key: cord-0934885-wo3x7ua1
authors: Anita, S.; Banerjee, M.; Ghosh, S.; Volpert, V.
title: Vaccination in a two-group epidemic model
date: 2021-01-13
journal: nan
DOI: 10.1101/2021.01.10.21249557
sha: cf7e1d7c321406f24b983b8f74b8d637f34b3ec1
doc_id: 934885
cord_uid: wo3x7ua1

Epidemic progression depends on the structure of the population. We study a two-group epidemic model with the difference between the groups determined by the rate of disease transmission. The basic reproduction number, the maximal and the total number of infected individuals are characterized by the proportion between the groups. We consider different vaccination strategies and determine the outcome of the vaccination campaign depending on the distribution of vaccinated individuals between the groups.

Epidemic progression in a heterogeneous population depends on the proportion of different groups characterized by the rate of disease transmission [3, 4, 5, 6] . In this work we study the efficacy of vaccination campaign for different vaccination strategies. We consider the epidemic progression in a two-group population consisting of susceptible (S 1 , S 2 ) and infected (I 1 , I 2 ) compartments and described by the following model:

where β ij are the rates of disease transmissions, σ j (j = 1, 2) are the clearance rates of infected individuals, and N is the total population. Here V 1 and V 2 denote the number 1 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted January 13, 2021. ; https://doi.org/10.1101/2021.01.10.21249557 doi: medRxiv preprint NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. of vaccinated in the first and second group, respectively. Let us note that the number of individuals in each class, which can be infected, is S j − V j , j = 1, 2. We suppose that vaccination is fully efficient in the sense that vaccinated individuals cannot become infected.

For this model we are intended to calculate the basic reproduction number, size of the epidemic and study the effect of vaccination to arrest the disease progression.

In this section we determine the basic reproduction number in the heterogeneous population, the total and the maximal number of infected individuals depending on the number of vaccinated individuals.

At the beginning of epidemic, let us assume that S 10 and S 20 denote the number of susceptible individuals in the two groups such that S 10 +S 20 = N , and we define S 10 N = k and S 20 N = 1−k, where 0 ≤ k ≤ 1. The Jacobian matrix of the system (1.1) -(1.2) evaluated at the disease free equilibrium point is

Clearly, two eigenvalues equal to zero, while the largest eigenvalue is given by

where we have assumed σ 1 = σ 2 = σ, β 12 = β 21 = (β 11 + β 22 )/2 for simplicity of presentation.

Equating the largest eigenvalue to zero, we find the basic reproduction number is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted January 13, 2021. ; https://doi.org/10.1101/2021.01.10.21249557 doi: medRxiv preprint the regions of epidemic growth and the region epidemic extinction can be obtained from the relation R 0 = 1:

The epidemic extinction occurs in E 1 and the epidemic progresses in E 2 where: 

Taking a sum of equations (1.1), (1.2), and then integrating between t = 0 and t = ∞, we obtain the equalities:

is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted January 13, 2021. ; under the assumption that I j (0) = I j (∞) = 0, j = 1, 2 and S f 1 , S f 2 are the final sizes of two susceptible groups. Next, we divide equation (1.1) by S j , and integrate from 0 to ∞:

using the equalities in (2.1). Introducing four new quantities

S 20 , and assuming that σ 1 = σ 2 = σ, β 12 = β 21 = (β 11 + β 22 )/2, we obtain the following equations

with respect to x 1 and x 2 . The positive solution of this system, satisfying the restriction 0 < x 1 , x 2 < 1, determines the final size of susceptible populations (Fig. 2, left) . We can now determine the number of infected individuals in each group at the end of epidemic and the total number of infected in both groups, is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted January 13, 2021. ; https://doi.org/10.1101/2021.01.10.21249557 doi: medRxiv preprint their maxima (j = 1, 2). Integrating equations S 1 + I 1 , S 2 + I 2 from 0 to t m , we obtain:

where I m j = I j (t m ), j = 1, 2. Next, we divide equation (1.1) by S j , and integrate over [0, t m ] to find:

Assuming that I j (t m ) = 0, we get from (1.2):

As before, we assume that σ 1 = σ 2 = σ and set γ ij = β ij /σ, u 1 = I m 1 /N , u 2 = I m 2 /N , and v 1 , v 2 as defined in the previous subsection. Using (2. 3), we can rewrite equation (2.2) as follows:

Solving this system of equations, we find u j and, consequently, I m j , j = 1, 2 (Fig. 2, right) . We then use formulas (2.3) to determine S m j , j = 1, 2.

Since there is a cost related to vaccination, an optimal control problem (optimal vaccination) is proposed:

meaning that we are interested to minimize total damage of the epidemics, which includes the cost related to the total number of infected individuals and the cost of the vaccine. Here,

is the control/vaccination strategy. Since, due to certain constraints, not all individuals may be vaccinated we have imposed the constraints on V 1 , V 2 : 0 ≤ V j ≤ S j0 , j = 1, 2, and γ is a positive constant. Let is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted January 13, 2021. Parameter values are the same as in Fig. 2 .

Consider the following adjoint problem

where A T V is the transposed of the matrix

Following the method presented in [1, 2] , we can prove that for any θ ∈ R 2 such that

for any sufficiently small > 0.

The evaluation of directional derivative of J allows us to derive a conceptual iterative algorithm (gradient type) which improves at each step the control V = (V 1 , V 2 ). On the other hand, we can prove as in [1] that there exists at least one optimal control V * is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted January 13, 2021. ; https://doi.org/10.1101/2021.01.10.21249557 doi: medRxiv preprint

Using the form of the directional derivative of J we deduce that

This allows us to improve the above mentioned gradient-type algorithm. More detailed analysis and applications of this approach will be presented elsewhere.

Different vaccination strategies are illustrated in Fig. 3 . S 1 corresponds to people less than 60 years old, S 2 to people more than 60. Assume that 7 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted January 13, 2021. ; k = 0.8, that is, S 10 = 0.8N and S 20 = 0.2N . Consider the vaccination strategies where V 1 /V 2 =const and let us vary this ratio. The total numbers of infected individuals in each group at the end of epidemic are presented in the following table. The ratio V 1 /V 2 = 4 corresponds to a random choice of vaccinees. As before, increase of the proportion of the first group decreases the total number of infected I tot 1 + I tot 2 . However, the total number of deaths depends on the mortality rate in each group. In the application to the Covid-19, we assume that the mortality rate of infected individuals in the second group is of the order of magnitude 10 times larger than in the first group. In this case, the total number of deaths decreases for a larger proportion of vaccinees in the second group.

An Introduction to Optimal Control Problems in

Mathematical Methods in Optimization of Differential Systems, Kluwer, Dordrecht

Mathematical Tools for Understanding Infectious Disease Dynamics

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Extended SEIQR type model for COVID-19 epidemic and data analysis

The last author has been supported by the RUDN University Strategic Academic Leadership Program.