A dynamic reaction-restore-type transmission-rate model for COVID-19

FERNANDO CÓRDOVA-LEPE1, JUAN PABLO GUTIÉRREZ-JARA2

1Departamento de Matemática, Física y Estadística,

Universidad Católica del Maule,

Avenida San Miguel 3605, Talca,

CHILE

2Centro de Investigación de Estudios Avanzados del Maule,

Universidad Católica del Maule,

Avenida San Miguel 3605, Talca,

CHILE

Abstract: COVID-19 became a paradigmatic global pandemic for science, in a real laboratory inserted in reality

to understand how some dangerous virus spread can occur in human populations. In this article, a new strate-

gic epidemiological model is proposed, denoted β-SIR. It is because the transmission rate βfollows a proper

dynamic law, more precisely a reaction-restore type transmission rate model. Some analytical results associated

with dynamic consequences are presented for variables of epidemiological interest. It is concluded, observing the

geometry of variables plots, such as transmission rate, effective reproductive number, daily new cases, and actives,

that pandemic propagation is very sensible to the population behavior, e.g., by adherence to non-pharmaceutical

mitigations and loss of compliance levels.

Key-Words: Infection disease, SIR model, variable transmission rate, COVID-19.

Received: April 22, 2023. Revised: December 11, 2023. Accepted: January 19, 2024. Published: March 21, 2024.

1 Introduction

The use of ordinary differential equations (ODE) in

the epidemiological analysis of infectious diseases

presents a history of explosive growth, [1], [2], [3],

[4], [5], [6], [7], [8], [9], [10], [11], [12], far above

other mathematical possibilities, [13], [14], [15], [16],

[17], [18], [19], [20], [21]. As a tool, the ODEs stand

out for their ability, via interpretation, to describe, ex-

plain, and project (i.e., to model), dynamics of conta-

gion and population spread of diseases, which turns

them into virtual instruments for testing control ac-

tions. Thus, there is the possibility of establishing ex-

ante evaluation scenarios simulating population in-

terventions (e.g., in crisis contexts), such as mitiga-

tion measures or pharmaceutical-type solutions, [22],

[23], [24], [25]. There is also the possibility of ex-

post type analysis, that is, those that seek to deter-

mine the necessary conditions of the past to explain

the information and data of the present, [26], [27],

[28], [29]. Another important aspect is the possibility

of determining threshold parameter values with epi-

demiological significance, such as in herd immunity,

knowing the immune fraction to stop an epidemic.

They are also an important instrument for optimiza-

tion (according to certain objectives and restrictions)

between alternative health control strategies that have

been used in the past or are even in the design stage,

[30], [31], [32], [33].

Every expansion has a beginning; let us briefly re-

fer to the initial uses of ODEs as an instrument for

epidemiological analysis. Leaving aside the work of

the considered founder of demography [34], by intro-

ducing the first life table presenting mortality as sur-

vival rates [35], a properly mathematical perspective,

for the study of the spread and control of infectious

(integrating an ODE) appears with the work of [36],

in 1786 [37], [38], on smallpox and the effect of vario-

lation. However, it is a chronologically isolated mile-

stone concerning subsequent methodological devel-

opments. To our knowledge, it was not until a period

at the beginning of the twentieth century that the idea

of using mathematical analysis as a methodological

possibility for epidemiology began more clearly. A

sketch, mainly focused on the contributions of [39],

in 1906 [40], the introducer of the law of mass action

in a childhood infections paper, including measles, as

well as the essential reference to the contribution of

[41], with a focus on a mosquito population as vec-

tors, who in the second edition of The Prevention

of Malaria, published in 1911, builds mathematical

models of malaria transmission, [42]. Nevertheless,

if we are to seek the characterization of foundational

work, that first scientific article that succeeds, by ex-

panding its potential effectively, inaugurating a disci-

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pline, Mathematical Epidemiology, is the SIR model

(Susceptible - Infected - Recovered) as we know it

today, a purification by simplification of the triad of

works by [43], [44], [45], in the years 1927, 1932, and

1933, respectively.

To a large extent, the importance and widespread

use of the SIR model, as a model of the population

dynamics of infectious disease spread, lies in the effi-

cient and effective concurrence of a mechanistic and

generic explanatory perspective, on the one hand, and

the fair mathematical technicality on the other, [46].

The plausibility of this model has at least two sources:

a theoretical argumentation of hypotheses about the

process in the abstract and a certain geometric corre-

spondence of the numerical projections with the data.

It is this minimalist character, or low cost in complex-

ity, to represent the essentials of the epidemiological

process, which allows it to be classified as a strate-

gic model, [47], in contrast to a tactical model that

aspires to high fidelity with a specific reference of

reality (a certain disease and population, considering

many variables and precise relationships, always as-

piring to a high predictive capacity).

In the standard SIR model, that is, with constant

coefficients, if R0>1, we see that the asset curve

has a bell-shaped or unimodal shape. That is, it

has a single maximum that occurs when the suscep-

tible fraction (a decreasing function) is equal to the

value 1/R0, with R0the basic reproductive number

or the average number (at the beginning of the dis-

ease) of the new infective directly infected by an in-

fective while in said condition. The first estimates

(February 2020) for CO-VID-19 of R0were in the

range 1.40 −6.49, see, [48], already around May

of the same year, a systematic review reduced it to

2.81 −3.82, [49]. Another study for Western Eu-

ropean countries, with data until mid-March 2020,

places it at 1.90 −2.60, [50]. So, for example, with

aR0equal to 1.5or 3for COVID-19, we have that

said maximum would imply a susceptible percentage

between 33.3% and 66.6%; which respectively im-

plies between 2/3 and 1/3 of the population already

infected, which is absolutely far above what was ob-

served for the first wave, in many countries. Thus,

the conclusion is that the countries or communities

observed in general managed to lower the contagion

rate by changing behavior regarding contact with oth-

ers, obviously in voluntary terms or by mandate of

the health authority, through, for example, mitigating

measures, non-pharmaceutical type.

A strategic model, that is, a simple and generic

one, must have the potential to capture with a good

degree of precision the dynamics of the disease. For

example, the fact that the intensity of transmission can

change over time in response to behavioral changes in

the population due to official control interventions. It

is the case, for example, in models based on the SIR

model, that intervenes in the transmission rate (nor-

mally denoted β), which is constant in the standard

model, to represent some types of changes, e.g. in

the number of close contacts, environmental condi-

tions, or blockades in the passage of pathogens. In

this regard, two possibilities are visualized: using an

explicit function to represent the dependence of βon

time or another variable or indicator, [51], [52], or an

implicit one, [53]. In the latter, we represent changes

in beta, for example, through a dynamic law for the

derivative of beta. The novelty of our paper, inspired

by [54], is to introduce a new type of beta-SIR model

and to continue to study its dynamical consequences.

There are other possibilities to introduce natural

variability. For instance, probability distributions for

the intensity of transmission in a given period, have

been useful for modeling outbreaks in small-sized

populations or for accounting for the volubility of hu-

man behavior. In the case of non-uniform transmis-

sion, there are also network models, to represent inter-

actions in local groups, e.g., family, friends, or work,

which differ in the magnitude of the distance and the

interaction times. Another way to incorporate vari-

ability is to collect the experience of a type of trans-

mission behavior (adjusting to the data), so that it is

an input to project outbreaks in another period in pop-

ulations with some degree of similarity to that which

the experience had. In conclusion, if it is necessary

to represent more complex behaviors, one way is to

consider the variable beta by abstracting and inter-

preting the forces and conditioning that intervene in

determining the epidemic dynamics.

2 The model

2.1 Some preliminaries

Let us consider an SIR model, i.e., the population is

theoretically divided into susceptible, infectious and

removed groups, of respective sizes, at a time t, defin-

ing functions S(t),I(t)and R(t), such that







S′=−β(t)SI/N

I′=β(t)SI/N−γI

R′=γI,

(1)

in which a temporally variable transmission rate β(·)

stands out, but with parameters γ(recovery rate) and

N(total population S+I+R) constant.

The first order condition necessary for a minimum

or maximum of the active group, that is, I′= 0 at

some instant τ, since I′=γ{Re(t)−1}I, with Re=

(β/γ)S/N, implies Re(τ) = 1, or equivalently

s(τ) = γ/β(τ),(2)

where s(·) = S(·)/N, by (1), is a decreasing func-

tion.

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Then, this equality can occur at several moments,

depending on the variability of β(·). Thus, the model

can break the unimodal character of the typical active

SIR curve with βas a constant parameter. Thus, if

τ1and τ2, with τ1< τ2, are two consecutive critical

points of I(·), as s(τ1)> s(τ2), the relation (2) im-

plies β(τ1)< β(τ2). Moreover, if there is a peak at

τ1, then there is the possibility of a second peak at a

certain τ3,τ3> τ2, to the extent that βincreases from

a minimum at τ2.

In the literature associated with the mathemati-

cal modeling of infectious diseases, several ways are

found to incorporate changes in the transmission rate

due to environmental variations and human behav-

ior throughout a process of serious epidemic devel-

opment in a certain population. One way is to as-

sume β(·)as a specific and predetermined function

of time, as is the case in [55], [56], [57]. However,

we also find models that innovate by introducing a

dependence on other compromised variables, but still

explicitly, as in [58], [59]. However, some works do

not predetermine it: In [60], it is assumed that the in-

trinsic transmission rate changes as a stochastic pro-

cess with values limited to a limited range of possibil-

ities. In [61], the transmission rates involved are solu-

tions of a differential equation with stochastic noise.

In [62], a time-varying transmission rate is consid-

ered, assuming loss of adhesion to control actions, as

lockdown measures. Here, we will consider an evolv-

ing rate β(·)according to the theoretical dynamics of

the process itself; this is the novelty context of the

present work.

In [54], a dynamic law is introduced to express the

variability of the transmission rate of the type

β′(t) / β(t) = restitution

| {z }

f(·)

−

g(·)

z }| {

reaction (3)

that is, expressing the variability of β(·)as a constant

tension between the factors that tend to decrease (re-

action) and increase (restitution). Observe that, hav-

ing β(·)[time−1], it corresponds to β′(·)[time−2],

that is, (3) expresses acceleration changes. In this

sense, f(·)[time−1] and g(·)[time−1] correspond to

proportions of decrease and increase in transmission

rate per time unit.

Both the reaction factor and the restoration factor

operate through the way that individuals in the popu-

lation in question change their behavior regarding as-

pects that mean a change in the rate of contact with

infectious diseases or in the intensity of pathogenic

blockade. From the experience with COVID-19, we

know that population sectors stopped adhering to mit-

igation measures for various reasons, many of them

sensitive to economic and cultural situations. Al-

though the most widespread would be psychological

(pandemic fatigue, [63],[64]), economic (economic

insecurity, [65], [66]) and informational (low-risk

perception, [67], [68], [69], [70]) types. The exis-

tence or expectation of a vaccine and its implemen-

tation also alters risk perception.

Regarding the reaction factor, we observe that the

recommendations suggest that the health authority,

normally advised by a technical team of several ex-

perts, use various sources of information and indica-

tors to define the health status of the population and

implement mitigation measures, as occurred due to

the COVID-19 pandemic. Key information sources

include numerical epidemiological data (prevalence,

daily cases, reproductive numbers, etc.) that track the

strength of the infection and its components, such as

causal variables that explain the intensity of the virus

spread. It is about diagnosing and projecting to plan

an intervention, a deployment of mandates that aim to

guarantee the system’s medical care capacity for the

most serious cases.

A key element in understanding the restitution fac-

tor is that, although there may be a significant social

effort to reduce the natural (or intrinsic) rate of trans-

mission, the sustainability of this becomes difficult

over time. In fact, the greater and longer the exigen-

cies for behavior changes in the population, the more

likely it is to be lost in compliance.

2.2 The β(·)-SIR model

The main novelty of this work is to consider what we

denote by β-SIR model, that is, we have the suscep-

tible - infective - removed compartmental model de-

fined by (1), but now, this is coupled with a variable

transmission rate β(·)following the structure (3), via

the differential law:

β′(t) / β(t) = ν(β∗−β(t))

| {z }

f(β(t))

−

g(I′(t))

z }| {

µI′(t),(4)

with the initial condition β0:= β(0),S0:= S(0) =

N−I0,I0:= I(0) >0and R0:= R(0) = 0. An

important assumption is that R0=β0/γis greater

than one, so I′(0) >0. The parameters νand µare

called the coefficient of restitution and the reaction

coefficient, respectively.

Observe that in the formulation of (4) we consider

that the relative change in the rate is given by the ten-

sion between a reaction factor g(·)which is propor-

tional to a difference in each unit of time, the num-

ber of people who become infectious minus those who

leave the condition. While, the restitution factor f(·)

fulfills the function of putting pressure on the trans-

mission rate to return to a value, called potential or

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cultural (the one that would exist if there was no re-

action), of the rate equal to β∗. Therefore, if β < β∗

(resp. >) the restitution factor is positive (resp. neg-

ative) exerting upward pressure (resp. downward).

An important aspect is that the β∗rate is consid-

ered the intrinsic rate, which would occur in a virgin

population of both infectives and information or ac-

tions that have changed the natural (habitual) behav-

ior of people. In this sense, we see that β∗does not

necessarily coincide with β0:= β(0) and, in general,

it can be considered that β0≤β∗.

3 Analytical results

3.1 Case ν= 0

Let us note that equation (4) takes the form β′=

−µI′β, which deduces the existence of automatism in

the control, that is, βdecreases if and only if Igrows.

As the system starts with I(·)increasing, we see that

the peak of the assets occurs simultaneously with the

minimum value that the transmission rate can reach.

Observing that at critical times τ(I′(τ)and β′(τ)

null), we have the equality β′′(τ) = −µ I′′(τ)β(τ),

we deduce that temporally maxima (resp. minima) of

I(·)correspond to minimums (resp. maximums) of

β(·).

The literature offers several examples that assume

an exponential-type functional expression to estab-

lish a decrease in the contagion rate (see, [51], [52]).

However, several of them do not offer a return to the

original rate, but rather an asymptotic approximation

to a new level, although lower than the initial one. In

this case, we are assuming, by direct integration of

(4), that

β(t) = β0e−µ(I(t)−I0), t ≥t0.(5)

Thus, it is also exponential, but it correlates inversely

with the number of assets. In this formulation, which

only considers a reaction factor, it also differs in that

if the active group is expressed with values below I0

(e.g., a certain reaction threshold), the value of βin-

creases to β0. By the way, this is quite an optimistic

possibility, in that an arithmetic increase in the active

group translates into a geometric reduction in the con-

tagion rate.

In the idea of interpreting the reaction coefficient,

let us note that the reaction factor g(·)is equal to µ

when I′(τ)=1for some instant τ. Linearizing I(t)

at this moment, we have I(t)∼I′(τ)(t−τ)+I(τ) =

(t−τ) + I(τ), so if t=τ+ 1, we have I(τ+ 1) ∼

I(τ) + 1. Then µis the fraction by which βdecreases

so that the system does not incorporate one more in-

fectious agent per unit of time. Thus, µcan be con-

sidered a uniparametric measure of the intensity of

the effort required to lower the transmission rate. In

that sense, at the beginning of the propagation (while

the susceptible group is approximateable by the entire

population), assuming R0>1, if we require the asset

differential not exceed a quantity ∆Ibefore Re= 1,

according to (5), such as Re>R0e−µ∆I, you must

have µ≥ln (R0)/∆I.

Given a real number xwe define its sign, σ(x),

as −1,0or +1 depending on whether xis negative,

zero, or positive. Now, since I′/I=γ{Re(t)−1},

we have σ(Re−1) = σ(I′) = −σ(β′). So, at the

beginning of infectious development, at a certain in-

stant t0, if R0=Re(t0)>1, then β(·)will drop until

I(·)reaches its maximum. Furthermore, β(·)will re-

cover its initial value β0, only when I(·)decreases to

I0, that is, at time twhen

I(t)/I0=exp Zt

{Re(a)−1}da}= 1,

this is, Z[t0,t]

{Re(a)−1}da = 0.

That is, in practical terms, on a graph of the effec-

tive reproductive number versus time, βreturns to its

intrinsic value β0, when the amount of area that ac-

cumulates above and below the level Re= 1 is the

same.

The following result points towards control, as it

provides a range for the value of the effective repro-

ductive number, which depends on those eliminated

(recovered plus deaths) and on the susceptible frac-

tion of the population also being within a certain range

of values.

Theorem 1 The effective reproductive number, de-

fined by Re(t) = β(t)s(t)/γ, with s(t) = S(t)/N,

for t≥t0, while 0≤p < s(t)< q ≤1, satisfies

Λq,p(t)≤ Re(t)/R0≤Λp,q(t),(6)

where Λx,y(t) = y

xR0+[1−xR0]e−µR(t).

Proof: See Appendix A.

Let us observe that Λx,y(·)is a function such that

Λx,y(t)≤1,Λx,y(t0) = yand, furthermore Λ′

x,y(t) =

−{x/y}Λx,y(t)µR0e−µR(t)R′(t), that is, it is de-

creasing since always R′(t)≥0. See Figure 1.

3.2 Case ν > 0

Assuming that in (4) the function I(·)is known, its

integration as a linear equation leads us to:

Theorem 2 If we consider in (4) an initial time t0and

respective conditions β0=β(t0)and I0=I(t0),

then:

β(t) = β0

E(t0, t)

1 + νβ0Zt

E(t0, τ)dτ

,(7)

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0 5 10 15 20

Time

0.2

0.4

0.6

0.8

Trajectories

p,q

q,p

Re/R0

Fig. 1: It can be seen that when s(t)is between pand

q, quotient Re/R0is necessarily between Λq,p and

Λp,q. It is considered β0= 0.3,γ= 1/14 ν= 0,

µ= 0.5,p= 0.3and q= 0.7. The initial condition

is s(0) = 0.9,i(0) = 0.1and r(0) = 0.

where t≥t0and E(t0, t)is equal to exp{νβ∗(t−

t0)−µ(I(t)−I0)}.

Proof: See Appendix B.

The importance of Theo. 2 is in the possibility

of knowing the transmission rate from the actives’s

data. Let us note that the function E(t0,·)in (7), the

variable I(·)is equal to R′(·)/γ, that is, it could be

estimated from the daily removed individuals. This

rate defines the calculation of the effective reproduc-

tive number Re(t) = β(t)s(t)/γ, in which s(t)is the

susceptible fraction. That is, we have an evaluation

of the transmission speed.

Taking into account that R0>1and that the con-

tagion process starts with a value of βin its intrin-

sic value β∗, that is, β(t0) = β∗. Since β′(t0) =

−µγ[R0−1]I(t0)is negative, we have that β(·)

will decrease until it reaches a minimum at a time τ,

τ > t0, in which

β(τ) = β∗−µ

νI′(τ)and moreover

β′′(τ) = −µI′′(τ)β(τ).

(8)

For example, these equalities tell us how much β(·)

reduces its value in the period [t0, τ],(µ/ν)I′(τ)

units. Furthermore, this minimum β(τ)implies an

active curve, with, necessarily, a decelerated growth

(concave) in a neighborhood of τ, that is, I′′(τ)<0.

Since I′<0implies β′>0, this first minimum for

β(·)is never reached after I(·)presents a first peak

or, equivalently, the effective reproductive number

reaches or decreases from the desired level.

What can we say about the transmission rate at the

time when the effective reproductive number reaches

the value one? We know that Re(t) = 1+γ−1I′/I, so

Re= 1 if and only if I′= 0, i.e., when β′=ν(β∗−

β)β. This proves that the peak Imof I(·)is reached

in an instant τmafter a first minimum of β(·), that is,

when β(·)is retrieving the value. Furthermore, that

at that moment β′(τm)≤νβ2

∗/4. On the other hand,

note that R′

e(t) = γ−1[I′′I−(I′)2]/I2and R′

e(t) =

(γN)−1(β(t)S(t))′equals to (γN)−1(β′(t)S(t) +

β(t)S′(t)). Using S′=−βSI/N, by equating and

evaluating at τm, we obtain I′′/I= [ν(β∗−β)−

βI/N]βS/N. Since I′′ <0, it follows that

β(τm)> β∗

νN

νN +Im

4 Numerical results

The graphical possibilities of transmission rate, effec-

tive reproductive number, daily new cases, and infec-

tious are presented matrixly by the rows in Figure 2

and Figure 3, depending on the value of the restora-

tion coefficient: low, medium, and high. This, con-

sidering two orders of magnitude for the pair (ν, µ)

according to Table 1. Moreover, all these cases re-

garding the initial condition at instant t0= 0, with a

population of size Nthat is considered to be decom-

posed into (S0, I0, R0) = (N−1,1,0).

Table 1: Parameter values for two orders of magni-

tude (A and B) of the restoration and reaction coef-

ficients, total population, and intrinsic transmission

rate.

Case ν µ N β∗

A2/101,5/101,8/1013/101,4/101,5/101,6/1011020.65

B2/102,5/102,8/1023/104,4/104,5/104,6/1041050.45

Now, regarding regularities or patterns in the

graphs, let us observe that:

• Without going into detail, the corresponding

forms for cases A and B are similar, except that

case A represents a comparatively rapid epidemic

spread.

• Over time, the transmission rate β(·)always de-

creases towards a flat valley and then begins to

recover value as soon as daily cases drop, even

exceeding the intrinsic value. Finally, it reaches

a peak and ends up asymptotically decaying to

β∗. In case A (Figure 2(a-c)) the initial decline

is convex, but in case B (Figure 3(a-c)) the de-

crease begins concavely. We also note that in

both cases, A or B, the minimum value to which

β(·)can drop is seemingly more sensitive to the

parameter νthan to µ.

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ν= 0.2ν= 0.5ν= 0.8

0 50 100 150 200

Time

0.5

1.5

2.5

Transmission rate

= 0.3

= 0.4

= 0.5

= 0.6

0 50 100 150 200

Time

0.5

1.5

2.5

Transmission rate

= 0.3

= 0.4

= 0.5

= 0.6

0 50 100 150 200

Time

0.5

1.5

2.5

Transmission rate

= 0.3

= 0.4

= 0.5

= 0.6

(a) (b) (c)

0 50 100 150 200

Time

= 0.3

= 0.4

= 0.5

= 0.6

0 50 100 150 200

Time

= 0.3

= 0.4

= 0.5

= 0.6

0 50 100 150 200

Time

= 0.3

= 0.4

= 0.5

= 0.6

(d) (e) (f)

0 50 100 150 200

Time

0.5

1.5

2.5

3.5

Daily new cases

= 0.3

= 0.4

= 0.5

= 0.6

0 50 100 150 200

Time

0.5

1.5

2.5

3.5

Daily new cases

= 0.3

= 0.4

= 0.5

= 0.6

0 50 100 150 200

Time

0.5

1.5

2.5

3.5

Daily new cases

= 0.3

= 0.4

= 0.5

= 0.6

(g) (h) (i)

0 50 100 150 200

Time

Infectious

= 0.3

= 0.4

= 0.5

= 0.6

0 50 100 150 200

Time

Infectious

= 0.3

= 0.4

= 0.5

= 0.6

0 50 100 150 200

Time

Infectious

= 0.3

= 0.4

= 0.5

= 0.6

(j) (k) (l)

Fig. 2: Simulations of equation (1)+(4) for different values of νand µ. Conditions initials: S(0) = 99,I(0) = 1,

R(0) = 0 and β(0) = 0.65.γ= 1/14 and N= 100 was considered.

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ν= 0.02 ν= 0.05 ν= 0.08

0 100 200 300 400

Time

0.2

0.4

0.6

0.8

1.2

1.4

Transmission rate

= 0.0003

= 0.0004

= 0.0005

= 0.0006

0 100 200 300 400

Time

0.2

0.4

0.6

0.8

1.2

1.4

Transmission rate

= 0.0003

= 0.0004

= 0.0005

= 0.0006

0 100 200 300 400

Time

0.2

0.4

0.6

0.8

1.2

1.4

Transmission rate

= 0.0003

= 0.0004

= 0.0005

= 0.0006

(a) (b) (c)

0 100 200 300 400

Time

= 0.0003

= 0.0004

= 0.0005

= 0.0006

0 100 200 300 400

Time

= 0.0003

= 0.0004

= 0.0005

= 0.0006

0 100 200 300 400

Time

= 0.0003

= 0.0004

= 0.0005

= 0.0006

(d) (e) (f)

0 100 200 300 400

Time

200

400

600

800

Daily new cases

= 0.0003

= 0.0004

= 0.0005

= 0.0006

0 100 200 300 400

Time

200

400

600

800

Daily new cases

= 0.0003

= 0.0004

= 0.0005

= 0.0006

0 100 200 300 400

Time

200

400

600

800

Daily new cases

= 0.0003

= 0.0004

= 0.0005

= 0.0006

(g) (h) (i)

0 100 200 300 400

Time

2000

4000

6000

8000

10000

12000

Infectious

= 0.0003

= 0.0004

= 0.0005

= 0.0006

0 100 200 300 400

Time

2000

4000

6000

8000

10000

12000

Infectious

= 0.0003

= 0.0004

= 0.0005

= 0.0006

0 100 200 300 400

Time

2000

4000

6000

8000

10000

12000

Infectious

= 0.0003

= 0.0004

= 0.0005

= 0.0006

(j) (k) (l)

Fig. 3: Simulations of equation (1)+(4) for different values of νand µ. Conditions initials: S(0) = 99999,

I(0) = 1,R(0) = 0 and β(0) = 0.45.γ= 1/14 and N= 100000 was considered.

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• The initial decrease in the rate βis in correspon-

dence with the fall of Re, but unlike β(·)which

recovers value, we have that Recontinues to

decrease and tends towards zero, which is only

explainable by a significant relative decrease in

the susceptible fraction, since R′

e<0implies

β′/β+S′/S < 0. Another clear pattern of the

effective reproductive number is to present a very

planar zone when it is close to the value one, a

zone that is wider when the coefficient of restitu-

tion νis smaller, compare Figure 2(d-f) or Figure

3(d-f). Furthermore, by fixing νin each case, the

flat zone is wider if the reaction coefficient µis

higher.

• In case B, the daily case curves, Figure 3(g-i), are

bimodal with the possibility that the highest peak

is the first or the last depending on whether the

restoration coefficient νis respectively higher or

lower. Thus, the unimodality inherent to the clas-

sic SIR (constant parameters) may not be met. In

case A, Figure 2(h,i) the bimodality is not ex-

pressed. It is conjectured that there must exist

a bound for ν/µbefore which these curves are

necessarily unimodal.

• For infectious count curves, the effect of a

restoration coefficient ν, when fixing the reaction

coefficient µ, e.g., the black line for Figure 3(j-i),

is a flattened curve for a longer time at lower val-

ues of ν. Now, if we set ν, a higher and more ad-

vanced asset curve is observed at lower values of

µ, that is, at smaller reactions. Observations that

are also valid for the curve of new daily cases.

On the other hand, let us note that the total de-

crease (by reaction) achieved for the beta rate during

a certain initial time interval seems to be at the cost

of recovering this rate the rest of the time and above

the intrinsic value, although approaching it asymptot-

ically, see Figure 4. Indeed, if there exists a unique

t∗> t0in which β(t∗) = β∗, by denoting by A−

(resp. A+) the area between β∗and β(·)over [t0, t∗]

(resp. [t∗,∞)) we have:

A−=Z[t0,t∗]

{β∗−β}

=ν−1Z[t0,t∗]β′/β+µI′

=µν−1{I(t∗)−I(t0)}

and similarly A+=µν−1I(t∗), so that A+−

A−=µν−1I(t0).

5 Discussion

The classical SIR model considers a constant trans-

mission rate. However, in the context of a high- and

0 100 200 300 400 500 600 700

Time

0.2

0.4

0.6

0.8

Transmission rate

= 0.02 and = 0.0003

A-

A+

Fig. 4: Initials Conditions: S(0) = 99999,I(0) = 1,

R(0) = 0 and β(0) = 0.45.γ= 1/14 and N=

100000 was considered. In this case A+− A−=

ν/µ= 0.015.

immediate-risk pandemic (without pharmaceutical al-

ternatives), one of the main sources, if not the main,

of the changes in the transmission rate that the pop-

ulation may present during an epidemic is relates to

human behavior in terms of adherence and compli-

ance with mitigating measures. In the search for a

model that brings together the mechanistic aspects

of the SIR but with a phenomenological perspective

in terms of simplicity, the assumption of a reaction-

restoration tension for the variation of the transmis-

sion rate seems reasonable.

How can the transmission rate, as a potential cul-

tural and physical-environmental determinant of a

specific population, vary, as it did with COVID-19

before the advent of vaccines? In the first pandemic

stage, the general observation (based on pandemic

data) is that there was a reaction from authorities and

individuals that allowed, based on perception or indi-

cators of the danger of the disease, to reduce its worth.

The information that the public received or that was

analyzed to decree specific measures is directly or

indirectly a reading of the status of individuals con-

cerning the disease, particularly those of population

scope, such as the variation of the infectious group

size, which is our option in the present work.

In this sense, the main contribution of the work

is to model the epidemiological consequences of

achieving the health authority and the media to main-

tain pressure on the system, which manifests itself in

a percentage change of decrease in the transmission

rate that, as has been said, is proportional to the vari-

ation of the infectious group. In other words, having

a first dimension or evaluation of the behavior of the

system’s state variables, both in the case of no resis-

tance to control and in the opposite case.

In particular, in the case of zero resistance to con-

trol (ν= 0), for an estimation interval of the remain-

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ing susceptible fraction, Theo. 1 provides a bound-

ing band for the effective reproductive number as a

fraction of the basic reproductive number and of the

removals count (which could be the variable with

good measurement possibilities) in principle some-

what crude, as shown in Figure 1, but which could

(and it is a mathematical challenge) be improved.

Now, by adding a social resistance to mitigation (ν=

0), Theo. 2 gives us the rate determination relation-

ship according to active cases through expression (7)

which is, accordingly, the generalization of (5). These

results, although elementary, have potential useful-

ness in the development of more tactical models, par-

ticularly those that are fed back through data for a

specific population. We consider it important to note

that when the active group varies downwards, the con-

trol acts not only canceling itself but also providing

a stimulus for its increase, above the intrinsic value,

which makes some sense within the social behavior of

reunion again between people. What the comparison

between Figure 2 and Figure 3 show is that lower or-

der restoration levels can mean a much slower return

to the original or intrinsic transmission rate.

6 Conclusion

The main hypothesis considered is to assume that the

reaction factor is proportional to the variation of the

active group. Note that this is the group, in terms of

size, that directly determines hospital occupancy and

the number of deaths. The assumption is that only as

this group grows does a downward reaction emerge.

So, if it remains constant for some period, such a reac-

tion disappears; finally, if it goes down, the tendency

for the rate to rise grows. Another possibility, some-

what close to our case, but which we leave open for

analysis in future work, is to assume that the reaction

factor g(·)takes the form g(I, I′) = µI′/I, that is,

the reaction is sensitive to the percentage variation of

the active group.

A general observation is that the β-SIR type mod-

els, such as the one analyzed or the β-SEIR versions,

maintain a degree of simplicity that helps to under-

stand the mechanics (in terms of the response of the

system), not only the contagion of the disease, but

also the corresponding “social mechanics”, both of

the population in general (mainly in the restoration

factor), and the work of the health authority (in the re-

action factor), the above as long as there are no phar-

maceutical possibilities that alter the variables. Let

us observe that the type of information or indicators

that the authority collects or uses to implement miti-

gation, directly from the triad (S, I, R)or its deriva-

tives, such as reproductive numbers, is an aspect that

could become key, mainly concerning the times and

levels that ensure effective management, for example

hospital management, in severe cases.

Another key aspect, which could explain a good

part of the variability that experience tells us in the

data, mainly at the beginning of the epidemic, could

be the time delay between the value of the considered

indicator(s), the implementation, and the citizen re-

sponse. In other words, more research is required, as-

sociated with the management of health emergencies

and human behavior in specific populations, to pro-

pose, with greater predictive zeal, functional forms to

express variations in the rate of reactive or restoration

order.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present re-

search, at all stages from the formulation of the prob-

lem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

This work was supported by ANID, Fon-

decyt Regular, grant number 1231256.

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DOI: 10.37394/23208.2024.21.12

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E-ISSN: 2224-2902

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Conflicts of Interest

The authors have no conflicts of interest to

declare that are relevant to the content of this

article.

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(Attribution 4.0 International , CC BY 4.0)

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_US

Appendix A

Note that replacing the second equation of (1) in (4)

with ν= 0, we obtain β′=−µ[βs −γ]I β, with

s(·) = S(·)/N, the susceptible fraction. By amplify-

ing, by −1/β2and changing the variables Z= 1/β,

we have the equation Z′+µ γ I Z =µ s I, where to-

gether with the expression for R′(t) = γI in (1), we

observe that µR′Z=µγIZ , achieving Z eµR′=

[Z′+ZµR′]eµR =µsIeµR. Integrating over [t0, t]

and considering R(t0) = 0, we have that,

Z(t) = 1

β0

+µZt

s(a)I(a)eµR(a)dae−µR(t).

Thus, if we consider that s(a) = λ, we can define

Zλ(t) := 1

β0

+µλ Zt

I(a)eµR(a)dae−µR(t).

From where, if we consider p≤s(t)≤q, we have:

Zp(t)≤Z(t)≤Zq(t). Then, continuing with the

integration for the expression of Zλ, we see that,

I(a)eµR(a)da =

µγ Zt

[eµR(a)]′da =1

µγ [eµR(t)−1],

finally

Zλ(t) = γ+λβ0[eµR(t)−1]

γβ0eµR(t)=

λR0+e−µR(t)[1 −λR0]

β0

Since, Z−1

q≤β(t)≤Z−1

p, we have

pR0

qR0+ [1 −qR0]e−µR(t)≤ Re(t)≤

qR0

pR0+ [1 −pR0]e−µR(t);

which ends the proof. ⋄

Appendix B

From (7), we define U(t0, t) = νβ∗(t−t0)−µ(I(t)−

I0), so that (4) is equal to β′={U′(t0, t)−νβ}β.

So the change of variables z(t) = 1/β(t), implies

z′=ν−U′(t0, t)z. Thus, one has

z(t) = z(t0)exp −Zt

U′(t0, τ)dτ+

νZt

exp −Zt

U′(t0, a)dadτ.

Now, since U(t0, t0) = 0 and U(t0, t)−U(t0, τ ) =

U(τ, t), by integrating U′into the expression z(t), we

obtain 1

β(t)=1

β0

exp (−U(t)) +

νZt

exp (−U(τ, t)) dτ.

Considering E(t0, t) = exp{U(t0, t)}, isolating β(·)

completes the proof of Theorem 2. ⋄

WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

DOI: 10.37394/23208.2024.21.12

Fernando Córdova-Lepe, Juan Pablo Gutiérrez-Jara

E-ISSN: 2224-2902

130

Volume 21, 2024