key: cord-0899317-0gpubg0d
authors: Buonomo, Bruno; Marca, Rossella Della; d’Onofrio, Alberto; Groppi, Maria
title: A behavioural modelling approach to assess the impact of COVID–19 vaccine hesitancy
date: 2021-12-08
journal: J Theor Biol
DOI: 10.1016/j.jtbi.2021.110973
sha: b8d312c984c3986e518a50913e7a0204b7253a19
doc_id: 899317
cord_uid: 0gpubg0d

We introduce a compartmental epidemic model to describe the spread of COVID–19 within a population, assuming that a vaccine is available, but vaccination is not mandatory. The model takes into account vaccine hesitancy and the refusal of vaccination by individuals, which take their decision on vaccination based on both the present and past information about the spread of the disease. Theoretical analysis and simulations show that voluntary vaccination can certainly reduce the impact of the disease but is unable to eliminate it. We also demonstrate how the information–related parameters affect the dynamics of the disease. In particular, vaccine hesitancy and refusal are better contained in case of widespread information coverage and short-term memory. Finally, the possible impact of seasonality on the spread of the disease is investigated.

r✉♠♦✉rs ❛❜♦✉t t❤❡ ❞✐s❡❛s❡ st❛t✉s ✐♥ t❤❡ ❝♦♠♠✉♥✐t②✳ ■♥ ❬✽(✱ t❤❡ ♠♦❞❡• ✐s ❛♣♣•✐❡❞ t♦ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❈❖❱■❉✕ ✶✾ ❡♣✐❞❡♠✐❝ ✐♥ ■t❛•②✳ ❚❤❡ ♣❛♣❡r ❛r❣✉❡s t❤❛t ❝✐t✐③❡♥ ❝♦♠♣•✐❛♥❝❡ ❛•♦♥❣ ✇✐t❤ ♠✐t✐❣❛t✐♦♥ ♠❡❛s✉r❡s ♣•❛②❡❞ ❛ ❞❡❝✐s✐✈❡ r♦•❡ ✐♥ ❝✉r❜✐♥❣ t❤❡ ❡♣✐❞❡♠✐❝ ❝✉r✈❡ ❜② ♣r❡✈❡♥t✐♥❣ ❛ ❞✉♣•✐❝❛t✐♦♥ ♦❢ ❞❡❛t❤s ❛s ✇❡•• ❛s ❛❜♦✉t 46% ♠♦r❡ ✐♥❢❡❝t✐♦♥s✳ ❚❤❡ ❈❖❱■❉✕✶✾ ♣❛♥❞❡♠✐❝ ❝❛✉s❡❞ ❛ ✇♦r•❞✇✐❞❡ r❡s❡❛r❝❤ ❡✛♦rt t❤❛t r❡s✉•t❡❞ ✐♥ t❤❡ r❛♣✐❞ ❞❡✈❡•♦♣♠❡♥t ♦❢ ♥❡✇ ✈❛❝❝✐♥❡s ❬✹✽✱ ✺✹(✱ s♦♠❡ ♦❢ ✇❤✐❝❤ ❜❡•♦♥❣ t♦ t❤❡ ♥❡✇ ❝•❛ss ♦❢ ♠|◆❆ ✈❛❝❝✐♥❡s ❬✷✱ ✻✻(✳ ■♥ •✐❣❤t ♦❢ t❤❡ s✐❣♥✐✜❝❛♥t ❝❤❛♥❣❡s ✐♥ t❤❡ •✐✈❡s ♦❢ ♠✐••✐❛r❞s ♦❢ ♣❡♦♣•❡ ❛♥❞ t❤❡ ❤✉❣❡ ♥❡❣❛t✐✈❡ ✐♠♣❛❝t ♦♥ t❤❡ ✇♦r•❞ ❡❝♦♥♦♠②✱ ♦♥❡ ❝♦✉•❞ ❤❛✈❡ ❡①♣❡❝t❡❞ t❤❛t ♦♥•② ❛ t✐♥② ♣r♦♣♦rt✐♦♥ ♦❢ ♣❡♦♣•❡ ✇♦✉•❞ r❡❛••② ❜❡ ❤❡s✐t❛♥t t♦✇❛r❞s ✈❛❝❝✐♥❛t✐♦♥✳ ❯♥❢♦rt✉♥❛t❡•②✱ t❤✐s ✇❛s ♥♦t t❤❡ ❝❛s❡✳ ❆s ❡❛r•② ❛s ❏✉♥❡ ✷✵✷✵✱ ◆❡✉♠❛♥♥✕❇ö❤♠❡ ❛♥❞ ❝♦✲ ✇♦r❦❡rs ❬✻✹( ✐♥✈❡st✐❣❛t❡❞ ❛tt✐t✉❞❡s ❛❜♦✉t ❛♥t✐✕❈❖❱■❉✕✶✾ ✈❛❝❝✐♥❛t✐♦♥ ❢r♦♠ ❛ r❡♣r❡s❡♥t❛t✐✈❡ s❛♠♣•❡ ♦❢ ❝✐t✐③❡♥s ❢r♦♠ s❡✈❡♥ ❊✉r♦♣❡❛♥ ❝♦✉♥tr✐❡s✳ ❙✉r♣r✐s✐♥❣•②✱ ❛•t❤♦✉❣❤ t❤❡ ✜rst ❡♣✐❞❡♠✐❝ ✇❛✈❡ ✐♥ ❊✉r♦♣❡ ❤❛❞ •✉st ❡♥❞❡❞✱ ❤❡s✐t❛♥❝② ❛♥❞ ♦♣♣♦s✐t✐♦♥ t♦✇❛r❞s t❤❡ ✈❛❝❝✐♥❡s ✇❡r❡ ❢♦✉♥❞ ❛♠♦♥❣ ❛ •❛r❣❡ ♣r♦♣♦rt✐♦♥ ✐♥ ❛•• ✐♥ ❛•• ❝•❛ss❡s✱ ❛❣❡ ❣r♦✉♣s✱ ❛♥❞ s❡①❡s✳ ■♥ ♣❛rt✐❝✉•❛r✱ 38% ♦❢ t❤❡ ❋r❡♥❝❤ r❡s♣♦♥❞❡♥ts ✇❡r❡ ❤❡s✐t❛♥t ✭28%✮ ♦r ✇❡r❡ str♦♥❣•② ❛❣❛✐♥st ✭10%✮ ❈❖❱■❉✕✶✾ ✈❛❝❝✐♥❡s✳ ❇❡❢♦r❡ ♠✐❞✕❉❡❝❡♠❜❡r ✷✵✷✵✱ ♣❤❛s❡ ✸ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥t❛t✐♦♥ ♦❢ ♠❛♥② ✈❛❝❝✐♥❡s ❡♥❞❡❞✱ ✐♥❞✐❝❛t✐♥❣ t❤❛t t❤❡② ❤❛✈❡ ♦✉tst❛♥❞✐♥❣ ❡✛❡❝t✐✈❡♥❡ss ✐♥ ♣r❡✈❡♥t✐♥❣ ❈❖❱■❉✕✶✾ ❬✷✱ ✺✹✱ ✻✻(✳ ❚②♣✐❝❛••②✱ ❞r✉❣ r❡❣✉•❛t♦r② ❛❣❡♥❝✐❡s ❞❡✜♥❡❞ ♣r✐♦r✐t② ❣r♦✉♣s ❢♦r ✈❛❝❝✐♥❛t✐♦♥ ✭❡•❞❡r•② ♣❡♦♣•❡ ✇✐t❤ s❡r✐♦✉s ❝♦✕♠♦r❜✐❞✐t✐❡s✱ ❤❡❛•t❤❝❛r❡ ✇♦r❦❡rs ✐♥ s❡♥✐♦r r❡s✐❞❡♥❝❡s✱ ❡t❝✳✮✳ ❋r♦♠ ❛ r❛t✐♦♥❛• ✈✐❡✇♣♦✐♥t✱ t❤❡r❡ ✇❡r❡ ❛•• t❤❡ ♣r❡♠✐s❡s t♦ ❜❡•✐❡✈❡ t❤❛t ✈❛❝❝✐♥❡ ❤❡s✐t❛♥❝② ❤❛❞ ❜❡❡♥ str♦♥❣•② r❡❞✉❝❡❞ ❛♥❞ t❤❛t ♠❛♥❞❛t♦r② ✈❛❝❝✐♥❛t✐♦♥ ❝❛♠♣❛✐❣♥s ❝♦✉•❞ ❤❛✈❡ ❜❡❡♥ ❝♦♥✲ ❞✉❝t❡❞✱ ❜✉t t❤✐s ✇❛s ♥♦t t❤❡ ❝❛s❡✳ ❈♦♥❝❡r♥✐♥❣ t❤❡ ♠❛♥❞❛t♦r② ♥❛t✉r❡ ♦❢ t❤❡ ✈❛❝❝✐♥❛t✐♦♥ ❝❛♠♣❛✐❣♥✱ ✐♥ ♠❛♥② ❝♦✉♥tr✐❡s✱ t❤❡ ✈❛❝❝✐♥❡s ❛r❡ ♥♦t ♠❛♥❞❛t♦r② ❬✺✵✱ ✺✻✱ ✼✵(✳ ❆s ❢♦r ✈❛❝❝✐♥❡ ❤❡s✐t❛♥❝②✱ ❛♥ ✐♥✈❡st✐❣❛t✐♦♥ ❝♦♥❞✉❝t❡❞ ✐♥ ❖❝t♦❜❡r ✷✵✷✵ ❬✸✽( s✉❣❣❡sts t❤❛t 46% ♦❢ ❋r❡♥❝❤ ❝✐t✐③❡♥s ❛r❡ ✈❛❝❝✐♥❡ ❤❡s✐t❛♥t✳ ❖t❤❡r ❝♦✉♥tr✐❡s ❡①❤✐❜✐t ♣❡r❝❡♥t❛❣❡s ♦❢ ♦♣♣♦s✐t✐♦♥ ❛♥❞ ❤❡s✐t❛♥❝② t❤❛t ❡①❝❡❡❞ 30%✿ 36% ✐♥ ❙♣❛✐♥ ❛♥❞ ❯❙❆✱ 35% ✐♥ ■t❛•②✱ 32% ✐♥ ❙♦✉t❤ ❆❢r✐❝❛✱ ❛♥❞ 31% ✐♥ ❏❛♣❛♥ ❛♥❞ •❡r♠❛♥②✳ ••♦❜❛••②✱ t❤❡ ❤❡s✐t❛♥❝② ❛♥❞ ♦❜•❡❝t✐♦♥ ❛r❡ ❛s •❛r❣❡ ❛s 27%✳ •✐✈❡♥ t❤❡s❡ •❛r❣❡ ♣❡r❝❡♥t❛❣❡s ♦❢ ❤❡s✐t❛♥❝❡ ❛♥❞ ♦♣♣♦s✐t✐♦♥ t♦ ❈❖❱■❉✕✶✾ ✈❛❝❝✐♥❡s✱ ✇❡ t❤✐♥❦ t❤❛t ❡♠♣•♦②✐♥❣ ❛ ❜❡❤❛✈✐♦✉r❛• ❡♣✐❞❡♠✐♦•♦❣② ❛♣♣r♦❛❝❤ t♦ ♠♦❞❡• t❤❡ ✐♠♣•❡♠❡♥t❛t✐♦♥ ♦❢ ❛ ✈❛❝❝✐♥❛t✐♦♥ ❝❛♠♣❛✐❣♥ ❢♦r ❈❖❱■❉✕ ✶✾ ✐s ❛♣♣r♦♣r✐❛t❡✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ❛❞♦♣t ❛ str❛t❡❣② s✐♠✐•❛r t♦ t❤❡ ♦♥❡ ✉s❡❞ ✐♥ ❬✷✺(✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ✈❛❝❝✐♥❛t✐♦♥ r❛t❡ ✐s ❛ ♣❤❡♥♦♠❡♥♦•♦❣✐❝❛• ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣r❡s❡♥t ❛♥❞ ♣❛st ✐♥❢♦r♠❛t✐♦♥ t❤❛t t❤❡ ❝✐t✐③❡♥s ❤❛✈❡ ♦♥ t❤❡ s♣r❡❛❞ ♦❢ t❤❡ ❡♣✐❞❡♠✐❝✳ ◆♦t❡ t❤❛t✱ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❙■| ❛♥❞ ❙❊■| ✐♥❢❡❝t✐♦✉s ❞✐s❡❛s❡s✱ ♠♦r❡ ♠❡❝❤❛♥✐st✐❝ ♠♦❞❡•s ❜❛s❡❞ ♦♥ ❡✈♦•✉t✐♦♥❛r② ❣❛♠❡ t❤❡♦r✐❡s ❬✸✱ ✷✸✱ ✷✹✱ ✼✹( ❡①✐st✱ ❜✉t t❤❡② ❛r❡ r❡❞✉❝❡❞ t♦ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❬✶✵✱ ✷✺( ✐♥ ❝❛s❡ ♦❢ ✈♦•❛t✐•❡ ♦♣✐♥✐♦♥ s✇✐t❝❤✐♥❣ ❬✶✽✱ ✷✸✱ ✼✹(✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❝♦♥s✐❞❡r ❛ ❈❖❱■❉✕✶✾ ❛✛❡❝t❡❞ ♣♦♣✉•❛t✐♦♥ ❝♦♥tr♦••❡❞ ❜② ✈❛❝❝✐♥❛t✐♦♥✱ ✇❤❡r❡ t❤❡ ✜♥❛• ❝❤♦✐❝❡ t♦ ❣❡t ✈❛❝❝✐♥❛t❡❞ ♦r ♥♦t ✐s ♣❛rt✐❛••② ❞❡t❡r♠✐♥❡❞ ♦♥ ❛ ✈♦•✉♥t❛r② ❜❛s✐s ❛♥❞ ❞❡♣❡♥❞❡♥t ♦♥ ♣✉❜•✐❝•② ❛✈❛✐•❛❜•❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ s♣r❡❛❞✐♥❣ ♦❢ t❤❡ ❞✐s❡❛s❡ ✐♥ t❤❡ ❝♦♠♠✉♥✐t② ✐♥ t❤❡ ❜♦t❤ t❤❡ ♣r❡s❡♥t ❛♥❞ r❡❝❡♥t ♣❛st✳ ❖✉r ♠♦❞❡• ✐s ✐♥s♣✐r❡❞ ❜② t❤❡ ❝♦♠♣❛rt♠❡♥t❛• ❡♣✐❞❡♠✐❝ ♠♦❞❡• ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✽(✱ ✇❤❡r❡❜② ❈❖❱■❉✕ ✶✾ tr❛♥s♠✐ss✐♦♥ ❞✉r✐♥❣ t❤❡ ✷✵✷✵ •♦❝❦❞♦✇♥ ✐♥ ■t❛•② ✇❛s st✉❞✐❡❞✳ ❆♥ ❛♥❛•♦❣♦✉s s✐t✉❛t✐♦♥ ✇❛s ❝♦♥s✐❞❡r❡❞ ❜② •✉♠❡• ❛♥❞ ❝♦✲✇♦r❦❡rs ❢♦r t❤❡ ❙❆|❙ ❡♣✐❞❡♠✐❝ ✇❤❡♥ t❤❡② ❡①❛♠✐♥❡❞ ❛ ❙❆|❙ ♠♦❞❡• ✐♥ ❬✸✼( ❛♥❞ t❤❡♥ ❝♦♥s✐❞❡r❡❞ ✈❛❝❝✐♥❛t✐♦♥ ✐♥t❡r✈❡♥t✐♦♥ ✐♥ ❬✸✻(✳ ❲❡ ♣❡r❢♦r♠❡❞ ❛ q✉❛•✐t❛t✐✈❡ ❛♥❛•②s✐s ❜❛s❡❞ ♦♥ st❛❜✐•✐t② ❛♥❞ ❜✐❢✉r❝❛t✐♦♥ t❤❡♦r✐❡s✳ ❚❤❡ ❛♥❛•②s✐s s❤♦✇s t❤❛t✱ ✇❤❡♥ t❤❡ ❝♦♥tr♦• r❡♣r♦❞✉❝t✐♦♥ ♥✉♠❜❡r✱ R V ✱ ✐s •❡ss t❤❛♥ ✶✱ ♦♥•② ❛ ❞✐s❡❛s❡✕❢r❡❡ ❡q✉✐•✐❜r✐✉♠ ✭❉❋❊✮ t❤❛t ✐s ❣•♦❜❛••② st❛❜•❡ ❡①✐sts❀ ✇❤❡♥ R V > 1✱ t❤❡ ❉❋❊ ✐s ✉♥st❛❜•❡✱ ❛♥❞ ❛♥ ❡♥❞❡♠✐❝ ❡q✉✐•✐❜r✐✉♠ ❛r✐s❡s✳ ❚❤❡ ♠♦❞❡• ✐s t❤❡♥ ♣❛r❛♠❡t❡r✐s❡❞ ❜❛s❡❞ ♦♥ t❤❡ ❈❖❱■❉✕✶✾ ❡♣✐❞❡♠✐❝ ✐♥ ■t❛•② ❛♥❞ ♣r❡•✐♠✐♥❛r② r❡♣♦rts ♦♥ ❈❖❱■❉✕✶✾ ✈❛❝❝✐♥❡s✳ ■♥ ♥✉♠❡r✐❝❛• s✐♠✉•❛t✐♦♥s✱ ✇❡ ❝♦♥s✐❞❡r t✇♦ ♣♦ss✐❜•❡ st❛rt✐♥❣ t✐♠❡s ❢♦r ❛ ♦♥❡✕ ②❡❛r ✈❛❝❝✐♥❛t✐♦♥ ❝❛♠♣❛✐❣♥✳ ❲❡ ❛ss❡ss t❤❡ r♦•❡ ♦❢ ❝r✐t✐❝❛• ♠♦❞❡• ♣❛r❛♠❡t❡rs ❜② ❡✈❛•✉❛t✐♥❣ ❤♦✇ t❤❡② ❛✛❡❝t s✉✐t❛❜•❡ ❡♣✐❞❡♠✐♦•♦❣✐❝❛• ✐♥❞✐❝❛t♦rs✳ ❋✐♥❛••②✱ t❤❡ ❡✛❡❝ts ♦❢ s❡❛s♦♥❛•✐t② ❛r❡ ✐♥✈❡st✐❣❛t❡❞ ❜② ♠❛❦✐♥❣ ✸ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❞✐s❡❛s❡ tr❛♥s♠✐ss✐♦♥ ❛♥❞ s❡✈❡r✐t② ❛s ✇❡•• ❛s ✈❛❝❝✐♥❛t✐♦♥ r❛t❡s ❛r❡ •♦✇❡r ❞✉r✐♥❣ t❤❡ ✇❛r♠❡r ♠♦♥t❤s✳ ❚❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐s❡❞ ❛s ❢♦••♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷✱ t❤❡ ♠♦❞❡• ✐s ✐♥tr♦❞✉❝❡❞✱ ❛♥❞ ✐♥ ❙❡❝t✐♦♥ ✸✱ t❤❡ q✉❛•✐t❛t✐✈❡ (0)) ❬✸✾(✳ ❆s ❢❛r ❛s t❤❡ 

✇❤❡r❡ Θ a ❬r❡s♣✳ Θ s ( ✐s t❤❡ ❡①♣❡❝t❡❞ t✐♠❡ ✉♥t✐• r❡❝♦✈❡r② ❢♦r ❛s②♠♣t♦♠❛t✐❝ ❬r❡s♣✳ s②♠♣t♦♠❛t✐❝( ✐♥❞✐✈✐❞✉❛•s✳ ❲❡ ❛ss✉♠❡ Θ a = 6 ❛♥❞ Θ s = 18 ❞❛②s ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ❝♦♥s✐❞❡r❛t✐♦♥s ♠❛❞❡ ✐♥ ❬✽(✳ 

2/1, 000 ②❡❛rs −1 ✐s t❤❡ ❝♦✉♥tr②✕•❡✈❡• ❜✐rt❤ r❛t❡ ❬✹✵( ❛♥❞ ✐✐✮ t❤❡ ✐♥✢♦✇ t❡r♠ ❞✉❡ t♦ ✐♠♠✐❣r❛t✐♦♥✱ Λ m = 4, 000/7 ❞❛②s −1 ✱ ❡st✐♠❛t❡❞ ❢r♦♠ t❤❡ ♥✉♠❜❡r ♦❢ ❝✐t✐③❡♥s r❡♣❛tr✐❛t❡❞ ❞✉r✐♥❣ ❛❜♦✉t t❤❡ ✜rst ✶✶ ✇❡❡❦s ♦❢ t❤❡ ❈❖❱■❉✕✶✾ ❡♣✐❞❡♠✐❝ ✐♥ ■t❛•② ❬✹✶(✳ ❈♦♥❝❡r♥✐♥❣ a ❛♥❞ k✱ ♥✉♠❡r✐❝❛• s♦•✉t✐♦♥s ❜② ✈❛r②✐♥❣ a ∈ [1/60, 1] ❞❛②s −1 ❛♥❞ k ∈ [0.2, 1] ❛r❡ ❣✐✈❡♥ ✐♥ ❙❡❝t✐♦♥ ✺ ✭❢♦r ❞❡t❛✐•❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r❛♥❣❡s ♦❢ ✈❛•✉❡s ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ♣❛r❛♠❡t❡rs✱ s❡❡ ❬✽(✮✳ ❋✐♥❛••②✱ t♦ ♦❜t❛✐♥ ❛♥ ❛♣♣r♦♣r✐❛t❡ ✈❛•✉❡ ❢♦r t❤❡ ❜❛s❡•✐♥❡ tr❛♥s♠✐ss✐♦♥ r❛t❡ β✱ ✇❡ ❝♦♥s✐❞❡r ♠♦❞❡• ✭✸✮✕✭✶✹✮ ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ✈❛❝❝✐♥❛t✐♦♥ str❛t❡❣✐❡s ✭ϕ 0 = 0 ❞❛②s −1 ✱ D = 0✮ ❛♥❞ s❡❛r❝❤ ❢♦r t❤❡ ✈❛•✉❡ t❤❛t ❜❡st ✜ts ✇✐t❤ t❤❡ ✐♥✐t✐❛• ❵✉♥❝♦♥tr♦••❡❞✬ ♣❤❛s❡ ♦❢ t❤❡ s❡❝♦♥❞ ❡♣✐❞❡♠✐❝ ✇❛✈❡ ✐♥ ■t❛•②✳ ▼♦r❡ ♣r❡❝✐s❡•②✱ ✇❡ ❛❝❝♦✉♥t ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❈❖❱■❉✕✶✾✕✐♥❞✉❝❡❞ ❞❡❛t❤s ✐♥ ■t❛•② ❢r♦♠ ✶✻ ❆✉❣✉st✱ ❛ss✉♠❡❞ ❛s t❤❡ st❛rt✐♥❣ ❞❛t❡ ♦❢ t❤❡ s❡❝♦♥❞ ✇❛✈❡ ✭s❡❡ ❙❡❝t✐♦♥ ✹✳✶✮✱ ❛♥❞ ✶✸ ❖❝t♦❜❡r ✷✵✷✵ ❬✹✸(✳ ■♥❞❡❡❞✱ ♦♥ ✶✸ ❖❝t♦❜❡r✱ t❤❡ ❈♦✉♥❝✐• ♦❢ ▼✐♥✐st❡rs ❛♣♣r♦✈❡❞ ❛ ❞❡❝r❡❡ t♦ r❡✐♥tr♦❞✉❝❡ str✐❝t❡r r✉•❡s t♦ •✐♠✐t t❤❡ s♣r❡❛❞ ♦❢ t❤❡ ❞✐s❡❛s❡ ❬✹✷(✳ ❲❡ ✉s❡ ❞❛t❛ r❡❣❛r❞✐♥❣ ❞❡❛t❤s ✇❤✐❝❤ ❛r❡ ♠♦r❡ ❛❝❝✉r❛t❡ ❝♦♠♣❛r❡❞ ✇✐t❤ ♦t❤❡r ♦♥❡s✳ ❆♥②✇❛②✱ ❜② s❡tt✐♥❣ β = 2.699 · 10 −8 ❞❛②s −1 ✱ ✇❡ ♦❜t❛✐♥ ❛ ❣♦♦❞ ✜t ♥♦t ♦♥•② ✇✐t❤ t❤❡ ❝✉♠✉•❛t✐✈❡ ❞❡❛t❤s ✭s❡❡ ❋✐❣✳ ✷❇✮ ❜✉t ❛•s♦ ✇✐t❤ t❤❡ t♦t❛• ✐♥❢❡❝t✐♦✉s ❝❛s❡s✱ I a + I s ✭s❡❡ ❋✐❣✳ ✷❆✮✳ ❆•• t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡• ❛s ✇❡•• ❛s t❤❡✐r ❜❛s❡•✐♥❡ ✈❛•✉❡s ❛r❡ •✐st❡❞ ✐♥ ❚❛❜•❡ ✶✳ ✺ ◆✉♠❡r✐❝❛• s✐♠✉•❛t✐♦♥s ◆✉♠❡r✐❝❛• s✐♠✉•❛t✐♦♥s ❛r❡ ♣❡r❢♦r♠❡❞ ✐♥ ▼❆❚▲❆❇ ❬✺✾(✳ ❲❡ ✉s❡ t❤❡ ✹t❤ ♦r❞❡r |✉♥❣❡✕❑✉tt❛ ♠❡t❤♦❞ ✇✐t❤ ❝♦♥st❛♥t st❡♣ s✐③❡ ❢♦r ✐♥t❡❣r❛t✐♥❣ t❤❡ s②st❡♠ ❛♥❞ t❤❡ ♣•❛t❢♦r♠✕✐♥t❡❣r❛t❡❞ ❢✉♥❝t✐♦♥s ❢♦r ❣❡tt✐♥❣ t❤❡ ♣•♦ts✳ 

1.33 · 10 7 1.09 · 10 7 −2.39 · 10 6 V (t f ) 4.58 · 10 7 4.77 · 10 7 1.92 · 10 6 ❈❱(t f ) 4.62 · 10 7 4.82 · 10 7 2.00 · 10 6 max(I s ) 5.76 · 10 4 9.14 · 10 4 3.38 · 10 4 arg max(I s ) ✶✵✺✳✶✹ ✶✶✵✳✻✶ ✺✳✹✼ ❈((t f ) 4.42 · 10 5 6.44 · 10 5 2.02 · 10 5 ❈■(t f ) 1.03 · 10 6 1.51 · 10 6 4.80 · 10 5 ❈❉(t f ) 5.04 · 10 3 7.30 · 10 3 2.26 · 10 3 ❚❛❜•❡ ✷✿ ■♥❢♦r♠❛t✐♦♥✕❞❡♣❡♥❞❡♥t ✈❛❝❝✐♥❛t✐♦♥ ❝❛s❡ ✭ϕ 0 = 0. ❼ ❈♦♥st❛♥t ✈❛❝❝✐♥❛t✐♦♥ ✭D = 0✮✱ ✇✐t❤ r❛t❡ ϕ 0 = ϕ p1 0 = 4.25 · 10 −3 ❞❛②s −1 ✭r❡❞ •✐♥❡s✮✱ ✇❤✐❝❤ ✐s s✉❝❤ t❤❛t t❤❡ ♣❡❛❦ ✈❛•✉❡ ♦❢ I s ✐s ❡q✉❛• t♦ t❤❡ ♣❡❛❦ ✈❛•✉❡ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✐♥❢♦r♠❛t✐♦♥✕❞❡♣❡♥❞❡♥t ✈❛❝❝✐♥❛t✐♦♥✳ ❖♥❡ ❝❛♥ ♥♦t❡ t❤❛t✱ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ ❡♣✐❞❡♠✐❝ ♣❡❛❦ ♦❝❝✉rs ❡❛r•✐❡r✱ ❛t t = 119 ❞❛②s✱ ❛♥❞ t❤❡ ✜♥❛• ❝✉♠✉•❛t✐✈❡ ♥✉♠❜❡r ♦❢ ❞❡❛t❤s ✐s s♠❛••❡r✿ ❈❉(t f ) = 5, 948❀ ❼ ❈♦♥st❛♥t ✈❛❝❝✐♥❛t✐♦♥ ✭D = 0✮✱ ✇✐t❤ r❛t❡ ϕ 0 = ϕ p2 0 = 7.87 · 10 −3 ❞❛②s −1 ✭❣r❡❡♥ •✐♥❡s✮✱ ✇❤❡r❡ t❤❡ ♣❡❛❦ ♦❢ I s ✐s ❤❛•✈❡❞ ✇✳r✳t✳ t❤❡ ❝❛s❡ ♦❢ ✐♥❢♦r♠❛t✐♦♥✕❞❡♣❡♥❞❡♥t ✈❛❝❝✐♥❛t✐♦♥✳ ❚❤❡ ❡♣✐❞❡♠✐❝ ♣❡❛❦ ♦❝❝✉rs ✈❡r② ❡❛r•②✱ ❛t t = 72 ❞❛②s✱ ❛♥❞ t❤❡ ✜♥❛• ❝✉♠✉•❛t✐✈❡ ♥✉♠❜❡r ♦❢ ❞❡❛t❤s ✐s r❡•❛t✐✈❡•② ♠♦❞❡st✿ ❈❉(t f ) = 2, 203✳ ❙✐♠✉•❛t✐♦♥s ❢♦r t❤❡ ❝❛s❡ ❱❆❳✲✸✵ ❛r❡✱ ♦❢ ❝♦✉rs❡✱ ❣r❛♣❤✐❝❛••② s✐♠✐•❛r t♦ t❤♦s❡ ✐♥ ❋✐❣✳ ✹❀ ❤❡♥❝❡✱ ❝♦rr❡s♣♦♥❞✐♥❣ ♣•♦ts ❛r❡ ❤❡r❡ ♦♠✐tt❡❞✳ ❋r♦♠ ❛ q✉❛♥t✐t❛t✐✈❡ ♣♦✐♥t ♦❢ ✈✐❡✇✱ t♦ ❝♦♠♣❛r❡ t❤❡ r❡s✉•ts ✐♥ t❤❡ ❝❛s❡ ❱❆❳✲✸✵ ✇✐t❤ t❤❡ ❝❛s❡ ❱❆❳✲✵✱ ✇❡ ❢♦❝✉s ♦♥ t❤❡ s❝❡♥❛r✐♦ 

❼ t❤❡ ♦❝❝✉rr❡♥❝❡ t✐♠❡ ♦❢ t❤❡ s②♠♣t♦♠❛t✐❝ ♣r❡✈❛•❡♥❝❡ ♣❡❛❦✱ ❛r❣max(I s )❀ ❼ t❤❡ ❝✉♠✉•❛t✐✈❡ ❞✐s❡❛s❡✕✐♥❞✉❝❡❞ ❞❡❛t❤s ❛t t f = 365 ❞❛②s✱ ❈❉(t f )✳ ❲❡ st❛rt ❜② ✐♥✈❡st✐❣❛t✐♥❣ ❤♦✇ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ♣❛r❛♠❡t❡rs✱ ♥❛♠❡•② t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❝♦✈❡r❛❣❡✱ k✱ ❛♥❞ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❞❡•❛②✱ T a = a −1 ✱ ♠❛② ❛✛❡❝t t❤❡ ❡♣✐❞❡♠✐❝✬s ❝♦✉rs❡✱ s❡❡ ❋✐❣✳ ✻✳ ❲❡ ♦❜s❡r✈❡ t❤❛t ❢♦r ❛r❣max(I s ) ❛♥❞ ❈❉(t f ) ✭❛s ✇❡•• ❛s max(I s )✱ ❈■(t f ) ❛♥❞ ❈((t f )✮✱ t❤❡ ♣❛tt❡r♥s ♦❢ t❤❡ ❝♦♥t♦✉r ♣•♦ts ❛r❡ s✐♠✐•❛r❀ ✐♥ ♣❛rt✐❝✉•❛r✱ ❢♦r s♠❛•• k = 0.2✱ t❤❡ r❛♥❣❡ ♦❢ t❤❡ s✐♠✉•❛t❡❞ ✈❛r✐❛❜•❡ ✇❤❡♥ T a ✐♥❝r❡❛s❡s ✐s •❛r❣❡✱ ✇❤❡r❡❛s ❢♦r k = 1✱ t❤❡ r❛♥❣❡ ✐s r❡str✐❝t❡❞ ❛♥❞ •♦✇✳ ❚❤❡ ✐♥✈❡rs❡ ♣❤❡♥♦♠❡♥♦♥ ✐s ♦❜s❡r✈❡❞ ❢♦r ❈❱(t f )✱ ✇❤❡r❡ t❤❡ r❛♥❣❡ ✐s r❡str✐❝t❡❞ ❛♥❞ s♠❛•• ❢♦r •♦✇ k = 0.2✱ ✇❤✐•❡ ✐t ✐s •❛r❣❡r ❢♦r k = 1✳ ❚❤❡♥✱ ✇❡ ✐♥✈❡st✐❣❛t❡ ❤♦✇ t❤❡ ❢❛❝t♦r ♦❢ ✈❛❝❝✐♥❡ ✐♥❡✛❡❝t✐✈❡♥❡ss✱ σ✱ ❛♥❞ t❤❡ ✐♥❢♦r♠❛t✐♦♥✲✐♥❞❡♣❡♥❞❡♥t ❝♦♥st❛♥t ✈❛❝❝✐♥❛t✐♦♥ r❛t❡✱ ϕ 0 ✱ ❛✛❡❝t t❤❡ s❛♠❡ q✉❛♥t✐t✐❡s ❝♦♥s✐❞❡r❡❞ ❛❜♦✈❡✳ ❚❤❡ r❡s✉•ts ❛r❡ s❤♦✇♥ ✐♥ t❤❡ ❝♦♥t♦✉r ♣•♦ts ✐♥ ❋✐❣✳ ✼ ❢♦r t❤❡ ❝❛s❡ ♦❢ ❝♦♥st❛♥t ❜❛s❡•✐♥❡ ✈❛❝❝✐♥❛t✐♦♥ ✭ϕ 0 = 0.002 ❞❛②s −1 ❛♥❞ D = 0✮ ❛♥❞ ✐♥ ❋✐❣✳ ✽ ❢♦r t❤❡ ❝❛s❡ ♦❢ ✐♥❢♦r♠❛t✐♦♥✕❞❡♣❡♥❞❡♥t ✈❛❝❝✐♥❛t✐♦♥ ✭ϕ 0 = 0.002 ❞❛②s −1 ❛♥❞ D = 500µ/Λ✮✳ ❲❡ ♠❛② ♦❜s❡r✈❡ t❤❛t t❤❡ q✉❛♥t✐t❛t✐✈❡ ✐♠♣❛❝t ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥✕❞❡♣❡♥❞❡♥t ✈❛❝❝✐♥❛t✐♦♥ ✐s r❡♠❛r❦❛❜•❡ ✭❜✉t t❤✐s ✇❛s ❡①♣❡❝t❡❞✮✳ ❆s ❢♦r t❤❡ s❤❛♣❡s ♦❢ t❤❡ ♣•♦ts✱ ✇❡ ♥♦t❡ t❤❛t t❤❡ ♣•♦ts ❢♦r ❈❱(t f ) ✭♣❛♥❡•s ❆✮ ❛♥❞ ❢♦r t❤❡ t✐♠❡ ❛t s②♠♣t♦♠❛t✐❝ ♣r❡✈❛•❡♥❝❡ ♣❡❛❦s ✭♣❛♥❡•s ❇✮ ❛r❡ r❡♠❛r❦❛❜•② ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♦t❤❡r ♣•♦ts✳ ▼♦r❡♦✈❡r✱ t❤❡ ♣•♦t ❢♦r ❈❱(t f ) ✐s q✉❛•✐t❛t✐✈❡•② ❞✐✛❡r❡♥t ✐♥ t❤❡ ✐♥❢♦r♠❛t✐♦♥✕❞❡♣❡♥❞❡♥t ✈❛❝❝✐♥❛t✐♦♥ ❝❛s❡ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ❝❛s❡ ♦❢ ❝♦♥st❛♥t ✈❛❝❝✐♥❛t✐♦♥✳ ❆s ❞✐s❝✉ss❡❞ ✐♥ ❙❡❝t✐♦♥ ✹✳✷✱ t❤❡ ❜❛s❡•✐♥❡ ✈❛•✉❡ ♦❢ t❤❡ tr❛♥s♠✐ss✐♦♥ r❛t❡ β ❤❛s ❜❡❡♥ ❡st✐♠❛t❡❞ ✉s✐♥❣ t❤❡ ♦✣❝✐❛• ❞❛t❛ ❢r♦♠ ✶✻ ❆✉❣✉st t♦ ✶✸ ❖❝t♦❜❡r ✷✵✷✵ r❡•❡❛s❡❞ ❜② t❤❡ ■t❛•✐❛♥ ❛✉t❤♦r✐t✐❡s✳ ❚❤❡r❡❢♦r❡✱ ✐t t❛❦❡s 1.45 · 10 7 1.18 · 10 6 V (t f ) 4.47 · 10 7 −1.09 · 10 6 ❈❱(t f ) 4.51 · 10 7 −1.12 · 10 6 max(I s ) 5.03 · 10 4 −7.34 · 10 3 arg max(I s ) ✶✶✺✳✻✻ ✶✵✳✺✷ ❈((t f ) 4.02 · 10 5 −3.97 · 10 4 ❈■(t f ) 9.44 · 10 5 −8.91 · 10 4 ❈❉(t f ) 4.59 · 10 3 −444.88 ❚❛❜•❡ ✸✿ ■♥❢♦r♠❛t✐♦♥✕❞❡♣❡♥❞❡♥t ✈❛❝❝✐♥❛t✐♦♥ ❝❛s❡ ✭ϕ 0 = 0. 

✐❝s✿ s✐♥❣✉•❛r •✐♠✐ts✱ ♦✛✲❡q✉✐•✐❜r✐✉♠ ❛♥❞ tr❛♥s✐t✐♦♥s ✭P|■◆ ✷✵✶✼(❇❑◆❈❊✮✳ |✳❉✳▼✳ t❤❛♥❦s t❤❡ s✉♣♣♦rt ❜② •◆❋▼ t❤r♦✉❣❤ t❤❡ ❣r❛♥t (♦✉♥❣ |❡s❡❛r❝❤❡r ✷✵✷✵✳ ❚❤❡ ❛✉t❤♦rs t❤❛♥❦ t❤❡ ❤❛♥❞•✐♥❣ ❡❞✐t♦r ❛♥❞ t✇

❙✳ ❇❤❛tt✳ ❊st✐♠❛t✐♥❣ t❤❡ ❡✛❡❝ts ♦❢ ♥♦♥✕♣❤❛r♠❛❝❡✉t✐❝❛• ✐♥t❡r✈❡♥t✐♦♥s ♦♥ ❈❖❱■❉✕✶✾ ✐♥ ❊✉r♦♣❡✳ ◆❛t✉r❡✱ ✺✽✹✿✷✺✼✕✷✻✶✱ ✷✵✷✵✳ ❬✸✷( ❋r❡♥❝❤ P✉❜•✐❝ ❍❡❛•t❤ ❆❣❡♥❝②✳ ❉♦♥♥é❡s ❤♦s♣✐t❛•✐èr❡s r❡•❛t✐✈❡s á •✬è♣✐❞è♠✐❡ ❞❡ ❈❖❱■❉✕✶✾✳ ❤tt♣s✿✴✴✇✇✇✳❞❛t❛✳❣♦✉✈✳❢r✴❡♥✴❞❛t❛s❡ts✴❞♦♥♥❡❡s✲❤♦s♣✐t❛•✐❡r❡s✲r❡•❛t✐✈❡s✲❛✲•❡♣✐❞❡♠✐❡✲

②s✐s t♦ ❡st✐♠❛t❡ ♣♦t❡♥t✐❛• s♣r❡❛❞ ❛♥❞ s❡❛s♦♥❛•✐t② ♦❢ ❝♦r♦♥❛✈✐r✉s ❞✐s❡❛s❡ ✷✵✶✾ ✭❈❖❱■❉✕✶✾✮✳ ❏❆▼❆ ◆❡t✇♦r❦ ❖♣❡♥✱ ✸✭✻✮✿❡✷✵✶✶✽✸✹✕❡✷✵✶✶✽✸✹✱ ✷✵✷✵✳

✷✵✷✵✳ ❤tt♣s✿✴✴✇✇✇✳✇❤♦✳✐♥t✴❞♦❝s✴❞❡❢❛✉•t✲s♦✉r❝❡✴❝♦r♦♥❛✈✐r✉s❡✴s✐t✉❛t✐♦♥✲r❡♣♦rts✴ ✷✵✷✵✵✶✷✶✲s✐tr❡♣✲✶✲✷✵✶✾✲♥❝♦✈✳♣❞❢❄s❢✈rs♥❂✷✵❛✾✾❝✶✵❴✹✱ ✷✵✷✵✳ ✭❆❝❝❡ss❡❞ ♦♥ ▼❛r❝❤ ✷✵✷✶✮✳

✐s ✐t tr❛♥s♠✐t✲ t❡❞❄ ❤tt♣s✿✴✴✇✇✇✳✇❤♦✳✐♥t✴♥❡✇s✲r♦♦♠✴q✲❛✲❞❡t❛✐•✴❝♦r♦♥❛✈✐r✉s✲❞✐s❡❛s❡✲❝♦✈✐❞✲✶✾✲❤♦✇✲✐s✲ ✐t✲tr❛♥s♠✐tt❡❞✱ ✷✵✷✶✳ ✭❆❝❝❡ss❡❞ ♦♥ ❆♣r✐• ✷✵✷✶✮✳

We introduce a COVID-19 epidemic model in presence of non mandatory vaccination The choice to vaccinate depends on present and past information about the disease The analyses show that a voluntary vaccination is unable to eliminate the infection Numerical simulations point out the deep impact of information-related parameters We find that the influence of seasonality on disease dynamics is not neglectable