Equilibrium Solutions of a Modified SIR Model with Vaccination and

Several Levels of Immunity

FLAVIUS GUIAŞ

Department of Mechanical Engineering

Dortmund University of Applied Sciences and Arts

Sonnenstr. 96, 44139 Dortmund

GERMANY

Abstract: We consider a system of ordinary differential equations which extends the well-known SIR model for

the dynamics of an epidemic. The main feature is that the population is divided in several subgroups according

to their immunity level, which has as a consequence different infection rates. The maximum level of immunity

can be achieved either by recovering from an infection, or by possible vaccination. We consider the cases that

the vaccination rate is independent on the size of infected population, or that it depends also on this value by a

power law. In addition, we assume that the immunity level can decay in time. The goal of this paper is to analyze

the existence and uniqueness of equilibrium solutions, which can be either a trivial (disease-free) equilibrium,

with no infections, or an endemic equilibrium, with a certain amount of infected individuals. Moreover, we give

conditions for the local asymptotic stability of the unique trivial equilibrium solution. It will turn out that, if

this is the case, then there exists no endemic equilibrium, which means that the epidemic can be eradicated, by

arriving at herd immunity. On the other hand, if the trivial equilibrium is unstable, then we prove the existence

of an endemic equilibrium which, under natural conditions, turns out to be unique. The stability of the endemic

equilibrium remains still an open problem.

Key-Words: ordinary differential equations, epidemic model, SIR, equilibrium solutions, waning immunity,

vaccination

Received: March 15, 2023. Revised: December 2, 2023. Accepted: December 18, 2023. Published: December 31, 2023.

1 Introduction

The SIR model is a standard system of ordinary differ-

ential equations used for the modeling of epidemies,

see, [1], [2], [3]. It is a compartmental model, since

the population is divided into several compartments:

S - susceptible, I - infected and R - recovered. The

dynamics can be described by transitions between dif-

ferent stages. Susceptible individuals can get the dis-

ease by contact with infected ones. The correspond-

ing rate is directly proportional to the numbers Sand

I, while the transition from Ito Roccurs at a rate

proportional to the number Iof infected individuals.

An individual in this last stage is already immune and

cannot contribute anymore to the spread of the dis-

ease. By scaling the total population to 1, the system

of ordinary equations corresponding to the SIR model

is the following:

dt =−β·I·S

dt =β·I·S−α·I

dt =α·I(1)

The coefficients of the linear terms, i.e. the rates of

passing from one state to another, are inversely pro-

portional to the average time spent in the correspond-

ing state. With Tinf the average time spent in the in-

fectious state, we can therefore assume that α=T−1

inf ,

while for βwe can consider the form β=R(t)·T−1

inf .

The term R(t)denotes the time dependent effective

reproduction number, i.e. the average number of fur-

ther infections produced by the contacts with one in-

fectious individual. At the beginning of the epidemic

its value is equal to R0, the so called basic repro-

duction number, but after that it may vary due to re-

strictions, social distancing, increased frequency of

testing, etc. Possible variations within this simplest

framework are the SIRS and the SIS models, where in

the first case the recovered individuals become again

susceptible after some time, whereas in the latter situ-

ation an individual moves after infection directly into

the susceptible state. In [3], it is shown that for these

models the local asymptotic stability of the disease-

free equilibrium holds if R0<1, which means that

the disease will die out due to arriving at herd immu-

nity. If R0>1the disease will become endemic,

reaching an endemic equilibrium which turns out to

be asymptotically stable. In [4], the global stability of

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equilibrium solutions of the SIR, SIRS and SIS mod-

els is proven by constructing appropriate Lyapunov

functions.

The SIR model can be also extended to a SEIR

model, by introducing the intermediate state E - ex-

posed, or SVEIS with the state V - vaccinated, see,

[5], where also the local stability of equilibrium so-

lutions is analyzed. The stability of extensions of the

classic SIR model is analyzed also in [3], where it is

shown that in some cases there may be an asymptoti-

cally stable endemic equilibrium even if R0<1and

other situations in which there is an endemic equilib-

rium that is unstable for certain values of R0>1.

The phenomenon of waning immunity, where the

rate of loss of immunity depends on the time since

recovery, is modeled by delay differential equations

like in [6], by systems of integro-differential equa-

tions like in [7], [8], or by coupling ordinary and par-

tial differential equations like in [9], [10], the latter

reference considering also vaccination, which is also

considered in [11].

Other papers consider also age-structured popu-

lations, like: [12], with five levels of immunity and

a discrete age structure, while, [13], considers the

age and the immunity level as continuous variables

and the dynamics is modeled by a system of integro-

differential equations.

The paper, [14], considers a model which

describes two competing diseases, influenza and

COVID-19 and analyzes the dynamics of the corre-

sponding non-linear system of differential equations

under vaccination strategy and immunity waning.

In this paper we will modify the above SIR model

by considering, besides the infected state I, several

levels of immunity. The full immunity against infec-

tion can be obtained either by passing through the dis-

ease, or by vaccination. We further assume that the

level of immunity decays with time. Therefore we

introduce the states S0, . . . Sm, m ≥2, where S0de-

scribes the lowest possible level of immunity, while

Smstands for the maximum level, or full immunity.

The system of ordinary differential equations corre-

sponding to this model is the following:

dSk

dt =−βk·I·Sk−γk·Ip·Sk−ηk·Sk

+ηk+1Sk+1,0≤k≤m−1

dSm

dt =α·I+

m−1

k=0

γk·Ip·Sk−ηm·Sm

dt =

m−1

k=0

βk·I·Sk−α·I(2)

The assumption is that the total population remains

constantly equal to 1, since birth and death phenom-

ena are not considered. The infection mechanism is

similar to that of the SIR model, but with different

transmission rates βk≥0, depending on the immu-

nity level. We consider the assumption βm= 0, that

is, at the maximum level mthere exists full immunity.

This full immunity can be achieved either by pass-

ing through the disease, which means that infected in-

dividuals, which recover at rate α > 0, change from

state Ito the state Sm, or by vaccination from any

other level 0≤k≤m−1. The term which models

this transition is considered to be of the form γk·Ip·Sk

with the vaccination rate γk>0(we assume that

γm= 0, that is, the vaccination does not take place

at the highest immunity level) and with the parameter

p∈[0,1]. This choice is motivated by the fact that

the interest of individuals for vaccination might de-

pend on the size of the infected population Ithrough

the factor of the form Ip. For p= 0 the corresponding

term is considered as γk·Sk, i.e. the vaccination rate

is independent on the size of the infected population.

We will consider also the situation of no vaccination

at all, with γ≡0.

The decay of immunity is modeled by the transi-

tion from state Sk+1 to Skwith 0≤k≤m−1at rate

ηk>0, while η0= 0 (the immunity cannot decay

further at this lowest possible level).

Similarly to the SIR model we assume that α=

T−1

inf and βk=Rk·α, with Rkbeing the basic repro-

duction number, that is, the average number of sec-

ondary infections generated from an infected individ-

ual on immunity level k. We consider the reproduc-

tion numbers corresponding to each immunity level

to be constant in time.

We may consider natural monotonicity assump-

tions on the coefficients Rk, γk, which are decreas-

ing with k(reproduction numbers and vaccination rate

are lower at higher immunity levels, being =0 on the

highest level), but in our mathematical model we will

make use of them only if it is strictly necessary, other-

wise we keep the assumptions as general as possible.

One such example is the assumption Rk<1for

k≥1(the values of R0>0can be arbitrary).

This is no significant restriction, since we are free to

choose the compartments of our model. By recalling

the property of such epidemic models, which states

that, if the basic reproduction number is <1, then the

disease will eventually die out, we can interpret the

condition Rk<1for k≥1in the sense that we can

define the (partially) immune population groups in

our model as those for which the epidemic vanishes,

if the immunity would remain forever on that given

level. This assumption is needed only for the unique-

ness of the endemic equilibrium solution, otherwise

we don’t make use of it. Only for the basic reproduc-

tion number of the population without any immunity

we will consider also the possibility that R0>1.

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The system (2) was introduced by the present au-

thor in [15], by using a stochastic approach based

on Markov jump processes. A convergence result

was proved and also numerical simulations were per-

formed.

A related work is reported in [16], where the model

is denoted as SIR(k)Smodel, corresponding to klev-

els of immunity. The decay of immunity follows a

similar mechanism as described previously, but for

the decay rates two possible specific assumptions are

made: linear or exponential decay, while in [15], or

in this paper, the rates are kept in a general form. In

[16], there are considered two possible vaccination

schemes: a general one, similar to the one considered

in this paper with constant rates, but also a so called

rational scheme, where only the susceptible popula-

tion (with no immunity at all) and only one additional

compartment with partial immunity are subjected to

vaccination. The mathematical results of the men-

tioned paper are the following:

1. The existence of a unique trivial equilibrium is

shown for R0<1(no other equilibrium exists in

this case).

2. The model without vaccination has a unique en-

demic equilibrium if and only if R0>1.

3. The model with vaccination for m= 2 has a

unique endemic equilibrium if and only if Rη>

1, where Rηis a term which involves R0and the

coefficients of the system of equations. It is con-

jectured that this result holds also for arbitrary

m > 2.

4. The paper discusses also the connection between

the ODE model and an ODE-PDE model for

m→ ∞.

Compared to the aforementioned paper, which

considered only a particular form of the immunity de-

cay rates, the present work comes with new results

from several points of view. First, we keep the vacci-

nation and immunity decay rates as general as possi-

ble. Secondly, beyond constant vaccination rates, we

consider also the scenario that the vaccination rates

are proportional to some power Ipof the size of the

infected population. This models the situation that if

the incidence of the disease is low, the interest for vac-

cination is also low, and vice-versa.

Regarding the results of this paper, we show exis-

tence and uniqueness of the trivial and endemic equi-

librium (for all m, not only for m= 2) under similar

conditions as in [16], but in a more general setting

and also in an additional scenario regarding the vac-

cination. Moreover, we analyze also the asymptotic

stability properties of the disease-free (trivial) equi-

librium, and show its stability for R0<1in the case

that the vaccination rates depend on Ipand give also

necessary and sufficient conditions for stability if the

vaccination rates are independent on I, in the cases

m= 2 and m= 3. We conjecture that this stability

property holds for all k.

Concerning the endemic equilibrium, we show ex-

istence and uniqueness basically under the condition

that ensures that the disease-free equilibrium in both

scenarios regarding vaccination is unstable. Since we

give only a general existence proof and cannot com-

pute it by an exact formula as in the case of the trivial

equilibrium, the question of stability of this endemic

equilibrium is not addressed in this paper. The re-

sults are qualitatively similar to those corresponding

to some extensions to the SIR model described in [3],

since, in the case that the vaccination rates are inde-

pendent on I, the disease-free equilibrium can be sta-

ble also if R0>1, basically if the immunity does

not decay too fast and if the vaccination rates are high

enough. In the discussion section we will also point

out some practical limitations of this mathematical re-

sult.

The present paper is structured as follows. In Sec-

tion 2 we discuss the nonlinear system of equations

corresponding to the equilibrium state, that is, the

right hand sides of the ODE system (2) are set to be

equal to 0. We also compute the corresponding Jaco-

bian matrix, since the local asymptotic stability of an

equilibrium solution is ensured by the negative sign

of the real parts of the eigenvalues of the Jacobian

matrix in this point. After describing this setup, in

the next sections we present the main results of this

paper. In Section 3 we discuss the existence, unique-

ness and stability of the disease-free equilibrium, with

no infections (I= 0) and in section 4 the existence

and uniqueness of the endemic equilibrium. In each

of these two sections we consider both scenarios re-

garding vaccination which are assumed in this paper.

We conclude with a summary of the results given in

Section 5 and with a discussion of their practical rel-

evance which is presented in Section 6.

2 The nonlinear system of equations

at equilibrium

The considered dynamics, which involve only transi-

tions between different states, but no changes in the

population size, which is assumed to be equal to 1,

is reflected also by the fact that the sum of the RHS

in (2) is equal to 0, which means a conservation of

the total population. Since the equations whose solu-

tions are the equilibria which we are interested in are

dependent, we replace the last equation with the as-

sumed conservation property. The nonlinear system

of equations which we consider is therefore the fol-

lowing:

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−βk·I·Sk−γk·Ip·Sk−ηk·Sk+

+ηk+1Sk+1 = 0

for 0≤k≤m−1

α·I+

m−1

k=0

γk·Ip·Sk−ηm·Sm= 0

k=0

Sk+I= 1 (3)

We will also use the relation corresponding to the

RHS of the equation for Iin (2):

−

m−1

k=0

βk·I·Sk+α·I= 0 (4)

Note that equation (4) can be also obtained by sum-

ming up the first m+ 1 equations in system (3).

Since the sum of the components remains constant,

for the stability analysis of the equilibrium points,

which are the solutions of system (3), we disregard the

equation for Smand replace Sm= 1 −Pm−1

k=0 Sk−I

in the equation for Sm−1.

For this reduced system, considering the variables

in the order S0, S1, . . . Sm−1, I, the Jacobian matrix,

for which the signs the real parts of its eigenvalues de-

termine the local stability property of the equilibrium

points of the system (2), is given by:







−β0I−γ0Ipη1. . . 0−β0S0−γ0pIp−1S0

0−β1I−γ1Ip−η1. . . 0−β1S1−γ1pIp−1S1

−ηm−ηm. . . −βm−1I−γm−1Ip− −βm−1Sm−1−ηm−

−ηm−1−ηm−γm−1pIp−1Sm−1

β0I β1I . . . βm−1IPm−1

k=0 βkSk−α







(5)

When analyzing the existence and possible

uniqueness of equilibrium solutions, we will consider

separately the following situations regarding the

dynamics of vaccination: either the form γk·Ip·Sk

with p∈(0,1] where γk≥0(the value γk= 0 for

all kis also possible), or of the form γk·Sk, where

at least γ0>0. That is, in this case the vaccination

rates are independent on the size of the infected

population. We will analyze the trivial equilibrium

with I= 0 and also endemic equilibria with I= 0.

3 Equilibrium solutions with I= 0

We first discuss the equilibrium solutions with no

infections under different assumptions regarding the

vaccination.

3.1 The case p > 0or γ≡0

We consider first the case that the vaccination rate de-

pends on the size of the infected population or that we

have no vaccination at all. Under these assumptions,

for I= 0 the system (3) reduces to:

−ηk·Sk+ηk+1Sk+1 = 0,0≤k≤m−1

−ηm·Sm= 0

k=0

Sk= 1 (6)

Starting from the equation for Smwe obtain suc-

cesssively Sk= 0 for 1≤k≤m(since ηk= 0 if

k≥1). By noting that η0= 0 and considering the

conservation property, we obtain that the only equi-

librium with I= 0 is given by S0= 1, Sk= 0 for

k≥1. Under the assumption that p= 0 or γ≡0

we replace all terms involving γkin the Jacobian (5)

with 0, therefore the value of Jtaken in the current

equilibrium point will be







0η10. . . 0−β0

0−η1η2. . . 0 0

−ηm−ηm−ηm. . . −ηm− −ηm

−ηm−1

0 0 0 . . . 0β0−α







(7)

Theorem 3.1. Assume p > 0or γ≡0.

If R0<1, the unique solution S0= 1, S1=. . . =

Sm=I= 0 of (3) is an asymptotically stable equi-

librium point of (2).

If R0>1, this trivial equilibrium point is unsta-

ble.

Proof. From (7) can be easily seen (by expansion

w.r.t the last row) that one eigenvalue of Jis λ0=

β0−α=α(R0−1) whose sign depends on the as-

sumption on R0. The other eigenvalues are the solu-

tions of p(λ) = 0 with

p(λ) =



−λ η10. . . 0

0−η1−λ η2. . . 0

0 0 −η2−λ . . . 0

0 0 0 . . . ηm−1

−ηm−ηm−ηm. . . −ηm−

−ηm−1−

−λ



Substracting the first column from all the others we

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obtain

p(λ) =



−λ η1+λ λ . . . λ

0−η1−λ η2. . . 0

0 0 −η2−λ . . . 0

0 0 0 . . . ηm−1

−ηm0 0 . . . −ηm−1−

−λ



We add next all other rows to row 1 and therefore we

have

p(λ) =



−ηm−λ0 0 . . . 0

0−η1−λ η2. . . 0

0 0 −η2−λ . . . 0

0 0 0 . . . ηm−1

−ηm0 0 . . . −ηm−1−

−λ



Expanding the determinant with respect to the first

row and noting that the minor which arises has a tri-

angular form, we conclude that the other eigenvalues

of Jare λk=−ηk<0, k = 1, . . . m.

If R0<1, then all eigenvalues of the Jacobian

at the equilibrium point are negative and we conclude

that this equilibrium is asymptotically stable (locally).

If R0>1then there exists a positive eigenvalue and

therefore the trivial equilibrium is unstable.

This result corresponds to the intuitive trivial ex-

pectation that, if R0<1, that is, the reproduction

number is less than 1 even for the lowest level of

immunity, the disease will vanish and the compo-

nents will approach the equilibrium state where all

individuals are on the lowest immunity level. More-

over, since the vaccination terms have either the form

γk·Ip·Skwith p∈(0,1] or γ≡0(no vaccina-

tion), we conclude also that near this equilibrium the

vaccination becomes irrelevant to its stability, which

is always given. For R0>1however, we will see

that the system has also at least one nontrivial (en-

demic) equilibrium with I= 0 which, under further

additional conditions, is unique.

3.2 The case p= 0 (vaccination independent

on I)

In this case, for I= 0 the system (3) reduces to:

−(γk+ηk)·Sk+ηk+1Sk+1 = 0,0≤k≤m−1

m−1

k=0

γk·Sk−ηm·Sm= 0

k=0

Sk= 1 (8)

There are m+ 2 equations for the m+ 1 unknowns

S0, . . . Sm, but we note that summing up the first

m+ 1 equations we obtain 0, so in fact we can con-

sider only the equations for 0≤k≤m−1and the

conservation property defined by the last equation.

Since all coefficients which appear are >0, we

can express Sk+1 in terms of Sk, that is, successively

S1, S2, . . . Smin terms of S0in the form Sk=θk·S0.

Taking into account that Pm

k=0 Sk= 1, we obtain

that the system (8) has a unique solution with all Sk>

The interpretation is that, by vaccination at the

given constant rates depending only on the immunity

level, combined with immunity decay, the system has

a trivial equilibrium. In this equilibrium point with no

infections (I= 0) the Jacobian has the value







−γ0η1. . . 0

0−γ1−η1. . . 0

−ηm−ηm. . . −γm−1−ηm−1−

−ηm

0 0 . . . 0

−β0S0

−β1S1

−βm−1Sm−1−ηm

Pm−1

k=0 βkSk−α







(9)

For a stability result we need a negative sign of the

real parts of the eigenvalues of this matrix. Due to the

structure of the last row we have that one eigenvalue

is λ0=Pm−1

k=0 βkSk−α=α(Pm−1

k=0 RkSk−1). For

the stability of the equilibrium given by the solution

of (8) together with I= 0, it is therefore necessary

that λ0<0.

Remark 3.1. The necessary condition

Pm−1

k=0 RkSk<1for the stability of the trivial

equilibrium solution with I= 0 in the case p= 0

(vaccination rate independent on I) and γ0>0is

equivalent to the inequality Ψ( ¯

R, ¯γ, ¯η)<0, where

Ψ( ¯

R, ¯γ, ¯η) = (R0−1)γ−1

0+ (R1−1)η−1

m−1

k=2

(Rk−1)η−1

k−1

j=1

η−1

j(γj+ηj)−

−η−1

m−1

j=1

η−1

j(γj+ηj).(10)

This inequality holds always if Rk<1for all k, but

it can be also fulfilled even if this is not the case, if

in addition the vaccination rates γkand the immunity

decay rates ηkare chosen properly.

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Proof. From the system (8) we use the relations

Sk+1 =η−1

k+1(γk+ηk)·Sk, k = 0, . . . , m −1. Since

η0= 0, we obtain therefore S1=η−1

1γ0·S0and

Sk=η−1

k−1

j=1

η−1

j(γj+ηj)·γ0·S0, k = 2, . . . , m.

The conservation property

k=0

Sk= 1 is thus

equivalent to:

S0+η−1

1γ0·S0+

k=2

η−1

k−1

j=1

η−1

j(γj+ηj)·γ0·S0= 1

and from this we obtain:

S−1

0= 1 + η−1

1γ0+

k=2

η−1

k−1

j=1

η−1

j(γj+ηj)·γ0.

(11)

The condition

m−1

k=0

RkSk<1is therefore equivalent

to R0S0+R1η−1

1γ0·S0+

m−1

k=2

Rkη−1

k−1

j=1

η−1

j(γj+

ηj)·γ0·S0<1.By multiplying with S−1

0using (11),

bringing all the terms on the LHS and dividing subse-

quently with γ0, we obtain that a necessary condition

for stability is given by

(R0−1)γ−1

0+ (R1−1)η−1

m−1

k=2

(Rk−1)η−1

k−1

j=1

η−1

j(γj+ηj)−

−η−1

m−1

j=1

η−1

j(γj+ηj)<0.

It is clear that this inequality is fulfilled if one of the

following conditions holds:

•Rk<1for all k= 0, . . . , m −1.

•R0≥1, Rk<1, k = 1, . . . , m −1(the par-

tially immune individuals do not contribute to an

exponential spread of the epidemic, which hap-

pens mainly due to the non-immune population).

In this case the necessary condition for stability

is fulfilled if R0−1< γ0·η−1

m−1

j=1

(η−1

jγj+ 1).

This global inequality involving all vaccination

and immunity decay rates holds for example if

the vaccination rates γjare sufficiently high, or

if the immunity decay rates ηjare sufficiently

small, a property which confirms also our intu-

ition.

•Rk≥1for k= 0, . . . , m −1, (or basically no

restriction on Rk’s at all) if in addition the im-

munity decay rate ηmfrom the highest immu-

nity level (full immunity) is sufficiently small. A

larger threshold for this parameter can be consid-

ered for example if the vaccination rate γ0of in-

dividuals with no immunity is sufficiently large.

Expanding det(J−λI)with respect to the last row

for Jas in (9), one can see that the other eigenval-

ues than λ0=α(Pm−1

k=0 RkSk−1) depend only on

γi, ηi>0. We will show that for m= 2 and m= 3

these eigenvalues always have negative real parts.

Lemma 3.1. If m= 2 or m= 3 the eigenvalues of

the matrix

J1=







−γ0η1. . . 0

0−γ1−η1. . . 0

−ηm−ηm. . . −γm−1−ηm−1−

−ηm







have negative real parts, provided γi, ηi>0.

Proof. For the moment we will keep m≥2arbitrary.

The characteristic polynomial of degree mis given

by:

pm(λ) =



−γ0−λ η1. . . 0

0−γ1−η1−λ . . . 0

0 0 . . . ηm−1

−ηm−ηm. . . −γm−1−ηm−1−

−ηm−λ



Expanding with respect to the first row we obtain:

pm(λ) = (−γ0−λ)·pm−1(λ)−

−η1



0η20. . . 0

0−γ2−η2−λ η3. . . 0

0 0 0 . . . ηm−1

−ηm−ηm. . . −γm−1−ηm−1

−ηm−λ



where pm−1(λ)is the characteristic polynomial of a

(m−1)×(m−1)- submatrix and has the same struc-

ture as pm(λ).

The second determinant can be expanded with re-

spect to the first column. By noting that eliminating

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the first column and the last row we obtain the deter-

minant of a lower triangular matrix with η2, . . . ηm−1

on the diagonal, we therefore have:

pm(λ)=(−γ0−λ)·pm−1(λ)+(−1)m

i=1

ηi

= (−1)m·(˜

pm(λ) +

i=1

ηi)

where the polynomial ˜pm(λ)has positive coefficients

and roots with negative real parts.

The stability property of this general polynomial

is still an open problem, but we will show next that

for m= 2 and m= 3 this polynomial has roots with

negative real parts.

For m= 2 we have

p2(λ) = 

−γ0−λ η1

−η2−γ1−η1−η2−λ

=λ2+ (γ0+γ1+η1+η2)λ+η1η2

+γ0(γ1+η1+η2)

Since all involved γi, ηjare >0, the quadratic equa-

tion p2(λ) = 0 has the form λ2+a1λ+a0= 0 with

a0, a1>0. This means that the sum of the roots −a1

is negative, while the product a0is positive. If both

roots are real (even if equal), they must be necessar-

ily negative. In the case of complex conjugate roots of

the form x±iy, their sum is equal with 2x=−a1<0

which means that both eigenvalues λ1,2have negative

real part.

For m= 3 we have

p3(λ) = 

−γ0−λ η10

0−γ1−η1−λ η2

−η3−η3−γ2−η2

−η3−λ



=−[λ3+ (γ0+γ1+η1+γ2 + η2+η3)λ2

+[γ0(γ1+η1) + γ0(γ2+η2+η3) +

+(γ1+η1)(γ2+η2+η3) + γ0η2η3]λ

+γ0(γ1 + η1)(γ2+η2+η3) + η1η2η3]

For the polynomial with positive coefficients

−p3(λ) = λ3+a2λ2+a1λ+a0we will use the

Routh-Hurwitz criterion in order to show that its

roots have negative real parts. According to Routh-

Hurwitz, this property is equivalent to the fact that

all principal minors of the following Hurwitz matrix

are >0:

H=



a2a00

1a10

0a2a0



This means that

a2>0,

a2a0

1a1

>0,

det H=a0·

a2a0

1a1

>0.

Since all coefficients are positive, it is therefore suffi-

cient to show that the principal minor of second order

is positive, i.e. a1a2−a0>0. By computing this

term from the above polynomial, it will turn out that

all negative terms from −a0cancel with correspond-

ing terms in a1a2and that the remaining difference is

positive. This proves the statement of the lemma.

By summarizing the previous results we obtain the

stability property in this case.

Theorem 3.2. Let p= 0,βi, γi, ηi>0and Skthe

solution of (8) for m= 2 or m= 3. If the condition

Pm−1

k=0 RkSk<1holds, then these Sktogether with

I= 0 are an asymptotically stable equilibrium point

of (2).

4 Existence of an endemic

equilibrium with I= 0

We will analyze now the equilibrium solutions of (2)

with I= 0. The main result is the following:

Theorem 4.1. Let α, βk, ηk>0, γk≥0for all k,

except βm=η0= 0.

(i) If p∈(0,1] (vaccination dependent on I) or

γ0= 0 (no vaccination for individuals on the

lowest immunity level) and β0> α ⇔R0>1,

(ii) If p= 0 (vaccination independent on I), γ0>0

and Ψ( ¯

R, ¯γ, ¯η)>0with Ψdefined in (10),

then the system (3) has at least a solution

S0, S1, . . . , Sm, I with all components in (0,1)

which corresponds to a nontrivial endemic equilib-

rium.

If in addition Rk<1for k≥1, then this solution

is unique.

Proof. The basic outline of the proof can be described

as follows. In order to compute the solution of the

nonlinear system (3), we show first that we can ex-

press all Skin terms of I. Inserting these terms into

the conservation property, it will turn out that the ex-

istence of a solution is equivalent to the existence of

a solution I∈(0,1) of this nonlinear equation. This

can be ensured if there is a sign change of the values

of this function between 0 and 1. In the case that this

function can be proven to be monotone, then the so-

lution, if it exists, turns out to be unique. Moreover,

in the case of monotonicity without sign change, we

will conclude that there exists no solution of the equa-

tion for I, which means that there exists no endemic

equilibrium of the system of differential equations.

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From the equations for 0≤k≤m−1in (3) we

obtain successively

Sk+1 =η−1

k+1(βkI+γkIp+ηk)Sk

and therefore Sk=Pk(I)·S0, where Pk(I)is a gen-

eralized polynomial of degree kin I, that is, a finite

linear combination of powers of I, the maximal ex-

ponent being equal to k(since p∈[0,1]). More-

over, all coefficients of Pkare positive and starting

with P0(I) = 1 and P1(I) = η−1

1(β0I+γ0Ip)(since

η0= 0) for 0≤k≤m−1we have the recursion

formula

Pk+1(I) = η−1

k+1(βkI+γkIp+ηk)·Pk(I)(12)

Dividing (4) by I= 0 we obtain Pm−1

k=0 βkSk=

α⇔Pm−1

k=0 βkPk(I)·S0=α⇔S0=α/Qm−1(I),

where Qm−1(I) = Pm−1

k=0 βkPk(I)is a generalized

polynomial of degree m−1in I.

Inserting the Sk’s computed above into the conser-

vation property Pm

k=0 Sk+I= 1 we obtain

k=0

Qm−1(I)·Pk(I) + I= 1 ⇔

k=0

Pk(I) + I·Qm−1(I)−Qm−1(I) = 0 ⇔

k=0

Pk(I)+(I−1)

m−1

k=0

βkPk(I) = 0 ⇔

Rm(I) = 0

where Rm(I)defined as above is a generalized poly-

nomial of degree min I.

We have that Rm(1) = αPm

k=0 Pk(1) >0since

all coefficients of Pkare >0.

We further have Rm(0) = Pm−1

k=0 (α−βk)Pk(0)+

αPm(0) = α(Pm−1

k=0 (1 −Rk)Pk(0) + Pm(0)).

Assuming first that p∈(0,1] or γ0= 0, we have

that P1(0) = 0 and therefore Pk(0) = 0 for all 1≤

k≤m. We thus have Rm(0) = α(1 −R0)<0if

R0>1.

Due to the sign change of Rmbetween 0 and 1, we

conclude that there exists at least a nontrivial solution

Iof the equation Rm(I) = 0.

If p= 0 then we have to consider system (8) and

for computing Rm(0) we need to compute the terms

Pk(0) which, by adapting (12) for the case p= 0

can be computed by Pk+1(0) = η−1

k+1(γk+ηk)Pk(0).

Since P0≡1, this is exactly the recursion for the Sk

from Remark 3.1. We therefore have P1(0) = η−1

1γ0

and Pk(0) = η−1

kQk−1

j=1 η−1

j(γj+ηj)·γ0for k=

2, . . . , m. Using the form

Rm(0) = α(

m−1

k=0

(1 −Rk)Pk(0) + Pm(0))

we thus have

α−1Rm(0) = 1 −R0+ (1 −R1)η−1

1γ0+

m−1

k=2

(1 −Rk)η−1

k−1

j=1

η−1

j(γj+ηj)·γ0+

+η−1

m−1

j=1

η−1

j(γj+ηj)·γ0

The condition Rm(0) <0is therefore equivalent to

Ψ( ¯

R, ¯γ, ¯η)>0with Ψdefined in (10). In this case,

due to the sign change of Rm, we have at least a

nonzero solution of the equation Rm(I) = 0.

Once the existence of a solution Iof the equation

Rm(I)=0is established, the corresponding values

Skcan be computed as follows: Pm

k=0 Sk+I=

1⇔Pm

k=0 Pk(I)S0+I= 1 and from this we

obtain S0= (1 −I)/ Pm

k=0 Pk(I). The other val-

ues can be computed then recursively by Sk+1 =

η−1

k+1(βkI+γkIp+ηk)Sk.

We will analyze next sufficient conditions for the

monotonicity of Rm, which implies the uniqueness of

this solution.

Indeed, deriving the function

Rm(I) = α

k=0

Pk(I)+(I−1)

m−1

k=0

βkPk(I)(13)

and taking into account that P0(I)≡1and thus

P′

0(I) = 0, we obtain

R′

m(I) = α

k=0

P′

k(I) +

m−1

k=0

βkPk(I) +

+(I−1)

m−1

k=0

βkP′

k(I)

=αP ′

m(I) +

m−1

k=0

βkPk(I) +

m−1

k=1

(α+ (I−1)βk)P′

k(I)

=α P′

m(I) +

m−1

k=0

RkPk(I)+

m−1

k=1

(1 −Rk+IRk)P′

k(I)!

Since Pk(I), P ′

k(I)>0for I > 0, we note that Rk<

1for k= 1, . . . , m −1is a sufficient condition for

R′

m(I)>0, i.e. for the uniqueness of the endemic

equilibrium.

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Note that, if Rm(0) >0and Rm(I)is increasing

(for example if Rk<1for k≥1), then there exists

no endemic equilibrium, but only the trivial one. In

the case p > 0or γ≡0we have Rm(0) >0if

R0<1, while if p= 0, γ0>0this is equivalent

to the inequality Ψ( ¯

R, ¯γ, ¯η)<0with Ψdefined in

(10). In both cases we can thus say that, if the local

stability condition for the trivial equilibrium with I=

0is fulfilled, then this is the unique equilibrium state.

5 Summary of the results

Consider the system (2) of ordinary differential equa-

tions modeling the spread of an epidemic into a popu-

lation subjected to possible vaccination, with several

levels of immunity which can decay in time. The ba-

sic reproduction numbers on these levels are given by

Rk=αβkand the vaccination rates are either of the

form γkIpwith p∈(0,1] or constant γk, i.e. inde-

pendent on the size of the infected population. The

results of this paper regarding existence, uniqueness

and stability of the equilibrium solutions can be sum-

marized as follows.

5.1 The case p > 0or γ≡0(vaccination

dependent on Ior no vaccination)

In this case we have a unique trivial equilibrium so-

lution with I= 0, with S0= 1, S1=. . . , Sm= 0.

If R0<1, this is the only equilibrium solution at all

and it is locally asymptotically stable. Note however

that this situation has no practical relevance, since the

reproduction numbers Rkshould logically decay with

increasing level of immunity k, therefore an epidemic

with such values of the spreading paremeters would

vanish by its natural dynamics, even without vaccina-

tion.

The relevant situation is therefore when we have

R0>1. In this case there exists at least one non-

trivial solution corresponding to an endemic equilib-

rium with I∈(0,1). If additionally Rk<1for

k≥1, then this nontrivial equilibrium solution is

unique. This assumption on the basic reproduction

numbers for higher levels of immunity is a natural

one, since we can define our compartments in a con-

venient way and can therefore consider that on any

immunity level larger than 0 the basic reproduction

number corresponds to a value for which the epidemic

will eventually die out.

Since we cannot compute explicitly the endemic

equilibrium solution, its stability under the conditions

considered in this paper can be only conjectured.

5.2 The case p= 0 (vaccination independent

on I)

In this case, which assumes implicitely that at least

γ0>0, there exists a unique trivial equilibrium so-

lution with I= 0 and Sk>0for all k. A necessary

(and at least for m= 2,3also sufficient) condition

for the local asymptotic stability of this solution is

Pm−1

k=0 RkSk<1or, equivalently, Ψ( ¯

R, ¯γ, ¯η)<0,

with

Ψ( ¯

R, ¯γ, ¯η) = (R0−1)γ−1

0+ (R1−1)η−1

m−1

k=2

(Rk−1)η−1

k−1

j=1

η−1

j(γj+ηj)−

−η−1

m−1

j=1

η−1

j(γj+ηj).(14)

In this case, if in addition the property Rk<1for

k≥1holds, by the proof of Theorem 4.1 and the af-

terward remark, we conclude that there exists no en-

demic equilibrium with I > 0.

By neglecting the negative terms with factors Rk−

1<0for k≥1, we have that a sufficient condition

such that the only equilibrium solution is the trivial

one with I= 0 is given by

R0−1< γ0·η−1

m−1

j=1

(η−1

jγj+ 1), together with

Rk<1for k≥1.

Note that the RHS of the former inequality is

monotone increasing in the vaccination rates γjand

monotone decreasing in the immunity decay rates ηj.

Assuming the natural condition R0>1, we obtain

that the unique equilibrium is the trivial one (which

implies the vanishing of the epidemic) if the vacci-

nation rates are sufficiently high and/or the immunity

decay rates are sufficiently low.

If, however, Ψ( ¯

R, ¯γ, ¯η)>0, then we have at least

one nontrivial (endemic) equilibrium with I > 0. The

uniqeness of this equilibrium is implied by the condi-

tion Rk<1for k≥1. The stability of such nontrivial

equilibria is still an open problem.

6 Discussion

In this final section we discuss the possible practical

relevance of the mathematical results of this paper. In

any epidemic model one is basically interested if the

epidemic can be eradicated, that is, if herd immunity

can be achieved by a suitable vaccination strategy, or

if its dynamics will allways stabilize in an endemic

state. The practical relevant case is R0>1, since

otherwise the disease will vanish by itself. We also

consider the assumption Rk<1for k≥1, which

was already explained in this paper.

Consider first the case that the vaccination rates

depend also on Ip. In this situation the interest for

vaccination is high only if the number of infections is

high and vice-versa. The case that we have no vacci-

nation can be also considered within the same frame-

work. In this situation the disease-free equilibrium

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turns out to be unstable and there exists a unique en-

demic equilibrium. This means that herd immunity

cannot be achieved under these assumptions on the

vaccination strategy.

However, in the case that the vaccination rates are

constant, independent on I, the picture turns out to be

different. We give a necessaray condition for stabil-

ity of the disease-free equilibrium which can be ful-

filled also if R0>1, if the vaccination rates are suffi-

ciently high and the immunity decay rates sufficiently

low. We proved that this condition is also sufficient

in the cases m= 2 and m= 3, which correspond

to the classes ’no immunity’, ’partial immunity’, ’full

immunity’ and ’no immunity’, ’low immunity’, ’high

immunity’, ’full immunity’, respectively. Moreover,

if the necessary condition for this trivial equilibrium is

fulfilled, then in turns out that that there exists no en-

demic equilibrium. That is, if stability is given, which

in our paper is shown for m≤3, then one can ar-

rive at herd immunity and therefore the disease can

be eradicated.

Nevertheless, this theoretical result has some prac-

tical limitations. The immunity decay rates are ba-

sically given, so they can not be changed, while the

vaccination rates cannot be increased arbitrarily, due

to medical and logistic reasons. So, depending on the

particular parameters of the model which correspond

to a given disease and on the values of the vaccina-

tion rates that can be assumed as realistic, in prac-

tice both outcomes are possible: either herd immu-

nity is achieved, according to the theoretical result,

or one arrives at an endemic equilibrium. This out-

come appears if the necessary condition for stability

of the disease-free equilibrium is not fulfilled, due to

fast decay of immunity and low vaccination rates, the

last fact being possible due to several reasons, also of

practical nature.

References:

[1] F. Brauer and C. Castillo-Chávez, Mathematical

Models in Population Biology and Epidemiology,

Springer, New York, 2001.

[2] F. Brauer, Compartmental Models in Epidemiol-

ogy, in F.Brauer, P.van den Driessche and J.Wu

(eds.) Mathematical Epidemiology, Springer,

Berlin, Heidelberg 2008, pp. 19-79.

[3] F. Brauer et al., Endemic Disease Models, Mathe-

matical Models in Epidemiology. Texts in Applied

Mathematics, vol 69. Springer, New York, 2019,

DOI: 10.1007/978-1-4939-9828-9_3

[4] A. Korobeinikov and G.C. Wake, Lyapunov

Functions and Global Stability for SIR, SIRS and

SIS Epidemiological Models, Appl.Math.Letters,

Vol.15, 2002, pp. 955-960.

[5] G.P.Sahu and J. Dhar, Analysis of an SVEIS epi-

demic model with partial temporary immunity

and saturation incidence rate, Applied Mathemat-

ical Modeling, Vol.36, 2012, pp.908-923.

[6] M.L. Taylor and T.W. Carr, An SIR epidemic

model with partial temporary immunity modeled

with delay, J. Math. Biol. Vol.59, 2009, pp.841–

880, DOI: 10.1007/s00285-009-0256-9

[7] S. Bhattacharya and F.R. Adler, A Time Since

Recovery Model with Varying Rates of Loss

of Immunity. Bulletin of Mathematical Biology,

Vol.74, 2012, pp.2810-2819.

[8] S. Nakata et al., Stability of epidemic models with

waning immunity, SUT Journal of Mathematics,

Vol.50, No.2, 2014, pp.205-245.

[9] M.V. Barbarossa and G. Röst, Mathematical mod-

els for vaccination, waning immunity and im-

mune system boosting: a general framework,J.

Math. Biol. Vol.71, No.6-7, 2015, 1737–1770.

[10] M. Ehrhardt et al., SIR-based mathematical

modeling of infectious diseases with vaccina-

tion and waning immunity. Journal of Compu-

tational Science, Vol. 37, 101027, 2019. DOI:

10.1016/j.jocs.2019.101027

[11] J.M. Heffernan and M.J. Keeling, Implications

of vaccination and waning immunity, Proc. R.

Soc. B Vol. 276, 2009, pp.2071–2080, DOI:

10.1098/rspb.2009.0057

[12] R-M. Carlsson et al., Modeling the waning and

boosting of immunity from infection or vaccina-

tion, Journal of Theoretical Biology, Vol. 497,

2020, 110265, DOI: 10.1016/j.jtbi.2020.110265

[13] K. Okuwa et al., An age-structured epidemic

model with boosting and waning of immune

status, Mathematical Biosciences and Engineer-

ing, Vol.18, No.5, 2021, pp.707–5736, DOI:

10.3934/mbe.2021289

[14] R. Musa et al., A non-linear differential equa-

tion model of COVID-19 and seasonal in-

fluenza co-infection dynamics under vaccina-

tion strategy and immunity waning, Health-

care Analytics, Vol.4, 2023, 100240, DOI:

10.1016/j.health.2023.100240

[15] F. Guiaş, Epidemic models with several levels

of immunity, in C.H. Skiadas, C. Skiadas (eds.),

Quantitative Demography and Health Estimates,

The Springer Series on Demographic Methods

and Population Analysis 55, Springer, 2023,

pp.163-174, DOI: 10.1007/978-3-031-28697-1

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

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[16] M. El Khalifi and Tom Britton, Extending

susceptible-infectious-recovered-susceptible epi-

demics to allow for gradual waning of immunity,

J. R. Soc. Interface Vol.20, 2023, 20230042, DOI:

10.1098/rsif.2023.0042

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