Progetto Mathematical biology

V

from the vaccinated or not vaccinated state: vaccinated individuals go to a class (E ), while sus-

i

U

ceptible and unvaccinated go to an other exposed compartment (E ). Since the purpose of this

i

project is to study the vaccine effects, the infected individuals are distinguished from the exposed

compartment in which they belonged and also from the symptomatic or asymptomatic clinical state

AU

SU

AV

SV , I

, I

, I ). As in the description of the equations for Group 1, the asymptomatic can go

(I i

i

i

i S D

only to survival compartment (R ) and the symptomatic can also finish in dead class (R ).

i i

In this study is important to highlight the fact that the parameters describing the path of a vacci-

nated individual, are different from an unvaccinated one. The parameters regarding the vaccinated

individuals are indicated with a letter V as superscript. The schemes, which describe the dynamic

of the various groups, are shown in Figure 4 and in Figure 5.

Figure 4: Dynamic of Group 1

5 6

i = 1

Figure 5: Dynamic of Group

The contact matrix used in the simulations is constructed starting from the values in the literature [1]

C C

and with some manipulations on the elements and . These modifications hold the proportions

ij ji

· ≈ ·

N C N C

with the purpose of having the assumption .

i ij j ji

 

4.3065 1.9556 3.7230 2.4661 0.8182

2.2128 7.9858 4.3652 3.6920 0.8648

 

 

3.1543 3.2686 6.6914 4.1419 1.1765

C =  

 

1.5578 2.0611 3.0880 4.7167 1.3536

 

0.8348 0.7798 1.4168 2.1863 1.3913

The force of infection is in relationship with the social contact matrix C and also from the param-

A S

r r

eters and , which their value is dependent from the basic reproduction number chosen in the

simulation. This function is imposed as a linear combination between the elements of the contact

S A

r r

matrix, the rates and and moreover from the proportions of the infected individuals in each

group, differing the symptomatic from the asymptomatic.

5 AU

S SU SV A AU

·

·

S S A A )

+ I ) + r (I + I

r (I

·

· + r I

r I X j j

j

j

1

1 + C

λ = C ij

i i1 N N

1 j

j=2

v(·),

The function which describes the vaccination, is obtained interpolating the data took from

T H

27

the literature[2]. The data concerns the period from the starting of the vaccination in Italy

T H

4

December 2020, until April 2021. The behaviour of the function in the after days is defined

considering a maximum of 30000, 400000, and 500000 vaccinations per day as declared by Italian

administration. In Figure 6 we can see the oscillations of these vaccinations which probably are due

to the lack of doses of the vaccine. 6

Figure 6: Vaccination function

The differential equations that describe the dynamic in Group 1, in which for etical consideration we

do not consider the vaccination, are given by: A

dS dI

1 1 A

A ·

−σ · · · − · − I

= λ S = +ν (1 π ) E γ

1 1 1 1 1 1

1

dt dt S

dE dR

1 1 A A S S

· · − · · · − ·

= +σ λ S ν E = +γ I + γ (1 α ) I

1 1 1 1 1

1 1 1 1

dt dt

S D

dI dR

1 1

S S S S

· · − · · ·

= +ν π E γ I = +γ α I

1 1 1

1 1 1 1

dt dt

The equations for the others groups, in which we consider the vaccination campaign, are:

dS

i −v(t) − · ·

= σ λ S

i i i

dt

dV i V

· − · ·

= +q v(t) σ λ V

i i

i

dt

dU

i − · − · ·

= +(1 q) v(t) σ λ U

i i i

dt U

dE i U

· · · · − ·

= +σ λ S + σ λ U ν E

i i i i i i i

dt

V

dE i V V

· · − ·

= +σ λ V ν E

i i

i i

dt

SU

dI

i U S SU

· · − ·

= +ν π E γ I

i i i

dt

AU

dI

i U A AU

· − · − ·

= +ν (1 π ) E γ I

i i i

dt

SV

dI

i V V S SV

· · − ·

= +ν π E γ I

i i i

dt

AV

dI

i V V A AV

· − · − ·

= +ν (1 π ) E γ I

i i i

dt

S

dR

i A AV A AU S V SV S SU

· · · − · · − ·

I + γ I + γ (1 α ) I + γ (1 α ) I

= +γ i

i i I i I

dt

D

dR

i S V SV S SU

· · · ·

= +γ α I + γ α I

i

i I i

dt

The overall system is composed by 50 differential equations, 6 for Group 1 and 44 for the others

groups. The computations are made by the mathematical software Matlab, in particular the solutions

are given by ODE45 build-in function.

In Table 3 we summarise the explanation of the various terms and parameters.

7

S susceptible

i

V vaccinated with success

i

U vaccinated without success

i

V

E exposed from vaccinated group

i U

E /E exposed from unvaccinated group

1 i

SV

I infected and symptomatic from vaccinated class

i

AV

I infected and asymptomatic from vaccinated class

i SU

S /I

I infected and symptomatic from unvaccinated class

1 I

A AU

I /I infected and asymptomatic from unvaccinated class

1 I

D

R dead

i

S

R survival

i

v(t) number of individual vaccinated at time t

q probability of success of vaccined

σ susceptibility of an unvaccinated individual [3]

i

V

σ susceptibility of a vaccinated individual

i

λ force of infection

i

ν rate from exposed to infected [4]

π probability of becoming symptomatic from unvaccinated class [5]

i

V

π probability of becoming symptomatic from vaccinated class

i

S

γ rate of recovery/death of the symptomatic individuals [6]

i

A

γ rate of recovery/death of the asymptomatic individuals [5]

i

α probability of symptoms becoming severe from unvaccinated class [5]

i

V

α probability of symptoms becoming severe from vaccinated class

i Table 3: Terms and Parameters

The subscript "i" means the relationship with Group i.

2.3 The basic reproduction number

In our simulations we set an initial basic reproduction number, which summarises in its value, the

condition for that the epidemic spreads or goes toward the extinction.

In the following computations we are going to show the theoretical computations that allow us to

obtain this value for our multi-group system.

The basic idea is to linearize the infectious component, evaluated in the disease free equilibrium.

Since we have 15 equations of the infection subsystem, the dimensions of the jacobian computed in

×

15 15.

the disease free equilibrium are With the purpose to see the matrix in the best way, we

define the following submatrices:  

 

S A S A

−ν σ C r σ C r 0 σ C r σ C r

i ii i ii i ij i ij

S

−γ

ν π 0 0 0 0

J = J =

i

i,i i,j

   

i A

− −γ

ν (1 π ) 0 0 0 0

i i

The matrices which the components regard only the same group (J ) can be splitted in two matrices:

i,i

∀

J = T + M i.

transition and transmission matrices: i,i i,i i,i

The elements of the transition matrix depends from the parameters involved the passage of an

individual from a compartment to an other. Instead the elements of the transmission matrix depends

from the parameters that are involved in the infection from an individual to an other.

8

After this splitting, the jacobian can be written as a sum of a matrix T and another matrix B.

   

T 0 0 0 0 B J J J J

1,1 1,1 1,2 1,3 1,4 1,5

J T 0 0 0 0 B J J J

2,1 2,2 2,2 2,3 2,4 2,5

   

   

J J T 0 0 0 0 B J J

J = T + B = +

3,1 3,2 3,3 3,3 3,4 3,5

   

   

J J J T 0 0 0 0 B J

4,1 4,2 4,3 4,4 4,4 4,5

   

J J J J T 0 0 0 0 B

5,1 5,2 5,3 5,4 5,5 5,5

where    

S A

−ν 0 0 0 σ C r σ C r

i ii i ii

S

−γ

ν π 0 0 0 0

T = B =

i

i,i i,i

   

i A

− −γ

ν (1 π ) 0 0 0 0

i i

From the Perron-Frobenius Theory for non-negative matrices, we can define the basic reproduction

−1

·

R K = B (−T )

number as the spectral ratio of the next generation matrix , in others words:

0 −1

R = ρ(K) = ρ(−BT )

0

From the literature we know that the infectivity of symptomatic and asymptomatic are proportional

S A

r

r and .

[7].Then it is reasonable to consider proportional the parameters

This result is useful because if we fixed the values of the others parameters in the system, we can

S

r

compute the basic reproduction number in function of just and vice versa. We can see this

S

R r

relationship between and in Figure 7.

0 S

R r

Figure 7: related to

0

3 Methods and Simulations

The deterministic mathematical model has been proved by various simulations in different scenarios.

We consider three type of vaccination effects:

(σ );

1. reduction of the susceptibility i (π );

2. reduction of the probability to get symptoms i (α ).

3. reduction of the probability of the symptoms to become severe i

9

As first question we ask which one of these type of vaccine is the best and which vaccination strategy

is the best considering the age groups.

Since the efficacy of the vaccine (e ) is nowadays unknown, because it can be computed only after

f f

enough vaccinations, we suppose three different maximum values: 60%, 80%, 95%. The efficacy

is related to the parameters of the vaccination terms considering a linear decreasing distribution in

the age-groups: maximum value for the Group 2, minimum value for the Group 5, as shown in Table 4.

Group 2 Group 3 Group 4 Group 5

− − −

e e 2.5% e 5% e 7.5%

f f f f f f f f