❧✐❞❛❞❡ ❞❡ ❧✉❣❛r ❡ ✈❛❝✐♥❛çã♦✳ ❆♠❜❛s ❛s ❋✐❣✉r❛s ✹✳✸❛ ❡ ✹✳✸❜ ❛♣r❡s❡♥✲ t❛♠ ❛ ♣r♦♣♦rçã♦ ❞❡ ✐♥❞✐✈í❞✉♦s s✉s❝❡tí✈❡✐s✱ ✐♥❢❡t❛❞♦s ❡ r❡❝✉♣❡r❛❞♦s ♣♦r ❞✐❛ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ♠❡s♠♦s ♣❛râ♠❡tr♦s ❦= 4✱ τ = 0.2✱δ = 0.2 ❡ν = 0.1✳ ✺✹
✹✳✹ ❋r❛çã♦ ✜♥❛❧ ❞❡ ✐♥❢❡t❛❞♦s r❡s✉❧t❛♥t❡ ❞❡ ✶✵✵✵ r❡♣❡t✐çõ❡s ❝♦♠ ♥= 6✱ ♠= 5✱
τ = 0.2 ❡ν ✈❛r✐á✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✹✳✺ ❊①❡♠♣❧♦s ❞❡ ✉♠❛ s✐♠✉❧❛çã♦ ♣❛r❛ ♦ ▼♦♥t❡ ❈❛r❧♦✳ Pr♦♣♦rçã♦ ❞❡ ✐♥❞✐✲ ✈í❞✉♦s ✐♥❢❡t❛❞♦s ♣♦r ❞✐❛ ♥✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ♥ = 6 ❡ ♠ = 5✱ ❝♦♠ ✉♠❛ ❧✐♠✐t❛çã♦ ❞❡ ✐♥❢❡t❛❞♦s ❞❡ ✷✵✪ ❞❛ ♣♦♣✉❧❛çã♦✱ ❝♦♠ τ = 0.2❡ δ= 0.2✳ ✳ ✳ ✺✼
✹✳✻ ❊①❡♠♣❧♦ ❞❡ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ♠♦❞❡❧♦ SIR ❡st♦❝ást✐❝♦✱ ❝♦♥s✐❞❡r❛♥❞♦
◆ = 1000✱ I0 = 1✱ β = 1.5 ❡ k = 4✳ ➚ ❡sq✉❡r❞❛✱ ✉♠❛ s✐♠✉❧❛çã♦ ❞❛
❡✈♦❧✉çã♦ ❞♦s ♥ú♠❡r♦s ❞❡ s✉s❝❡tí✈❡✐s✱ ✐♥❢❡❝✐♦s♦s ❡ r❡❝✉♣❡r❛❞♦s ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ s✉rt♦ ❞❡ ❣r✐♣❡✳ ➚ ❞✐r❡✐t❛✱ ❤✐st♦❣r❛♠❛ ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ ✶✵ ♠✐❧ s✐♠✉❧❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✼ ❙✐♠✉❧❛çã♦ ❝♦♠ ♦ ♠♦❞❡❧♦ SIR ❡st♦❝ást✐❝♦ s✐♠♣❧❡s ❝♦♠ ❦ = 4 ❡ β = 1✳
◆❛ ❋✐❣✉r❛ ✹✳✼❛ t❡♠✲s❡ I0 = 1 ❡ ♥❛ ❋✐❣✉r❛ ✹✳✼❜ t❡♠✲s❡ I0 = 10✳ ✳ ✳ ✳ ✳ ✳ ✻✶
✹✳✽ ❙✐♠✉❧❛çã♦ ❝♦♠ ♦ ♠♦❞❡❧♦ SIR ❡st♦❝ást✐❝♦ s✐♠♣❧❡s ❝♦♠ ❦ = 4✱ β = 2 ❡
I0 = 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✹✳✾ ❙✐♠✉❧❛çã♦ ❝♦♠ ♦ ♠♦❞❡❧♦ SIR ❡st♦❝ást✐❝♦ ❝♦♠ ✈❛❝✐♥❛çã♦✿ ❦ = 4✱ β = 2 ❡ I0 = 1✳ ◆❛ ❋✐❣✉r❛ ✹✳✾❛ t❡♠✲s❡ ν= 0.1❡ ♥❛ ❋✐❣✉r❛ ✹✳✾❜ t❡♠✲s❡ ν = 0.2✳ ✻✸
✹✳✶✵ ❙✐♠✉❧❛çã♦ ❝♦♠ ♦ ♠♦❞❡❧♦SIR ❡st♦❝ást✐❝♦ ❝♦♠ ✈❛❝✐♥❛çã♦ ❡ ❞✐♥â♠✐❝❛ ♣♦✲
♣✉❧❛❝✐♦♥❛❧✿ ❦ = 4✱ β = 2 ❡ ν = 0.1✳ ◆❛ ❋✐❣✉r❛ ✹✳✶✵❛ t❡♠✲s❡ δ = 4 ❡
γ = 0.01❡ ♥❛ ❋✐❣✉r❛ ✹✳✶✵❜ t❡♠✲s❡ δ= 2 ❡γ = 0.1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✹✳✶✶ ❙✐♠✉❧❛çã♦ ❝♦♠ ♦ ♠♦❞❡❧♦SIR❡st♦❝ást✐❝♦ ❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ❤❡t❡r♦❣é♥❡❛✿
❦ = 4✱ ν = 0.1 ❡ nh = 30✳ ◆❛ ❋✐❣✉r❛ ✹✳✶✶❛ t❡♠✲s❡ βh = 3 ❡ βg = 1❀ ♥❛ ❋✐❣✉r❛ ✹✳✶✶❜ t❡♠✲s❡ βh = 1 ❡ βg = 0.5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✶✷ ❙✐♠✉❧❛çã♦ ❝♦♠ ♦ ♠♦❞❡❧♦SIR❡st♦❝ást✐❝♦ ❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ❤❡t❡r♦❣é♥❡❛
❡ ✈❛r✐❛çã♦ ❞♦ ♣❡rí♦❞♦ ❞❡ ❝♦♥tá❣✐♦✿ βh = 3✱ βg = 1 ❡ nh = 30✳ ◆❛ ❋✐❣✉r❛ ✹✳✶✷❛ t❡♠✲s❡ ❦❂✹❀ ♥❛ ❋✐❣✉r❛ ✹✳✶✷❜ t❡♠✲s❡ ❦❂✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s
❊❉❖ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s
▼❙❊■❘ ▼♦❞❡❧♦ ❞♦ t✐♣♦ ■♠✉♥❡s✲❙✉s❝❡tí✈❡✐s✲❊①♣♦st♦s✲■♥❢❡t❛❞♦s✲❘❡❝✉♣❡r❛❞♦s ▼❙❊■❘❙ ▼♦❞❡❧♦ ❞♦ t✐♣♦ ■♠✉♥❡s✲❙✉s❝❡tí✈❡✐s✲❊①♣♦st♦s✲■♥❢❡t❛❞♦s✲❘❡❝✉♣❡r❛❞♦s✲
❙✉s❝❡tí✈❡✐s
❙❆❘❙ ❙í♥❞r♦♠❡ ❘❡s♣✐r❛tór✐❛ ❆❣✉❞❛
❙❊■❘ ▼♦❞❡❧♦ ❞♦ t✐♣♦ ❙✉s❝❡tí✈❡✐s✲❊①♣♦st♦s✲■♥❢❡t❛❞♦s✲❘❡❝✉♣❡r❛❞♦s ❙■◗❘ ▼♦❞❡❧♦ ❞♦ t✐♣♦ ❙✉s❝❡tí✈❡✐s✲■♥❢❡t❛❞♦s✲◗✉❛r❡♥t❡♥❛✲❘❡❝✉♣❡r❛❞♦s
❙■◗❙ ▼♦❞❡❧♦ ❞♦ t✐♣♦ ❙✉s❝❡tí✈❡✐s✲■♥❢❡t❛❞♦s✲◗✉❛r❡♥t❡♥❛✲❙✉s❝❡tí✈❡✐s ❙■❙ ▼♦❞❡❧♦ ❞♦ t✐♣♦ ❙✉s❝❡tí✈❡✐s✲■♥❢❡t❛❞♦s✲❙✉s❝❡tí✈❡✐s
❙■❘ ▼♦❞❡❧♦ ❞♦ t✐♣♦ ❙✉s❝❡tí✈❡✐s✲■♥❢❡t❛❞♦s✲❘❡❝✉♣❡r❛❞♦s
❙■❘❙ ▼♦❞❡❧♦ ❞♦ t✐♣♦ ❙✉s❝❡tí✈❡✐s✲■♥❢❡t❛❞♦s✲❘❡❝✉♣❡r❛❞♦s✲❙✉s❝❡tí✈❡✐s
▲✐st❛ ❞❡ ❙í♠❜♦❧♦s
S ❝❧❛ss❡ ❞♦s ❙✉s❝❡tí✈❡✐s I ❝❧❛ss❡ ❞♦s ■♥❢❡❝✐♦s♦s R ❝❧❛ss❡ ❞♦s ❘❡❝✉♣❡r❛❞♦s N ◆ú♠❡r♦ t♦t❛❧ ❞❛ ♣♦♣✉❧❛çã♦ m, s, e, i, r, q, s ❋r❛çõ❡s ❞❛ ♣♦♣✉❧❛çã♦
β ◆ú♠❡r♦s ♠é❞✐♦ ❞❡ ❝♦♥t❛❝t♦s ❛❞❡q✉❛❞♦s ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ α ❚❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❞♦ ♠♦❞❡❧♦SIS ❡ SIQS
µ ❚❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❞♦s r❡st❛♥t❡s ♠♦❞❡❧♦s λ ❚❛①❛ ❞❡ ✐♥❞✐✈í❞✉♦s q✉❡ ♣❡r❞❡♠ ✐♠✉♥✐❞❛❞❡ δ ❚❛①❛ ❞❡ ♥❛t❛❧✐❞❛❞❡
γ ❡ ❞ ❚❛①❛s ❞❡ ♠♦rt❛❧✐❞❛❞❡ ν ❚❛①❛ ❞❡ ❱❛❝✐♥❛çã♦
ρ ❚❛①❛ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ♣❛r❛ ❢♦r❛ ❞❛ ❝❧❛ss❡ ❞♦s ✐♠✉♥❡s ❞♦s ♠♦❞❡❧♦s
▼❙❊■❘❙ ❡ ▼❙❊■❘
ε ❚❛①❛ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ♣❛r❛ ❢♦r❛ ❞❛ ❝❧❛ss❡ ❞♦s ❡①♣♦st♦s ❞♦s ♠♦❞❡❧♦s
▼❙❊■❘❙✱ ▼❙❊■❘ ❡ ❙❊■❘
θ ❚❛①❛ ❞❡ r❡♠♦çã♦ ❞♦ ❝♦♠♣❛rt✐♠❡♥t♦ ■ ❞♦s ♠♦❞❡❧♦s ❙■◗❙ ❡ ❙■◗❘ ω ❚❛①❛ ❞❡ r❡❝✉♣❡r❛çã♦ ❡ r❡t♦r♥♦ ❞♦ ❝♦♠♣❛rt✐♠❡♥t♦ ◗ ♣❛r❛ ♦ ❙ ❞♦
♠♦❞❡❧♦ ❙■◗❙
τ ❚❛①❛ ❞❡ r❡♠♦çã♦ ❞♦ ❝♦♠♣❛rt✐♠❡♥t♦ ◗ ❞♦ ♠♦❞❡❧♦ ❙■◗❘
❦ P❡rí♦❞♦ ❞❡ ❝♦♥tá❣✐♦ ❞♦s ♠♦❞❡❧♦s ❊st♦❝ást✐❝♦s
R0 ◆ú♠❡r♦ ❞❡ r❡♣r♦❞✉çã♦ ❜ás✐❝♦
σ ◆ú♠❡r♦ ❞❡ ❝♦♥t❛❝t♦
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❯♠❛ ❡♣✐❞❡♠✐❛ é ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ✉♠❛ ❞♦❡♥ç❛ ✐♥❢❡❝✐♦s❛ ❡ ❝♦♥t❛❣✐♦s❛ q✉❡ s❡ ❞❡s❡♥✈♦❧✈❡ ♥✉♠ ❧♦❝❛❧ ♦✉ r❡❣✐ã♦✱ ❞❡ ❢♦r♠❛ rá♣✐❞❛✱ ❢❛③❡♥❞♦ ♥✉♠ ❝✉rt♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ✉♠ ♥ú♠❡r♦ ❡❧❡✈❛❞♦ ❞❡ ✈✐t✐♠❛s ❡♠ s✐♠✉❧tâ♥❡♦ ❬✸✱✷✹❪✳
❆ ❡♣✐❞❡♠✐♦❧♦❣✐❛ é ❛ ❝✐ê♥❝✐❛ q✉❡ ❡st✉❞❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❞♦❡♥ç❛s ✐♥❢❡❝✐♦✲❝♦♥t❛❣✐♦s❛s✱ ♦s ❢❛t♦r❡s ❝♦♥❞✐❝✐♦♥❛♥t❡s ❡ ❞❡t❡r♠✐♥❛♥t❡s ♥❛ ♣♦♣✉❧❛çã♦ ❤✉♠❛♥❛✳ ❯♠❛ ❞❛s s✉❛s ❢✉♥çõ❡s é ❞❡s❝r❡✈❡r ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ❞♦❡♥ç❛✱ ♦✉ s❡❥❛✱ ❞❡s❝♦❜r✐r ❞❡ q✉❡ ♣❛❞❡❝❡♠ ♦s ✐♥❞✐✈í❞✉♦s✱ ♦♥❞❡ ❡ q✉❛♥❞♦✳ ❆ ❡♣✐❞❡♠✐♦❧♦❣✐❛ ✐❞❡♥t✐✜❝❛ ❛✐♥❞❛✱ ❛s ❝❛✉s❛s ♦✉ ❢❛t♦r❡s ❞❡ r✐s❝♦ ♣❛r❛ ❛s ❞♦❡♥ç❛s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ s❡ ♦❜t❡r ❛ r❛③ã♦ ♣❡❧♦ q✉❛❧ ♥❡♠ t♦❞♦s ♦s ✐♥❞✐✈í❞✉♦s tê♠ ❛ ♠❡s♠❛ ❞♦❡♥ç❛ ❞❡ ♠❛♥❡✐r❛ ✉♥✐❢♦r♠❡✳ ❯♠❛ ♦✉tr❛ ❢✉♥çã♦ é ❝♦♥str✉✐r ❡ t❡st❛r t❡♦r✐❛s ❜❡♠ ❝♦♠♦ ♣❧❛♥❡❛r✱ ✐♠♣❧❡♠❡♥t❛r ❡ ❛✈❛❧✐❛r ♦ ❝♦♥tr♦❧♦✱ ❞❡t❡çã♦ ❡ ♣r♦❣r❛♠❛s ❞❡ ♣r❡✈❡♥çã♦ ❬✷✶❪✳ ❚❡♥❞♦ ❡♠ ❝♦♥t❛ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❡♣✐❞❡♠✐♦❧♦❣✐❛ ♥♦ â♠❜✐t♦ ❞❛ s❛ú❞❡✱ tê♠ s✐❞♦ ❢❡✐t♦s ✈ár✐♦s ❡st✉❞♦s ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ s❡ ❞❡s❡♥✈♦❧✈❡r ♥♦✲ ✈♦s ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s q✉❡ ♣♦ss❛♠ ❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❡rr❛❞✐❝❛çã♦ ❞❡ ❞♦❡♥ç❛s ✐♥❢❡❝✐♦s❛s✳
❆ss✐♠✱ ♦s ♣r✐♠❡✐r♦s ♠♦❞❡❧♦s ❡♣✐❞é♠✐❝♦s ✉s❛❞♦s ❡r❛♠ ❞❡t❡r♠✐♥íst✐❝♦s ❬✸✱✶✵✱✷✶❪✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ ♠❡❝❛♥✐s♠♦ ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❛s ❞♦❡♥ç❛s ♣♦r s✐ só ♥ã♦ é s✉✜❝✐❡♥t❡✳ ➱ ♥❡❝❡ssár✐♦ ❝♦♥s✐❞❡r❛r ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞♦❡♥ç❛ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ q✉❛♥t✐t❛t✐✈♦✱ ❞❡ ♠♦❞♦ ❛ q✉❡ s❡ ❞❡❝✐❞❛ s♦❜r❡ ✈❛❝✐♥❛çõ❡s ♦✉ ♦✉tr❛s ♠❡❞✐❞❛s ♣r❡✈❡♥t✐✈❛s ♣❛r❛ ❛ s✉❛ ❝♦♥t❡♥çã♦ ❬✸✱✷✶❪✳ P♦st♦ ✐ss♦✱ ♦s ♠♦❞❡❧♦s s✐♠♣❧❡s ❢♦r❛♠ ❣❡♥❡r❛❧✐③❛❞♦s ❡♠ ✈ár✐♦s ❝❛♠✐♥❤♦s ♣❛r❛ ♦s t♦r♥❛r ♠❛✐s r❡❛✐s✳ ❆❧❣✉♠❛s ❞❡st❛s ❣❡♥❡r❛❧✐③❛çõ❡s ❡r❛♠ ♣❛r❛ ♣❡r♠✐t✐r ❛ ✉♠❛
❝♦♠✉♥✐❞❛❞❡ ♦♥❞❡ ❡①✐st❡ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ✐♥❞✐✈í❞✉♦s✱ ❛ ♠✐st✉r❛ ❡♥tr❡ ✐♥❞✐✈í❞✉♦s ❞❡ ❢♦r♠❛ ♥ã♦ ✉♥✐❢♦r♠❡ ♣❛r❛ ♣❡r♠✐t✐r✱ ♣♦r ❡①❡♠♣❧♦✱ ✈❛r✐❛çõ❡s s❛③♦♥❛✐s✳
❙❡❣✉♥❞♦ ❍❡t❤❝♦t❡ ❬✶✹❪✱ ♦s ♠♦❞❡❧♦s ❝♦♠ ♣♦♣✉❧❛çã♦ ❞❡ t❛♠❛♥❤♦ ❝♦♥st❛♥t❡ ♥ã♦ sã♦ ❛❞❡q✉❛❞♦s q✉❛♥❞♦ ♦s ♣❛rt♦s ❡ ❛s ♠♦rt❡s ♥ã♦ sã♦ ❡q✉✐❧✐❜r❛❞❛s ♦✉ q✉❛♥❞♦ ❛s ♠♦rt❡s r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛ ❞♦❡♥ç❛ sã♦ s✐❣♥✐✜❝❛t✐✈❛s✳ ❆s ❞♦❡♥ç❛s ✐♥❢❡❝✐♦s❛s ♠✉✐t❛s ✈❡③❡s tê♠ ❣r❛♥❞❡ ✐♠♣❛❝t♦ s♦❜r❡ ♦ t❛♠❛♥❤♦ ❞❛s ♣♦♣✉❧❛çõ❡s ❡ ❡✈❡♥t♦s ❤✐stór✐❝♦s✳ ❆s ❞♦❡♥ç❛s ✐♥❢❡❝✐♦s❛s q✉❡ tê♠ ❞❡s❡♠♣❡♥❤❛❞♦ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ ❞❡❜✐❧✐t❛çã♦ ❡ r❡❣✉❧❛çã♦ ❞❛s ♣♦♣✉❧❛çõ❡s ❤✉♠❛♥❛s ✐♥❝❧✉❡♠ ♦ s❛r❛♠♣♦✱ ❛ t✉❜❡r❝✉❧♦s❡✱ ❛ ♠❛❧ár✐❛ ❡ ❛ ♣♥❡✉♠♦♥✐❛✳
❖s ♠♦❞❡❧♦s ❝♦♠ ✉♠ t❛♠❛♥❤♦ ❞❡ ♣♦♣✉❧❛çã♦ ✈❛r✐á✈❡❧ ♠✉✐t❛s ✈❡③❡s sã♦ ♠❛✐s ❞✐❢í❝❡✐s ❞❡ ❛♥❛❧✐s❛r ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣♦rq✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ♣♦♣✉❧❛çã♦ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❞✐❝✐♦♥❛❧ q✉❡ é ❣♦✈❡r♥❛❞❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✳
❖✉tr❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦s ♠♦❞❡❧♦s ❡♣✐❞é♠✐❝♦s ❞❡t❡r♠✐♥íst✐❝♦s ❢♦✐ ♦ ❡st✉❞♦ ❞❡ ♠♦❞❡❧♦s ❡♣✐❞é♠✐❝♦s ❡st♦❝ást✐❝♦s✳ ❯♠ ♠♦❞❡❧♦ ❡st♦❝ást✐❝♦ é ♣r❡❢❡r❡♥❝✐❛❧♠❡♥t❡ ✉s❛❞♦ q✉❛♥❞♦ s❡ ❡st✉❞❛ ✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ♣❡q✉❡♥❛✳ ❙ã♦ ♠♦❞❡❧♦s ♠❛✐s r❡❛❧✐st❛s ✉♠❛ ✈❡③ q✉❡ ✐♥❝❧✉❡♠ ✈ár✐❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛♠❜♦s ♦s ♠♦❞❡❧♦s ❡♣✐❞é♠✐❝♦s tê♠ ♣❛♣❡✐s ✐♠♣♦rt❛♥t❡s ❛ ❞❡s❡♠♣❡♥❤❛r❡♠✱ ❝♦♥❢♦r♠❡ ♦ q✉❡ s❡ ♣r❡t❡♥❞❡ ❬✶✵❪✳
❉❡st❡ ♠♦❞♦✱ ♥❡st❡ tr❛❜❛❧❤♦ ❝♦♠❡ç❛✲s❡ ♣♦r r❡s✉♠✐r ♦ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ✈á✲ r✐♦s ❛✉t♦r❡s✱ ❛♦ ❧♦♥❣♦ ❞❛s ú❧t✐♠❛s ❞é❝❛❞❛s✱ ♥❛ ár❡❛ ❞♦s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡ ❡st♦✲ ❝ást✐❝♦s✳ ❉❡ s❡❣✉✐❞❛✱ ❢❛③✲s❡ ✉♠❛ s✐st❡♠❛t✐③❛çã♦ ❞♦s ♣r✐♥❝✐♣❛✐s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡ ❢❛③✲s❡ ❛ s✉❛ ❝♦♥✈❡rsã♦ ❡♠ ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s✳
P❛r❛ ❛❧❣✉♥s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s✱ ♦s q✉❡ ✐♥❝❧✉❡♠ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧ ♣r♦❝❡❞❡✉✲ s❡ ❛ ❛❧t❡r❛çõ❡s ❞❡ ❢♦r♠❛ ❛ ❛♣❡r❢❡✐ç♦❛✲❧♦s✳ ❊st❛s ❛❧t❡r❛çõ❡s ♣❡r♠✐t❡♠ ❢❛③❡r ✈❛r✐❛r ♦s ♣❛✲ râ♠❡tr♦s ❞❡ ♥❛t❛❧✐❞❛❞❡ ❡ ♠♦rt❛❧✐❞❛❞❡✳ P❛r❛ t♦❞♦s ♦s ♠♦❞❡❧♦s ❢❡③✲s❡ ❛♣❧✐❝❛çõ❡s ❞❡ ❝❛s♦s ♣rát✐❝♦s ❞❡ ❡st✉❞♦✳ ❈♦♠♣❛r❛✲s❡ t❛♠❜é♠ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❝♦♠ ❞✐❢❡r❡♥t❡s ♠♦❞❡❧♦s✳ ❖s r❡s✉❧t❛❞♦s ♠♦str❛♠ q✉❡ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠❛ t❛①❛ ❞❡ ✈❛❝✐♥❛çã♦ ❡ ✉♠ ❝♦♠♣❛rt✐♠❡♥t♦ ❞❡ q✉❛r❡♥t❡♥❛ ❛❥✉❞❛ ❛ ❞✐♠✐♥✉✐r ♦ ❛❧❝❛♥❝❡ ❞❡ ✉♠❛ ❡♣✐❞❡♠✐❛✳ ❯♠ ❡st✉❞♦ s❡♠❡❧❤❛♥t❡ ❢♦✐ ❢❡✐t♦ ♣♦r ❘♦❝❤❛ ❬✷✶❪ q✉❡ r❡s✉♠✐✉ ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡①✐s✲ t❡♥t❡s ♥❛ ❧✐t❡r❛t✉r❛✳ Pr♦♣ôs ♠♦❞❡❧♦s ♠❛✐s ❝♦♠♣❧❡t♦s ♣❛r❛ ❡st✉❞❛r ❛ ♠❛❧ár✐❛ ❡ ❛ ❙■❉❆✳ ❚s✉❥✐❣✉❝❤✐ ❬✷✹❪ ❡st✉❞♦✉ ♠♦❞❡❧♦s ❛♣❧✐❝❛❞♦s à ❡♣✐❞❡♠✐♦❧♦❣✐❛✱ ❝♦♠♦ ♦s ♠♦❞❡❧♦sSIS✱SIR
❡SIRS✱ ❞❡ ♠♦❞♦ ❛ ❞❡s❡♥✈♦❧✈❡r ✉♠ ♠♦❞❡❧♦ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ❞✐♥â♠✐❝❛ ❞❛ ❞✐s♣❡rsã♦ ❞❛
■♥✢✉❡♥③❛ ❆ ✭❍✶◆✶✮ ♥✉♠❛ ♣♦♣✉❧❛çã♦✳
P❛r❛ ♦s ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s ❡st✉❞♦✉✲s❡ ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ✉♠❛ ❡♣✐❞❡♠✐❛ ♥✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛✱ ❛♣❧✐❝❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ❊st❛ s✐♠✉❧❛çã♦ t❡♠ ❡♠ ❝♦♥t❛ ❛ ❝♦♥s✐❞❡r❛çã♦ ❞❡ ♠♦❜✐❧✐❞❛❞❡✱ ❛ ✈❛❝✐♥❛çã♦✱ ❛ ❡st✐♠❛t✐✈❛ ❞♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✐♥❢❡t❛❞♦s ❡ ❛ ❡st✐♠❛t✐✈❛ ❞❛ t❛①❛ ❞❡ ✈❛❝✐♥❛çã♦✳
Pr♦❝❡❞❡✉✲s❡ t❛♠❜é♠ ❛♦ ❡st✉❞♦ ❞♦ ❝♦♥tá❣✐♦ ❞❛ ●r✐♣❡ ♥✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ❛ ♣❛rt✐r ❞❛ s✐♠✉❧❛çã♦ ❞♦ ♠♦❞❡❧♦ SIR ✐♥❝♦r♣♦r❛♥❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡st♦❝ást✐❝♦✳ ❆ ♣♦♣✉❧❛çã♦
❡st✉❞❛❞❛ é ❝♦♥st✐t✉í❞❛ ♣♦r ✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ❡st✉❞❛♥t✐❧✱ ❝♦♠♦ ❛ ❞❛ ❊❙❚✐●✱ ❝♦♠♣♦st❛ ♣♦r ✷✵✵✵ ✐♥❞✐✈í❞✉♦s✳ ❆s s✐♠✉❧❛çõ❡s ❡❢❡t✉❛❞❛s tê♠ ❡♠ ❝♦♥t❛ ❛ ✈❛r✐❛çã♦ ❞♦ ♥ú♠❡r♦ ✐♥✐❝✐❛❧ ❞❡ ✐♥❢❡t❛❞♦s✱ ❛ ✈❛r✐❛çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❝♦♥t❛❝t♦s ❛❞❡q✉❛❞♦s✱ ❛ ✈❛❝✐♥❛çã♦✱ ❛ ❞✐♥â♠✐❝❛ ♣♦♣✉❧❛❝✐♦♥❛❧✱ ❛ ❤❡t❡r♦❣❡♥❡✐❞❛❞❡ ❞❛ ♣♦♣✉❧❛çã♦ ❡ ❛ ✈❛r✐❛çã♦ ❞♦ ♣❡rí♦❞♦ ❞❡ ❝♦♥tá❣✐♦ ♥✉♠❛ ♣♦♣✉❧❛çã♦ ❤❡t❡r♦❣é♥❡❛✳ ❯♠ ❡st✉❞♦ s❡♠❡❧❤❛♥t❡ ❢♦✐ r❡❛❧✐③❛❞♦ ♣♦r ❆♥❛❜❡❧❛ ❆❣✉✐❛r ❡ P❛✉❧❛ ❆❧✈❡s ❬✶❪ q✉❡ ❡st✉❞❛r❛♠ ♦ ❝♦♥tá❣✐♦ ❞❛ ❣r✐♣❡ ♣❡ss♦❛ ❛ ♣❡ss♦❛✱ ♥❛ ❝♦♠✉♥✐❞❛❞❡ ❞❛ ❊❙❚✐●✱ ❛♣❧✐❝❛♥❞♦ ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s ❡ ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s t❡♥❞♦ ❡♠ ❝♦♥t❛ ❛ ✈❛r✐❛çã♦ ❞♦ ♥ú♠❡r♦ ✐♥✐❝✐❛❧ ❞❡ ✐♥❢❡t❛❞♦s✱ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥t❛❝t♦s ❛❞❡q✉❛❞♦s ❡ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✈❛❝✐♥❛çã♦✳ ❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣❡❧♦s ❛✉t♦r❡s ✈ã♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❞❡st❡ tr❛❜❛❧❤♦✳
✶✳✶ ❖❜❥❡t✐✈♦s
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ♠♦❞❡❧❛r ❝❛s♦s ♣r♦♣♦st♦s ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ♠♦❞❡❧♦s ♠❛t❡✲ ♠át✐❝♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡ ❡st♦❝ást✐❝♦s ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✉rt♦ ❡♠ ❢✉♥çã♦ ❞❡ ✈ár✐♦s ❢❛t♦r❡s ❝♦♠♦ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ♠❡❞✐❞❛s ♣r❡✈❡♥t✐✈❛s ❝♦♠♦ ❛ q✉❛r❡♥t❡♥❛ ❡ ❛ ✈❛❝✐♥❛çã♦✱ ❛ ✈❛r✐❛çã♦ ❞❛ ❞✐♥â♠✐❝❛ ❞❛ ♣♦♣✉❧❛çã♦ ❡ ❛ ❤❡t❡r♦❣❡♥❡✐❞❛❞❡✳
✶✳✷ ❖r❣❛♥✐③❛çã♦ ❞♦ ❘❡❧❛tór✐♦
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ é ❢❡✐t❛ ✉♠❛ ♣❡q✉❡♥❛ ✐♥tr♦❞✉çã♦ ❛♦ t❡♠❛ ❞❡ ♠♦❞❡❧❛çã♦ ❞❡ ❞♦❡♥ç❛s ❡♣✐❞❡♠✐♦❧ó❣✐❝❛s✳ ❋❛③✲s❡ t❛♠❜é♠ ✉♠❛ ❞❡s❝r✐çã♦ ❞♦s ♦❜❥❡t✐✈♦s ♣r♦♣♦st♦s ♣❛r❛ ❡st❡ tr❛✲ ❜❛❧❤♦ ❜❡♠ ❝♦♠♦ r❡❢❡rê♥❝✐❛ à s✉❛ ♦r❣❛♥✐③❛çã♦✳ ◆♦ ❝❛♣ít✉❧♦ ✷ é ❢❡✐t❛ ✉♠❛ ❜r❡✈❡ ❞❡s❝r✐çã♦
❞♦s tr❛❜❛❧❤♦s ❝✐❡♥tí✜❝♦s ♠❛✐s r❡❧❡✈❛♥t❡s ♥♦ â♠❜✐t♦ ❞❛ ♠♦❞❡❧❛çã♦ ♠❛t❡♠át✐❝❛ ❞❛ ♣r♦♣❛✲ ❣❛çã♦ ❞❡ ❞♦❡♥ç❛s ✐♥❢❡❝✐♦s❛s ❝❡♥tr❛❞♦s ❡♠ ♠♦❞❡❧♦s ❞♦ t✐♣♦ ❞❡t❡r♠✐♥íst✐❝♦ ❡ ❡st♦❝ást✐❝♦✳ ➱ t❛♠❜é♠ ✐♥tr♦❞✉③✐❞♦ ✉♠❛ s❡❝çã♦ s♦❜r❡ ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ q✉❛❧✐t❛t✐✈❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳
◆♦ ❝❛♣ít✉❧♦ ✸ é ❢❡✐t❛ ✉♠❛ s✐st❡♠❛t✐③❛çã♦ ❞♦s ♣r✐♥❝✐♣❛✐s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s✳ ❈♦♠❡ç❛✲s❡ ♣♦r ❢❛③❡r r❡❢❡rê♥❝✐❛ ❛♦ ♠♦❞❡❧♦ ❞❡ ▲♦t❦❛✲❱♦❧t❡rr❛✳ ❉❡♣♦✐s sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ♣r✐♥❝✐♣❛✐s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s ❡st✉❞❛❞♦s✳ ❋❛③❡♠✲s❡ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ t♦❞♦s ♦s ♠♦❞❡❧♦s ❡♠ ♣r♦❜❧❡♠❛s ♣rát✐❝♦s✱ ♣r♦♣♦st♦s ♣♦r ♥ós ♦✉ r❡❝♦❧❤✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛✳ P♦r ✈❡③❡s ❝♦♠♣❛r❛♠✲s❡ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❝♦♠ ❞✐❢❡r❡♥t❡s ♠♦❞❡❧♦s✳
◆♦ ❝❛♣ít✉❧♦ ✹ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ♠♦❞❡❧♦s ❡st♦❝ást✐❝♦s✳ ◆❛ s❡❝çã♦ ✹✳✷ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ❡①❡♠♣❧♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❡ ✉♠❛ ❡♣✐❞❡♠✐❛ ♥✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ◆❛ s❡❝çã♦ ✹✳✸ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ♠♦❞❡❧♦ ❙■❘ ❡st♦❝ást✐❝♦ ❜❛s❡❛❞♦ ♥♦ ❛rt✐❣♦ ❞❡ ❇r✐tt♦♥ ❬✶✵❪✳ ◆❛ s❡❝çã♦ ✹✳✹ é ❢❡✐t❛ ✉♠❛ s✐♠✉❧❛çã♦ ❞♦ ❝♦♥tá❣✐♦ ❞❛ ❣r✐♣❡ ♣❡ss♦❛ ❛ ♣❡ss♦❛ ♥✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ❡s❝♦❧❛r t❡♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ❡❢❡✐t♦ ❞❡ ✈ár✐♦s ♣❛râ♠❡tr♦s✳ P❛r❛ t♦❞♦s ♦s ❡①❡♠♣❧♦s sã♦ ❛♣r❡s❡♥t❛❞❛s ❝♦♥❝❧✉sõ❡s ❡ ❢❡✐t❛s ❝♦♠♣❛r❛çõ❡s ❝♦♠ ♦✉tr♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳
P♦r ✜♠✱ ♥♦ ❝❛♣ít✉❧♦ ✺ ❞❡s❝r❡✈❡♠✲s❡ ❛s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❧✉sõ❡s ❞♦ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✲ ✈✐❞♦✳
❈❛♣ít✉❧♦ ✷
▼♦❞❡❧❛çã♦ ♠❛t❡♠át✐❝❛ ❞❡ ❡♣✐❞❡♠✐❛s
◆❡st❡ ❝❛♣ít✉❧♦ ♣r❡t❡♥❞❡✲s❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❜r❡✈❡ ❞❡s❝r✐çã♦ ❞♦s tr❛❜❛❧❤♦s ❝✐❡♥tí✜❝♦s ♠❛✐s r❡❧❡✈❛♥t❡s ♥♦ â♠❜✐t♦ ❞❛ ♠♦❞❡❧❛çã♦ ♠❛t❡♠át✐❝❛ ❞❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❞♦❡♥ç❛s ✐♥❢❡❝✐♦s❛s ❝❡♥tr❛❞♦s ❡♠ ♠♦❞❡❧♦s ❞♦ t✐♣♦ ❞❡t❡r♠✐♥íst✐❝♦ ❡ ❡st♦❝ást✐❝♦✳
❖ ❡st✉❞♦ q✉❛♥t✐t❛t✐✈♦ ❞❡ ❞♦❡♥ç❛s ❤✉♠❛♥❛s✱ ❡ ❞❛ ♠♦rt❛❧✐❞❛❞❡ r❡s✉❧t❛♥t❡ ❞❡ss❛s ❞♦❡♥ç❛s✱ t❡rá t✐❞♦ ✐♥í❝✐♦ ♥♦ sé❝✉❧♦ ❳❱■■✱ ❛ ♣❛rt✐r ❞♦s r❡❣✐st♦s ❡❧❛❜♦r❛❞♦s ♣♦r ❏♦❤♥ ●r❛✉♥t ♥❛ s✉❛ ♦❜r❛ ◆❛t✉r❛❧ ❛♥❞ P♦❧✐t✐❝❛❧ ❖❜s❡r✈❛t✐♦♥s ♠❛❞❡ ✉♣♦♥ t❤❡ ❜✐❧❧s ♦❢ ♠♦rt❛❧✐t②✱ ♣✉❜❧✐❝❛❞❛ ❡♠ ✶✻✻✷ ✭✈❡❥❛✲s❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✶✷❪✮✳
❈❡r❝❛ ❞❡ ✉♠ sé❝✉❧♦ ❞❡♣♦✐s✱ ♦ ♠❛t❡♠át✐❝♦ s✉íç♦ ❉❛♥✐❡❧ ❇❡r♥♦✉❧❧✐ ❞❡s❡♥✈♦❧✈❡✉ ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ♣❛r❛ ❛ ♣r♦♣❛❣❛çã♦ ❞❛ ✈❛rí♦❧❛✳ ◆♦ sé❝✉❧♦ ❳❱■■■✱ ❛ ✈❛rí♦❧❛ ❡r❛ ✉♠❛ ❞♦❡♥ç❛ ♣r❡✈❛❧❡♥t❡ ♥❛ ❊✉r♦♣❛ ❡ r❡s♣♦♥sá✈❡❧ ♣♦r ❝❡r❝❛ ❞❡ ✶✵✪ ❞❛ ♠♦rt❛❧✐❞❛❞❡ ✐♥❢❛♥t✐❧✳
❊♠ ✶✼✻✵✱ ❇❡r♥♦✉❧❧✐ ❛♣r❡s❡♥t♦✉ à ❆❝❛❞❡♠✐❛ ❘❡❛❧ ❢r❛♥❝❡s❛ ♦ s❡✉ tr❛❜❛❧❤♦ ❊ss❛✐ ❞✬✉♥❡ ♥♦✉✈❡❧❧❡ ❛♥❛❧②s❡ ❞❡ ❧❛ ♠♦rt❛❧✐té ❝❛✉sé❡ ♣❛r ❧❛ ♣❡t✐t ✈ér♦❧❡ ❡ ❞❡s ❛✈❛♥t❛❣❡s ❞❡ ❧✬✐♥♦❝✉❧❛t✐♦♥ ♣♦✉r ❧❛ ♣ré✈❡♥✐r✳ ❈♦♠ ❡st❡ tr❛❜❛❧❤♦✱ ❇❡r♥♦✉❧❧✐ ♥ã♦ só ❛♣r❡s❡♥t♦✉ ❛s s✉❛s ❝♦♥❝❧✉sõ❡s s♦❜r❡ ❛ ♠♦❞❡❧❛çã♦ ❞❛ ❡♣✐❞❡♠✐❛ ❞❛ ✈❛rí♦❧❛ ❝♦♠♦ t❛♠❜é♠ ❛r❣✉♠❡♥t♦✉ q✉❡ ❛ ✈❛r✐♦❧❛çã♦✱ ✐st♦ é✱ ❛ ✐♥♦❝✉❧❛çã♦ ❝♦♠ ♦ ✈ír✉s ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ ❞♦❡♥t❡s ✐♥❢❡t❛❞♦s ❝♦♠ ✈❛rí♦❧❛✱ tr❛r✐❛ ✐♠✉♥✐❞❛❞❡ ❛♦ ✐♥❞✐✈í❞✉♦✳ ❉❡st❛ ❢♦r♠❛✱ ❝♦♥❝❧✉✐✉ ❇❡r♥♦✉❧❧✐✱ r❡❞✉③✐r✲s❡✲✐❛ ❛ t❛①❛ ❞❡ ♠♦rt❛❧✐❞❛❞❡ ♣r♦✈♦❝❛❞❛ ♣❡❧❛ ❞♦❡♥ç❛ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛✉♠❡♥t❛r✲s❡✲✐❛ ❛ ♣♦♣✉❧❛çã♦ ❞❛ ❋r❛♥ç❛✳
✷✳✶ ▼♦❞❡❧❛çã♦ ❞❡t❡r♠✐♥íst✐❝❛
◆❡st❛ s❡❝çã♦ ❛♣r❡s❡♥t❛✲s❡ ✉♠❛ ❜r❡✈❡ ❞❡s❝r✐çã♦ ❞♦s tr❛❜❛❧❤♦s ♠❛✐s r❡❧❡✈❛♥t❡s ♥❛ ♠♦❞❡✲ ❧❛çã♦ ❞❡ ❞♦❡♥ç❛s ✐♥❢❡❝✐♦s❛s ❛tr❛✈és ❞❡ ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s✳
❆ ♠♦❞❡❧❛çã♦ ❞❡t❡r♠✐♥íst✐❝❛ é ❛❞❡q✉❛❞❛ à ♠♦❞❡❧❛çã♦ ♠❛t❡♠át✐❝❛ ❞❡ ❞♦❡♥ç❛s ♣r❡✲ ✈❛❧❡♥t❡s ❡♠ ❣r❛♥❞❡s ♣♦♣✉❧❛çõ❡s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛ t✉❜❡r❝✉❧♦s❡ ♦✉ ❛ ✈❛rí♦❧❛✱ r❡❢❡r✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡✳ ❚✐♣✐❝❛♠❡♥t❡✱ ♥♦s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ♠♦❞❡❧♦s ❝♦♠♣❛rt✐♠❡♥t❛✐s✱ ♦s ✐♥❞✐✈í❞✉♦s ❞❛ ♣♦♣✉❧❛çã♦ ❡♠ ❡st✉❞♦ sã♦ ❞✐str✐❜✉í❞♦s ❡♠ ❣r✉♣♦s ♦✉ ❝♦♠♣❛rt✐♠❡♥t♦s ❞✐st✐♥t♦s✱ ♦s q✉❛✐s r❡♣r❡s❡♥t❛♠ ❡stá❣✐♦s ❞✐❢❡r❡♥t❡s ❞❛ ❞♦❡♥ç❛ ✭✈❡❥❛✲s❡ ♦ ❈❛♣ít✉❧♦ ✸✮✳
❖s ❞✐❢❡r❡♥t❡s ❝♦♠♣❛rt✐♠❡♥t♦s sã♦ ❣❡r❛❧♠❡♥t❡ ❞❡s✐❣♥❛❞♦s ♣♦r ❧❡tr❛s ♠❛✐ús❝✉❧❛s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ S✱I ❡ R✳ ◆♦ ♠♦❞❡❧♦ SIR✱ S r❡♣r❡s❡♥t❛ ♦ ❣r✉♣♦ ❞❛ ♣♦♣✉❧❛çã♦ s✉s❝❡tí✈❡❧
❞❡ ❝♦♥tr❛✐r ❛ ❞♦❡♥ç❛✱ I r❡♣r❡s❡♥t❛ ♦ ❣r✉♣♦ ❞❛ ♣♦♣✉❧❛çã♦ q✉❡ ❝♦♥tr❛✐✉ ❛ ❞♦❡♥ç❛ ❡ q✉❡
♣♦❞❡ tr❛♥s♠✐t✐✲❧❛ ❛ ♦✉tr♦ ✐♥❞✐✈í❞✉♦✱ ❣r✉♣♦ ✐♥❢❡❝✐♦s♦✱ ❡R r❡♣r❡s❡♥t❛ ♦ ❣r✉♣♦ ❞❛ ♣♦♣✉❧❛✲
çã♦ q✉❡ r❡❝✉♣❡r♦✉ ❞❛ ❞♦❡♥ç❛✳ ❆s t❛①❛s ❞❡ tr❛♥s✐çã♦ ❞♦s ✐♥❞✐✈í❞✉♦s ❡♥tr❡ ♦s ❞✐❢❡r❡♥t❡s ❣r✉♣♦s ❡①♣r❡ss❛♠✲s❡ ❛tr❛✈és ❞❡ ❞❡r✐✈❛❞❛s✳ ❉❡st❡ ♠♦❞♦✱ ♦s ♠♦❞❡❧♦s ❞❡t❡r♠✐♥íst✐❝♦s sã♦ ❢♦r♠✉❧❛❞♦s ❡♠ t❡r♠♦s ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳
P❛r❛ t♦r♥❛r ♦ ♠♦❞❡❧♦ ♠❛✐s r❡❛❧✐st❛ é ✉s✉❛❧ ✐♥❝r❡♠❡♥t❛r✲s❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠♣❛rt✐✲ ♠❡♥t♦s✱ ❞❡ ❢♦r♠❛ ❛ ❝❛r❛❝t❡r✐③❛r ♠❛✐s ❡stá❣✐♦s ❞❛ ❞♦❡♥ç❛✳ P♦r ❡①❡♠♣❧♦✱ ♦s ♣r✐♠❡✐r♦s ❡st✉❞♦s s♦❜r❡ ♦ ❡❢❡✐t♦ ❞❛ q✉❛r❡♥t❡♥❛ ♥❛ tr❛♥s♠✐ssã♦ ❞❡ ✉♠❛ ❞♦❡♥ç❛ ✐♥❢❡❝✐♦s❛ ❢♦r❛♠ ♦s tr❛❜❛❧❤♦s ❞❡ ❋❡♥❣ ❡ ❚❤✐❡♠❡ ❬✷✼❪ ❡ ❋❡♥❣ ❡ ❲✉ ❬✷✽❪✳ ❊st❡s ❛✉t♦r❡s ✐♥tr♦❞✉③✐r❛♠ ♥♦ ♠♦❞❡❧♦ ❙■❘ ♦ ❝♦♠♣❛rt✐♠❡♥t♦ ❞❡ q✉❛r❡♥t❡♥❛✱Q✱ ❛ss✉♠✐♥❞♦ q✉❡ t♦❞♦s ♦s ✐♥❞✐✈í❞✉♦s ✐♥✲
❢❡t❛❞♦s ❞❡✈❡r✐❛♠ ♣❛ss❛r ♣❡❧♦ ❝♦♠♣❛rt✐♠❡♥t♦ ❛♥t❡s ❞❡ ✈♦❧t❛r❡♠ ♣❛r❛ ♦ ❝♦♠♣❛rt✐♠❡♥t♦ ❞♦s s✉s❝❡tí✈❡✐sS ♦✉ r❡❝✉♣❡r❛❞♦sR✳
❖✉tr♦s ❛✉t♦r❡s ❝♦♠♦ ❍❡t❤❝♦t❡✱ ❩❤✐❡♥ ❡ ❙❤❡♥❣❜✐♥❣ ❬✶✺❪ r❡❛❧✐③❛r❛♠ ✉♠❛ ❛♥á❧✐s❡ ❞❡ ❞♦✐s ♠♦❞❡❧♦s SIQS ❡ SIQR ❝♦♠ três t✐♣♦s ❞❡ ✐♥❝✐❞ê♥❝✐❛✿ ❛çã♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ s✐♠♣❧❡s
❡♠ ♠❛ss❛✱ ✐♥❝✐❞ê♥❝✐❛ ♣❛❞rã♦ ❡ ✐♥❝✐❞ê♥❝✐❛ ❞❡ q✉❛r❡♥t❡♥❛ ❛❥✉st❛❞❛✳ ◆❡st❡ ❛rt✐❣♦✱ ♦s ❛✉t♦r❡s ❛♥❛❧✐s❛♠ ❡ ❝♦♠♣❛r❛♠ ❢♦r♠✉❧❛çõ❡s ❞❡ ♠♦❞❡❧♦s ❝♦♠ ✉♠❛ ❝❧❛ss❡ ❞❡ q✉❛r❡♥t❡♥❛✳ ❯♠ ♦✉tr♦ ♣r♦♣ós✐t♦ ❞♦ tr❛❜❛❧❤♦ ❡r❛ ❞❡t❡r♠✐♥❛r s❡ ❛s s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s q✉❡ ♦❝♦rr❡♠ ❡♠ ♠♦❞❡❧♦sSIQS ❡ SIQR ❡r❛♠ s❡♠❡❧❤❛♥t❡s✳ ❊st❡s ♠♦❞❡❧♦s ❣❡♥❡r❛❧✐③❛♠ ♦s ♠♦❞❡❧♦s
❞❡ ❋❡♥❣ ❡ ❚❤✐❡♠❡ ❬✷✼❪✱ ♣❡r♠✐t✐♥❞♦ q✉❡ ❛❧❣✉♥s ✐♥❞✐✈í❞✉♦s s✐❣❛♠ ♦ ❝❛♠✐♥❤♦ ❤❛❜✐t✉❛❧ ❞❡ r❡❝✉♣❡r❛çã♦ ❡ ♦✉tr♦s s❡❥❛♠ ❞❡s✈✐❛❞♦s ❛tr❛✈és ❞❛ ❝❧❛ss❡ ❞❡ q✉❛r❡♥t❡♥❛✳ ❱❡r✐✜❝❛r❛♠ q✉❡ t♦❞♦s ♦s ♠♦❞❡❧♦s✱ ❡①❝❡t♦ ♦ ♠♦❞❡❧♦SIQR ❝♦♠ ✐♥❝✐❞ê♥❝✐❛ ❞❡ q✉❛r❡♥t❡♥❛ ❛❥✉st❛❞❛✱
❛♣r❡s❡♥t❛♠ ❡q✉✐❧í❜r✐♦ ❡♥❞é♠✐❝♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✱ ✐st♦ é ♥♦ ❧✐♠✐t❡ ♦ ♣r♦❝❡ss♦ ❡♥❞é♠✐❝♦ t❡♥❞❡ ♣❛r❛ ✉♠ ❡q✉✐❧í❜r✐♦ ❡stá✈❡❧✳ P❛r❛ ♦ ♠♦❞❡❧♦ SIQR ❝♦♠ ✐♥❝✐❞ê♥❝✐❛ ❞❡
q✉❛r❡♥t❡♥❛ ❛❥✉st❛❞❛✱ ♦s ❛✉t♦r❡s ✈❡r✐✜❝❛r❛♠ q✉❡ ❡①✐st❡♠ ♠✉✐t♦s ❛s♣❡t♦s q✉❡ ♣❛r❡❝❡♠ ❡st❛r ❡♠ s✐♥t♦♥✐❛ ❝♦♠ ❛s ♦s❝✐❧❛çõ❡s q✉❡ sã♦ ♦❜s❡r✈❛❞❛s ♥❛ ✐♥❝✐❞ê♥❝✐❛ r❡❛❧ ❞❛ ❞♦❡♥ç❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❛ ✐♥❝✐❞ê♥❝✐❛ t❡♠ ❡q✉✐❧í❜r✐♦s ❡♥❞é♠✐❝♦s q✉❡ sã♦ ❡s♣✐r❛✐s ✐♥stá✈❡✐s ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ ♣❛râ♠❡tr♦s✱ ♣❡❧♦ q✉❡ ❡st❡s ❞❛❞♦s s✉❣❡r❡♠ q✉❡ só ❤á s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s ♥❡st❡ t✐♣♦ ❞❡ ♠♦❞❡❧♦s ✭✈❡❥❛✲s❡ ❬✶✺❪✮✳
❊♠ ❍❡t❤❝♦t❡ ❬✶✸❪ ❛♥❛❧✐s❛♠✲s❡ três ♠♦❞❡❧♦s ❜ás✐❝♦s ✭SIS ❡♥❞é♠✐❝♦✱SIR ❡♣✐❞é♠✐❝♦
❡ SIR ❡♥❞é♠✐❝♦✮ ♣❛r❛ ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❞♦❡♥ç❛s ❡♠ ♣♦♣✉❧❛çõ❡s q✉❡ ❢♦r❛♠ ❛♣❧✐❝❛❞♦s
❛ ❞♦❡♥ç❛s ❡s♣❡❝í✜❝❛s✳ P❛r❛ ❡st❡s ♠♦❞❡❧♦s sã♦ ❛♣r❡s❡♥t❛❞♦s t❡♦r❡♠❛s ❞❡ ❧✐♠✐t❡ q✉❡ ❡♥✈♦❧✈❡♠ ♦ ♥ú♠❡r♦ ❞❡ r❡♣r♦❞✉çã♦ ❜ás✐❝♦ R0✱ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥t❛❝t♦ σ ❡ ♦ ♥ú♠❡r♦ ❞❡
s✉❜st✐t✉✐çã♦ ❘ ❜❡♠ ❝♦♠♦ ❛s s✉❛s ❡①t❡♥sõ❡s ❝♦♠♦ SEIR ❡ M SEIRS✳ ❖s ✈❛❧♦r❡s ❞❡ R0 ❡ σ sã♦ ❡st✐♠❛❞♦s ♣❛r❛ ❝♦♠♣❛r❛çã♦ ❞❛ ♣❡r❝❡♥t❛❣❡♠ ❞❡ ✈❛❝✐♥❛❞♦s ♣❛r❛ s❡ ♦❜t❡r
✐♠✉♥✐❞❛❞❡ ✭✈❡❥❛✲s❡ ❬✶✸✱✶✹❪✮✳
◆❡✐❧❛♥ ❡ ▲❡♥❤❛rt ❡♠ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ✇✐t❤ ❛♥ ❆♣♣❧✐❝❛t✐♦♥ ✐♥ ❉✐s❡❛s❡ ▼♦❞❡❧✐♥❣ ♣r♦♣✉s❡r❛♠ ✉♠❛ ♠♦❞✐✜❝❛çã♦ ♥♦ ♠♦❞❡❧♦ SEIR ✐♥tr♦❞✉③✐♥❞♦ ✉♠❛
❡str❛té❣✐❛ ❞❡ ✈❛❝✐♥❛çã♦✳ ❊st❛ ❡str❛té❣✐❛ ❛t✉❛ ❝♦♠♦ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❝♦♥tr♦❧♦ ❡ é ❢♦r♠✉✲ ❧❛❞♦ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥tr♦❧♦ ót✐♠♦ ♣❛r❛ ✐♥❝❧✉✐r ✉♠❛ r❡str✐çã♦ ✭✈❡❥❛✲s❡ ❬✶✽❪✮✳
✷✳✷ ▼♦❞❡❧❛çã♦ ❡st♦❝ást✐❝❛
❆ ♣r♦♣❛❣❛çã♦ ❞❡ ✉♠❛ ❞♦❡♥ç❛ ✐♥❢❡❝✐♦s❛ é ✉♠ ♣r♦❝❡ss♦ ❛❧❡❛tór✐♦✳ ◆✉♠ ❣r✉♣♦ ♣❡q✉❡♥♦ ❞❡ ✐♥❞✐✈í❞✉♦s✱ s❡ ✉♠ ❞❡❧❡s t❡♠ ♣♦r ❡①❡♠♣❧♦ ❣r✐♣❡✱ ❡♥tã♦ ❛❧❣✉♥s ❞♦s ✐♥❞✐✈í❞✉♦s s❡rã♦ ❝♦♥t❛❣✐❛❞♦s ❝♦♠ ♦ ✈ír✉s ❡♥q✉❛♥t♦ q✉❡ ♦✉tr♦s ♥ã♦ ❝❤❡❣❛rã♦ ❛ s❡r ❝♦♥t❛❣✐❛❞♦s✳
❊♠ ✶✾✻✼✱ ❇❛✐❧❡② ❬✼❪ ♣✉❜❧✐❝♦✉ ❚❤❡ ▼❛t❤❡♠❛t✐❝❛❧ ❚❤❡♦r② ♦❢ ■♥❢❡❝t✐♦✉s ❉✐s❡❛s❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✉♠ ❧✐✈r♦ s♦❜r❡ ❛ t❡♦r✐❛ ♠❛t❡♠át✐❝❛ ❞❡ ❡♣✐❞❡♠✐❛s✳ ❊st❡ ❧✐✈r♦ ❡r❛ ❞❡st✐♥❛❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❛r❛ ❛q✉❡❧❡s q✉❡ q✉❡r✐❛♠ s❛❜❡r ♠❛✐s s♦❜r❡ ♦ ✉s♦ ❞♦s ♠ét♦❞♦s ♠❛t❡♠á✲
