In this paper, we investigate the impact of parallel vaccination among different priority groups on vaccine procurement cost, social cost caused by infected individuals of different priority groups, and the economic cost imposed on the business owners, shopkeepers, and retail stores during lockdown policy implementation. Therefore, we propose a mathematical model to determine the best vaccine allocation to the supposed priority groups and implement the lockdown policy and other health protocols. The health workers at high to very high risk of acquiring and transmitting infection are excluded from the proposed model. The priority groups are adults above 65 years old, individuals between 50 and 64 years old, and retailers. The retailers could start their activity after vaccination, even during the lockdown policy implementation. The vaccination has the herd effect, which refers to the indirect protection of unvaccinated people
Proposed Susceptible–Exposed–Infectious–Recovered (SEIR) epidemic model of COVID-19 before vaccination.
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As shown in
Proposed SEIR epidemic model of COVID-19 after vaccination.
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i Index of suppliers (i = 1, … , n) j Index of priority groups (j = 1, … , m) k Index of businesses that are affected by lockdown policy (k = 1, . . . , P) pi Price of each vaccine dose from supplier i Capi Capacity of supplier i BCk Cost of locking down business k SCj Social cost caused by infected individuals of group j S1,jሺt) Number of susceptible individuals of the group j before vaccination (after vaccination) at time t (S2,jሺt)) E1,jሺt) Number of exposed individuals of the group j before vaccination (after vaccination) at time t (E2,jሺt)) Ia1,jሺt) Number of asymptomatic infected individuals of the group j before vaccination (after vaccination) (Ia ሺt)) at time t Is1,jሺt) Number of symptomatic infected individuals of the group j before vaccination (after vaccination) at (Is ሺt)) time t Hq1,jሺt) Number of home-quarantined individuals of the group j before vaccination (after vaccination) at time t ቀHq2,jሺt)ቁ H1,jሺt) Number of hospitalized individuals of the group j before vaccination (after vaccination) at time t ቀH2,jሺt)ቁ R1,jሺt) Number of recovered individuals of the group j before vaccination (after vaccination) at time t ቀR2,jሺt)ቁ ςj The recruitment rate of the priority group j βj The disease transmission rate of the priority group j δj The natural death rate of the priority group j ηj The death rate from COVID-19 in the priority group j μj Progression rate of COVID-19 based on its incubation period in the priority group j εj The proportion of exposed individuals who have progressed into asymptomatic individuals of the priority group j αj Rate of being home quarantined in the priority group j α′j Rate of being hospitalized in priority group j γj Rate of natural recovery of asymptomatic infected individuals in priority group j γ′j Rate of being symptomatic infected individuals from asymptomatic ones in priority group j λj Rate of recovery of hospitalized individuals in priority group j ℛ0,j Basic reproduction number of the priority group j Tj The time at which the epidemic in the priority group j ends C Fixed cost of the immunization program τj Start time of the priority group j’s vaccination ϕ Vaccine unit shortage cost FC $ The fixed unit cost of delivering a COVID-19 vaccine dose ( )
dose.distance
di0 The distance between the supplier i and the central distribution depot
The developed three-objective model aims to simultaneously minimize the vaccine procurement cost, the total social cost incurred by infected individuals of different priority groups (home quarantine or hospitalized ones) before and after vaccination, as well as the economic cost of shop closures. Therefore, finding out the best time to immunize each priority group and the number of vaccine doses assigned to each group is necessary.
The primary purpose of the first objective is to minimize the cost required to reduce or enhance the immunization intensity.
Min Z1 = ∑i pi. Vi + FC. di0. Vi. (1)
Eq. (1) denotes the total cost of vaccine procurement from different suppliers. Since the vaccine price depends on the vaccine type and its brand, we consider different prices for the supplied vaccines. Also, we consider that the vaccines supplied by different suppliers have different effectiveness. In addition, in Eq. (1), we consider the cost of vaccine delivery from the supplier to the central depot, which is shown in the second term.
τj Tj
Min Z2 = ∑[∫ SCjሺHq1,jሺt) + H1,jሺt))dt + ∫ ሺ SCjሺHq2,jሺt) + H2,jሺt)) + C uj ሺt))dt]. (2)
0 τ 2
The first term in Eq. (2) denotes the social cost of hospitalized and home-quarantined individuals of the priority group before vaccination. The second term is the cost, as mentioned earlier, after immunization, and the third is the vaccination cost considering the optimal variable. Notably, the third term aims to maximize the number of recovered individuals using the possible minimal control variable. Notably, the square of the control variable reflects the severity of vaccination side effects
τJ Tj
Min Z3 = ∑k[BCkሺτJ)ሺ∫ S1,Jሺt)dt) + ∫ ሺ BCkሺτJ)ሺHq2,Jሺt) + H2,Jሺt))]. (3)
0 τj
The last objective function minimizes the economic impact caused by shop closures. The first term in Eq. (3) denotes the cost imposed on businesses due to the lockdown before the vaccination of retailers as a priority group. The second term is the cost imposed on the business due to the infection of retailers. The cost imposed on the business is a function of the period of the lockdown policy. For the sake of simplicity, we suppose a linear function for the mentioned cost as follows:
BCk(τJ) = AτJ + B. (4)
Also, we consider a fixed cost of consumers losing to the shops. Thus, we would have Eq. (4) for the cost of business/shop closure.
Before the vaccination of each priority group, there is an epidemic model without a control tool, and the infected individuals recover naturally. The differential equations of the epidemic model would be as Eqs. (5)- (11).
dS1,jሺt)
= ςj − βjS1,jሺt) ∑ ቀIa1,jሺt) + Is1,jሺt)ቁ − δjS1,jሺt), 0 ≤ t < τj. (5)
dt j=1 dE1,jሺt)
= β S ሺt) ∑ ሺIa ሺt) + Is ሺt)) − μ E ሺt) − δ E ሺt), 0 ≤ t < τ . (6)
j 1,j j=1 1,j 1,j j 1,j j 1,j j dt dIa1,jሺt)
′ (7)
= εjμjE1,jሺt) − γjIa1,jሺt) − γ Ia1,jሺt) − ሺδj+ηj)Ia1,jሺt), 0 ≤ t < τj.
dt j dIs1,jሺt)
= ሺ1 − εj)μjE1,jሺt) + γ Ia1,jሺt) − ሺδj+ηj)Is1,jሺt) − αjIs1,jሺt) − α jIs1,jሺt), 0 ≤ t <
dt j
τj.
dHq1,jሺt)
= α Is ሺt) − ሺδ +η )Hq ሺt) − ν Hq ሺt), 0 ≤ t < τ . (9)
j 1,j j j 1,j j 1,j j dt dH1,jሺt)
= α′ Is ሺt) − ሺδ +η )H ሺt) − λ H ሺt), 0 ≤ t < τ . (10)
j 1,j j j 1,j j 1,j j dt dR1,jሺt)
= ν Hq ሺt) + λ H ሺt) + γ Ia ሺt) − δ R ሺt), 0 ≤ t < τ . (11)
j 1,j j 1,j j 1,j j 1,j j dt
When the required dose of vaccine is assigned to each priority group and the immunization program is started, the epidemic model would change by considering the vaccination as a control tool.
Constraint Error! Reference source not found. indicates that we should supply the number of vaccine doses for all the priority groups to control the epidemic. Notably, the demand for vaccine doses is a function of the value of the optimal control tool, the number of susceptible individuals in the group, and the time of vaccination.
Tj ∗
∑i Vi ≥ ∑j ∫ uj S2,jሺt)dt. (12)
τj
Finally, Eq. Error! Reference source not found. denotes that the maximum number of vaccine doses supplied by supplier i equals its capacity.
Vi ≤ Capi. (13)