# Modeling the effects of prosocial awareness on COVID-19 dynamics: A case
study on Colombia

Indrajit Ghosh 111Corresponding author. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Maia Martcheva  Agricultural and Ecological Research
Unit, Indian Statistical Institute, Kolkata - 700 108, West Bengal, India
Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

###### Abstract

The ongoing COVID-19 pandemic has affected most of the countries on Earth. It
has become a pandemic outbreak with more than 24 million confirmed infections
and above 840 thousand deaths worldwide. In this study, we consider a
mathematical model on COVID-19 transmission with the prosocial awareness
effect. The proposed model can have four equilibrium states based on different
parametric conditions. The local and global stability conditions for awareness
free, disease-free equilibrium is studied. Using Lyapunov function theory and
LaSalle Invariance Principle, the disease-free equilibrium is shown globally
asymptotically stable under some parametric constraints. The existence of
unique awareness free, endemic equilibrium and unique endemic equilibrium is
presented. We calibrate our proposed model parameters to fit daily cases and
deaths from Colombia. Sensitivity analysis indicate that transmission rate and
learning factor related to awareness of susceptibles are very crucial for
reduction in disease related deaths. Finally, we assess the impact of
prosocial awareness during the outbreak and compare this strategy with popular
control measures. Results indicate that prosocial awareness has competitive
potential to flatten the curve.

###### keywords:

COVID-19, Prosocial awareness, Mathematical model, Stability analysis, Data
analysis.

††journal: arXiv

##  1 Introduction

The ongoing outbreak of coronavirus disease 2019 (COVID-19), caused by SARS-
CoV-2 virus, a highly contagious virus, has been a massive threat for
governments of many affected countries. COVID-19 is causing obstacles for
public health organizations and is affecting almost every aspect of human
life. The outbreak was declared a pandemic of international concern by WHO on
March $11^{th}$, 2020 Who2020 . The virus can cause a range of symptoms
including dry cough, fever, fatigue, breathing difficulty, and bilateral lung
infiltration in severe cases, similar to those caused by SARS-CoV and MERS-CoV
infections huang2020clinical ; gralinski2020return . Many people may
experience non-breathing symptoms including nausea, vomiting and diarrhea
cdcgov2020 . Chan et. al chan2020familial  confirmed that the virus spreads
through close contact of humans. It has become an epidemic outbreak with more
than $24$ million confirmed infections and above $840$ thousand deaths
worldwide as of August $28^{th}$, 2020 Worldometer2020 .

Since first discovery and identification of coronavirus in 1965, three major
outbreaks occurred, caused by emerging, highly pathogenic coronaviruses,
namely the 2003 outbreak of Severe Acute Respiratory Syndrome (SARS) in
mainland China gumel2004modelling ; li2003angiotensin , the 2012 outbreak of
Middle East Respiratory Syndrome (MERS) in Saudi Arabia de2013commentary ;
sardar2020realistic , and the 2015 outbreak of MERS in South Korea
cowling2015preliminary ; kim2017middle . These outbreaks resulted in SARS and
MERS cases confirmed by more than $8000$ and $2200$, respectively
kwok2019epidemic . The COVID-19 is caused by a new genetically similar corona
virus to the viruses that cause SARS and MERS. Despite a relatively lower
death rate compared to SARS and MERS, the COVID-19 spreads rapidly and infects
more people than the SARS and MERS outbreaks. In spite of strict intervention
measures implemented in various affected areas, the infection spread around
the globe very rapidly. Due to nonavailability of vaccines and specific
medications, non-pharmaceutical control measures such a social distancing,
lockdown, use of mask, use of PPE kits, awareness through media are studied
using different theoretical frameworks ngonghala2020mathematical .

Mathematical modeling based on differential equations may provide a
comprehensive mechanism for the dynamics of the disease and also to test the
efficacy of the control strategies to reduce the burden of COVID-19. Several
studies were performed using real-life data from the affected countries and
analyzed various features of the outbreak as well as assess the impact of
intervention such as lockdown approaches to suppress the outbreak in the
concerned countries kucharski2020early ; sardar2020assessment ;
tang2020estimation ; zhao2020preliminary ; asamoah2020global . There has been
a few mathematical models to assess the impact of awareness campaigns against
COVID-19 yan2020impact ; zhou2020effects ; chang2020studying ;
khajanchi2020dynamics ; kobe2020modeling ; mbabazi2020mathematical ;
mohsen2020global . These research articles mainly incorporate the awareness
through media campaigns. The media effect is modelled in two ways: by adding
media compartment to COVID-19 model chang2020studying ; khajanchi2020dynamics
; mbabazi2020mathematical ; zhou2020effects ; kobe2020modeling  and through
reduction in incidence function due to media campaigns yan2020impact ;
mohsen2020global . However, there is a scope of investigating pro-social
awareness on the dynamics of COVID-19 transmission. The idea is that the aware
susceptible persons will pass the information (regarding use face mask, social
distancing, mortality due to COVID-19 etc.) to the unaware susceptible
individuals. The unaware people become aware by contacting the aware
susceptible and practice the self-protection measures.

As a case study, we use daily notified cases and deaths in Colombia. With over
$50$ million inhabitants Colombia is the third-most-populous country in Latin
America. On March $6^{th}$, 2020, Colombia reported the first confirmed case
of COVID-19. On $17^{th}$ March, President Iván Duque spoke to the Colombians
and declared the state of emergency, announcing that he would take economic
measures that were announced the following day. The first measure taken
seeking the protection of the elderly is to decree mandatory isolation from
$20^{th}$ March, 2020 to $31^{st}$ May, 2020 for all adults over 70 years of
age. They must remain in their residences except to stock up on food or access
health or financial services. Government entities were instructed to make it
easier for them to receive their pensions, medicines, healthcare or food at
home. On the evening of $20^{th}$ March, President Iván Duque announced a
19-day nationwide quarantine, starting on $24^{th}$ March at midnight and
ending on $12^{th}$ April at midnight Wiki2020colombiacovid . As of August
$28^{th}$, 2020, there were more than $590$ thousand confirmed cases
(currently, the world’s $7^{th}$ highest) and above 18 thousand confirmed
deaths Worldometer2020 . As the outbreak of COVID-19 is expanding rapidly in
Colombia, real-time analysis of epidemiological data are required to increase
situational awareness and inform interventions. Mathematical modeling based on
dynamic equations pang2020transmission ; tang2020updated ; frank2020covid  may
provide detailed mechanism for the disease dynamics. A few studies were based
on the Colombia COVID-19 situation rojas2020mathematical ; bizet2020time ;
teheran2020epidemiological ; manrique2020sir . These studies have broadly
suggested that control measures could reduce the burden of COVID-19. However,
none of the studies has considered awareness as a control utilizing recent
epidemic data from the Colombia.

The main objectives of this study are to (i) propose and analyze a
compartmental model incorporating prosocial awareness, (ii) use available
current COVID-19 epidemic from Colombia and calibrate the proposed model and
(iii) compare prosocial awareness with other popular control measures in
Colombia.

Rest of the paper is organized as follows: A mathematical model which
incorporates the prosocial awareness is described in Section 2. The
equilibrium points of the model and their stability along with related
conditions are presented in Section 3. In Section 4, the transcritical
bifurcation phenomenon is presented between multiple equilibria. Next in 5, we
fit the proposed model to daily new COVID-19 cases and deaths from Colombia.
The impact of prosocial awareness and comparison with other control strategies
is also studied. Finally in Section 6, we discuss the results from our study.

##  2 Model formulation

A compartmental differential equation model for COVID-19 is formulated and
analyzed. We adopt a variant that reflects some key epidemiological properties
of COVID-19. The model monitors the dynamics of six sub-populations, namely
unaware susceptible $(S_{u}(t))$, aware susceptible $(S_{a}(t))$, exposed
$(E(t))$, un-notified infected $(I(t))$, notified infected $(J(t))$ and
recovered $(R(t))$ individuals. The total population size is
$N(t)=S_{u}(t)+S_{a}(t)+E(t)+I(t)+J(t)+R(t)$. Our model incorporates some
demographic effects by assuming a proportional natural death rate $\mu>0$ in
each of the six sub-populations of the model. In addition, our model includes
a net inflow of susceptible individuals into the region at a rate $\Pi$ per
unit time. This parameter includes new births, immigration and emigration.
Instead of constant awareness rate, we consider that the awareness will induce
a behavioral response in the person and this person will transmit the
knowledge to other hosts epstein2008coupled ; just2018oscillations . Thus,
unaware suscptibles can become aware through contact with aware susceptibles.
The functional response in this regard is assumed to be $\frac{\alpha
S_{u}S_{a}}{N}$.

The flow diagram of the proposed model is displayed in Fig. 1.

Figure 1: Compartmental flow diagram of the proposed model.

Both susceptible populations decrease due to infection through successful
contact with infectives who may be notified or un-notified. Note that un-
notified class contains both asymptomatic and symptomatic infected
individuals. We assume that the transmission co-efficient is less for notified
individuals as they are kept in special observations. Thus, a reduced risk for
notified COVID-19 patients is modelled as $\frac{\beta\nu J}{N}$
martcheva2015introduction . On the other hand for un-notified individuals, the
standard mixing force of infection is formulated as $\frac{\beta I}{N}$
may1991infectious . By making successful contact with infectives both
susceptible group members become exposed to the disease. The exposed
population may become notified or un-notified at a rates $p\gamma$ and
$(1-p)\gamma$ respectively. The recovery rate of un-notified and notified
infected individuals are $\sigma_{1}$ and $\sigma_{2}$ respectively. The un-
notified COVID-19 patients become notified at a rate $\eta$. The mortality
rates related to COVID-19 are assumed to be $\delta_{1}$ and $\delta_{2}$ for
un-notified and notified persons respectively. From the above considerations,
the following system of ordinary differential equations governs the dynamics
of the system:

$\displaystyle\displaystyle{\frac{dS_{u}}{dt}}$ $\displaystyle=$
$\displaystyle\Pi-\beta\frac{I+\nu J}{N}S_{u}-\alpha\frac{S_{a}}{N}S_{u}-\mu
S_{u}+\theta S_{a},$ $\displaystyle\displaystyle{\frac{dS_{a}}{dt}}$
$\displaystyle=$
$\displaystyle\alpha\frac{S_{a}}{N}S_{u}-\epsilon\beta\frac{I+\nu
J}{N}S_{a}-(\mu+\theta)S_{a},$ $\displaystyle\displaystyle{\frac{dE}{dt}}$
$\displaystyle=$ $\displaystyle\beta\frac{I+\nu J}{N}(S_{u}+\epsilon
S_{a})-(\gamma+\mu)E,$ $\displaystyle\displaystyle{\frac{dI}{dt}}$
$\displaystyle=$ $\displaystyle(1-p)\gamma
E-(\eta+\sigma_{1}+\mu+\delta_{1})I,$
$\displaystyle\displaystyle{\frac{dJ}{dt}}$ $\displaystyle=$ $\displaystyle
p\gamma E+\eta I-(\sigma_{2}+\mu+\delta_{2})J,$
$\displaystyle\displaystyle{\frac{dR}{dt}}$ $\displaystyle=$
$\displaystyle\sigma_{1}I+\sigma_{2}J-\mu R,$

All the parameters and their biological interpretation are given in Table 1
respectively.

Table 1: Description of parameters used in the model. Parameters |  Interpretation |  Value |  Reference
---|---|---|---
$\Pi$ |  Recruitment rate |  – |  –
$\beta$ |  Transmission rate |  (0-1) |  Estimated
$\nu$ |  Modification factor |  0.5 |  Assumed
$\alpha$ |  Learning factor related to aware susceptibles |  (0-1) |  Estimated
$\theta$ |  Rate of transfer of aware individuals to unaware susceptible class |  0.02 |  samanta2013effect
$\epsilon$ |  Reduction in transmission co-efficient for aware susceptibles |  0.4 |  Assumed
$\frac{1}{\gamma}$ |  Incubation period |  5 |  li2020early ; linton2020incubation
$p$ |  Proportion of notified individuals |  0.2 |  wu2020characteristics ; yang2020epidemiological
$\eta$ |  Transfer rate from un-notified to notified |  0.01 |  Assumed
$\sigma_{1}$ |  Recovery rate from un-notified individuals |  0.17 |  woelfel2020clinical ; tindale2020transmission
$\sigma_{2}$ |  Recovery rate from notified individuals |  0.072 |  lopez2020end
$\delta_{1}$ |  Disease induced mortality rate in the un-notified class |  0.01 |  Assumed
$\delta_{2}$ |  Disease induced mortality rate in the notified class |  (0-1) |  Estimated
$\mu$ |  Natural death rate |  – |  –
$N$ |  Total population |  – |  –

##  3 Model analysis

###  3.1 Positivity and boundedness of the solution

This subsection is provided to prove the positivity and boundedness of
solutions of the system (2) with following initial conditions

$\displaystyle(S_{u}(0),S_{a}(0),E(0),I(0),J(0),R(0))^{T}\in\mathbb{R}_{+}^{6}.$
(3.1)

######  Proposition 3.1.

The system (2) is invariant in $\mathbb{R}_{+}^{6}$.

###### Proof.

Consider initial conditions (3.1) and let $S_{u}$ becomes zero at time $t_{1}$
before other state variables become zero, then

$\displaystyle\frac{dS_{u}}{dt}|_{S_{u}=0}$ $\displaystyle=\Pi+\theta
S_{a}\geq 0,$

at $t_{1}$. This shows that $S_{u}$ is a non-decreasing function of time at
$t_{1}$. Hence it follows that $S_{u}$ stays non-negative (similar argument is
employed in Theorem 3.1 of Sun et al. sun2011effect ). For other state
variables we note that

$\displaystyle\frac{dS_{a}}{dt}|_{S_{a}=0}$ $\displaystyle=0\geq 0,$
$\displaystyle\frac{dE}{dt}|_{E=0}$ $\displaystyle=\beta\left[\frac{(I+\nu
J)}{S_{u}+S_{a}+I+J+R}\right](S_{u}+\epsilon S_{a})\geq 0,$
$\displaystyle\frac{dI}{dt}|_{I=0}$ $\displaystyle=(1-p)\gamma E\geq 0,$
$\displaystyle\frac{dJ}{dt}|_{J=0}$ $\displaystyle=p\gamma E\geq 0,$
$\displaystyle\frac{dR}{dt}|_{R=0}$ $\displaystyle=\sigma_{1}I+\sigma_{2}J\geq
0.$

Thus we obtain non-negativity of all the six state variables and it follows
that $\mathbb{R}_{+}^{6}$ is an invariant set for the model (2). ∎

######  Proposition 3.2.

The system (2) is bounded in the region
$\Omega=\\{(S_{u},S_{a},E,I,J,R)\in\mathbb{R}_{+}^{6}|S_{u}+S_{a}+E+I+J+R\leq\frac{\Pi}{\mu}\\}$

###### Proof.

Adding all the equations of the model (2), total human populations satisfy the
following equations,

$\displaystyle\frac{dN}{dt}=\Pi-\mu N-\delta_{1}I-\delta_{2}J\leq\Pi-\mu N$

Since $\frac{dN}{dt}\leq\Pi-\mu N$, it follows that $\frac{dN}{dt}\leq 0$ if
$N\geq\frac{\Pi}{\mu}$. Thus, by using standard comparison theorem, it can be
shown that $N\leq N(0)e^{-\mu t}+\frac{\Pi}{\mu}(1-e^{-\mu t})$. In
particular, $N(t)\leq\frac{\Pi}{\mu}$ if $N(0)\leq\frac{\Pi}{\mu}$. Thus, the
region $\Omega$ is positively-invariant. Further, if $N(0)>\frac{\Pi}{\mu}$,
then either the solution enters $\Omega$ in finite time, or $N(t)$ approaches
$\frac{\Pi}{\mu}$ asymptotically. Hence, the region $\Omega$ attracts all
solutions in $\mathbb{R}_{+}^{6}$ . ∎

###  3.2 Equilibrium points, threshold quantities and stability analysis

The system (2) has four type of equilibrium points: awareness-free disease-
free equilibrium (AFDFE), disease free equilibrium (DFE), awareness-free
endemic equilibrium (AFEE) and endemic equilibrium (EE). The awareness free,
DFE is given by $E_{0}=(\frac{\Pi}{\mu},0,0,0,0,0)$.

######  Lemma 3.1.

The awareness free, DFE $E_{0}$ of system (2) is locally asymptotically stable
whenever $max\left[R_{1},\frac{\alpha}{\mu+\theta}\right]<1$ and unstable
otherwise, where

$R_{1}=\frac{\beta\gamma}{(\mu+\gamma)(\sigma_{2}+\mu+\delta_{2})}\left[(1-p)\frac{\eta\nu+\sigma_{2}+\mu+\delta_{2}}{\eta+\sigma_{1}+\mu+\delta_{1}}+\nu
p\right]$

.

###### Proof.

We calculate the Jacobian of the system (2) at $E_{0}$, which is given by

$\displaystyle
J_{E_{0}}={\begin{pmatrix}-\mu&-\alpha+\theta&0&-\beta&-\beta&0\\\
0&\alpha-(\mu+\theta)&0&0&0&0\\\ 0&0&-(\mu+\gamma)&\beta&\nu\beta&0\\\
0&0&(1-p)\gamma&-(\eta+\sigma_{1}+\mu+\delta_{1})&0&0\\\
0&0&p\gamma&\eta&-(\sigma_{2}+\mu+\delta_{2})&0\\\
0&0&0&\sigma_{1}&\sigma_{2}&-\mu\\\ \end{pmatrix}},$

Let $\lambda$ be the eigenvalue of the matrix $J_{E_{0}}$. Then the
characteristic equation is given by $det(J_{E_{0}}-\lambda I)=0$.

Clearly, $-\mu$, $-\mu$ and $\alpha-(\mu+\theta)$ are three eigenvalues of the
Jacobian matrix $J_{E_{0}}$. The other three eigenvalues are given the
following cubic equation

$\displaystyle\begin{array}[]{lll}f(\lambda):=\lambda^{3}+a_{1}\lambda^{2}+a_{2}\lambda+a_{3}=0\\\
\end{array}$ (3.3)

where,

$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle\mu+\gamma+m_{1}+m_{2}$
$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle(\mu+\gamma)(m_{1}+m_{2})-\beta\gamma$ $\displaystyle a_{3}$
$\displaystyle=$ $\displaystyle m_{1}m_{2}(\mu+\gamma)(1-R_{1}).$

Here
$m_{1}=\eta+\sigma_{1}+\mu+\delta_{1}$,
$m_{2}=\sigma_{2}+\mu+\delta_{2}$ and
$R_{1}=\frac{\beta\gamma}{m_{2}(\mu+\gamma)}\left[(1-p)\frac{\eta\nu+m_{2}}{m_{1}}+p\nu\right]$.

It is straight forward to show that coefficients of (3.3) satisfies Routh-
Hurwitz criterion if $R_{1}<1$. Thus, all the eigenvalues are negative or have
negative real parts if in addition $\frac{\alpha}{\mu+\theta}<1$.

On the other hand, if $max\left[R_{1},\frac{\alpha}{\mu+\theta}\right]>1$ then
at least one eigenvalue of the Jacobian matrix is positive and $E_{0}$ become
unstable. Hence the proof is complete. ∎

######  Theorem 3.1.

The awareness free DFE $E_{0}$ is globally asymptotically stable for the
system (2) if $max\left[R_{1},\frac{\alpha}{\mu+\theta}\right]<1$.

###### Proof.

The system (2) can be represented as

$\displaystyle\frac{dX}{dt}$ $\displaystyle=F_{1}(X,V)$
$\displaystyle\frac{dV}{dt}$ $\displaystyle=G_{1}(X,V),G_{1}(X,0)=0$

where $X=(S_{u},S_{a},R)\in R_{3}^{+}$ (uninfected classes of people),
$V=(E,I,J)\in R_{3}^{+}$ (infected classes of people), and
$E{0}=(\frac{\Pi}{\mu},0,0,0,0)$ is the awareness free, DFE of the system (2).
The global stability of $E_{0}$ is guaranteed if the following two conditions
are satisfied:

  1. 1.

For $\frac{dX}{dt}=F_{1}(X,0)$, $X^{*}$ is globally asymptotically stable,

  2. 2.

$G_{1}(X,V)=BV-\widehat{G}_{1}(X,V),$ $\widehat{G}_{1}(X,V)\geq 0$ for
$(X,V)\in\hat{\Omega}$,

where $B=D_{V}G_{1}(X^{*},0)$ is a Metzler matrix and $\hat{\Omega}$ is the
positively invariant set with respect to the model (2). Following Castillo-
Chavez et al castillo2002computation , we check for aforementioned conditions.
For system (2),

$\displaystyle F_{1}(X,0)$ $\displaystyle=\begin{pmatrix}\Pi-\mu S_{u}\\\ 0\\\
0\end{pmatrix},$ $\displaystyle B$
$\displaystyle=\begin{pmatrix}-(\gamma+\mu)&\beta&\nu\beta\\\
(1-p)\gamma&-m_{1}&0\\\ p\gamma&\eta&-m_{2}\end{pmatrix}$

and

$\displaystyle\widehat{G}_{1}(X,V)=\begin{pmatrix}\beta(I+\nu
J)(1-\frac{S_{u}}{N})\\\ 0\\\ 0\end{pmatrix}.$

Clearly, $\widehat{G}_{1}(X,V)\geq 0$ whenever the state variables are inside
$\Omega$. Also it is clear that $X^{*}=(\frac{\Pi}{\mu},0,0)$ is a globally
asymptotically stable equilibrium of the system $\frac{dX}{dt}=F_{1}(X,0)$.
Hence, the theorem follows. ∎

The unique disease-free equilibrium of the system (2) is given by

$\displaystyle
E_{1}=\left(\frac{\Pi(\mu+\theta)}{\mu\alpha},\frac{\Pi[\alpha-(\mu+\theta)]}{\mu\alpha},0,0,0,0\right),$

which exists if $\frac{\alpha}{\mu+\theta}>1$. To obtain the basic
reproduction number $R_{0}$ of the system (2), we apply the next generation
matrix approach. The infected compartments of the model (2) consist of
$(E(t),I(t),J(t))$ classes. Following the next generation matrix method, the
matrix $F$ of the trransmission terms and the matrix, $V$ of the transition
terms calculated at $E_{1}$ are,

$\displaystyle F$ $\displaystyle=\begin{pmatrix}0&\beta m_{3}&\nu\beta
m_{3}\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix},$ $\displaystyle V$
$\displaystyle=\begin{pmatrix}\gamma+\mu&0&0\\\ -(1-p)\gamma&m_{1}&0\\\
-p\gamma&-\eta&m_{2}\\\ \end{pmatrix},$

where,

$\displaystyle m_{1}$ $\displaystyle=\eta+\sigma_{1}+\mu+\delta_{1}$
$\displaystyle m_{2}$ $\displaystyle=\sigma_{2}+\mu+\delta_{2}$ $\displaystyle
m_{3}$
$\displaystyle=\frac{1}{\alpha}\left[(\theta+\mu)+\epsilon\\{\alpha-(\theta+\mu)\\}\right]$

Calculating the dominant eigenvalue of the next generation matrix $FV^{-1}$ ,
we obtain the basic reproductive number as follows van2002reproduction

$\displaystyle
R_{0}=\frac{m_{3}\beta\gamma}{m_{2}(\mu+\gamma)}\left[(1-p)\frac{\eta\nu+m_{2}}{m_{1}}+p\nu\right]$
(3.4)

The basic reproduction number $R_{0}$ is defined as the average number of
secondary cases generated by one infected individual during their infectious
period in a fully susceptible population. The basic reproduction number
$R_{0}$ of (2) given in 3.4.

Using Theorem 2 in van2002reproduction , the following result is established.

######  Lemma 3.2.

The disease-free equilibrium $\varepsilon_{0}$ of system (2) is locally
asymptotically stable whenever $R_{0}<1$, and unstable whenever $R_{0}>1$.

######  Remark 3.1.

Note that the threshold quantities $R_{1}$ and $R_{0}$ are linearly dependent
by the relation $R_{0}=m_{3}R_{1}$.

######  Theorem 3.2.

The DFE $E_{1}$ of the model (2), is globally asymptotically stable in
$\Omega$ whenever
$max\left[R_{0},\frac{\beta\gamma}{(\sigma_{1}+\mu+\delta_{1})(\mu+\gamma)},\frac{\beta\nu\gamma}{m_{2}(\mu+\gamma)}\right]<1$.

###### Proof.

Consider the following Lyapunov function

$\displaystyle\mathcal{D}=\frac{E}{\gamma+\mu}+\frac{I}{\gamma}+\frac{J}{\gamma}$

We take the Lyapunov derivative with respect to $t$,

$\displaystyle\dot{\mathcal{D}}$
$\displaystyle=\frac{\dot{E}}{\gamma+\mu}+\frac{\dot{I}}{\gamma}+\frac{\dot{J}}{\gamma}$
$\displaystyle=\frac{1}{\mu+\gamma}\left[\beta\frac{I+\nu J}{N}(S_{u}+\epsilon
S_{a})\right]-\frac{(m_{1}-\eta)I}{\gamma}-\frac{m_{2}J}{\gamma}$
$\displaystyle\leq\frac{(\sigma_{1}+\mu+\delta_{1})I}{\gamma}\left[\frac{\beta\gamma}{(\sigma_{1}+\mu+\delta_{1})(\mu+\gamma)}-1\right]+\frac{m_{2}J}{\gamma}\left[\frac{\beta\nu\gamma}{m_{2}(\mu+\gamma)}-1\right]\text{
(Since $S_{u}+\epsilon S_{a}\leq N$ in $\Omega$)}$

Thus, $\dot{\mathcal{D}}\leq 0$, whenever
$max\left[\frac{\beta\gamma}{(\sigma_{1}+\mu+\delta_{1})(\mu+\gamma)},\frac{\beta\nu\gamma}{m_{2}(\mu+\gamma)}\right]<1$.

Since all the variables and parameters of the model (2) are non-negative, it
follows that $\dot{\mathcal{D}}\leq 0$ with $\dot{\mathcal{D}}=0$ at DFE if
$max\left[\frac{\beta\gamma}{(\sigma_{1}+\mu+\delta_{1})(\mu+\gamma)},\frac{\beta\nu\gamma}{m_{2}(\mu+\gamma)}\right]<1$.
Hence, $\mathcal{D}$ is a Lyapunov function on $\Omega$. Therefore, followed
by LaSalle’s Invariance Principle lasalle1976stability , that

$(E(t),I(t),J(t))\rightarrow(0,0,0)\text{ as }t\rightarrow\infty$ (3.5)

Since $\lim\limits_{t\rightarrow\infty}supI(t)=0$ and
$\lim\limits_{t\rightarrow\infty}supJ(t)=0$ (from 3.5), it follows that, for
sufficiently small $\xi_{1}>0,\xi_{2}>0$, there exist constants
$L_{1}>0,L_{2}>0$ such that
$\lim\limits_{t\rightarrow\infty}supI(t)\leq\xi_{1}$ for all $t>L_{1}$ and
$\lim\limits_{t\rightarrow\infty}supJ(t)\leq\xi_{2}$ for all $t>L_{2}$.
Hence, it follows that,

$\displaystyle\frac{dR}{dt}\leq\sigma_{1}\xi_{1}+\sigma_{2}\xi_{2}-\mu R$

Therefore using comparison theorem smith1995theory

$\displaystyle
R^{\infty}=\lim\limits_{t\rightarrow\infty}supR(t)\leq\frac{\sigma_{1}\xi_{1}+\sigma_{2}\xi_{2}}{\mu}$

Therefore, as $(\xi_{1},\xi_{2})\rightarrow(0,0)$,
$R^{\infty}=\lim\limits_{t\rightarrow\infty}supR(t)\leq 0$
Similarly by using $\lim\limits_{t\rightarrow\infty}infI(t)=0$ and
$\lim\limits_{t\rightarrow\infty}infJ(t)=0$, it can be shown that

$\displaystyle R_{\infty}=\lim\limits_{t\rightarrow\infty}infR(t)\geq 0$

Thus, it follows from above two relations

$\displaystyle R_{\infty}\geq 0\geq R^{\infty}$

Hence $\lim\limits_{t\rightarrow\infty}R(t)=0$

Substituting $E(t)=I(t)=J(t)=R(t)=0$ in the original system (2), we get

$\displaystyle\displaystyle{\frac{dS_{u}}{dt}}$ $\displaystyle=$
$\displaystyle\Pi-\frac{\alpha S_{u}S_{a}}{S_{u}+S_{a}}-\mu S_{u}+\theta
S_{a},$ $\displaystyle\displaystyle{\frac{dS_{a}}{dt}}$ $\displaystyle=$
$\displaystyle\frac{\alpha S_{u}S_{a}}{S_{u}+S_{a}}-(\mu+\theta)S_{a},$

Following vargas2011global , a suitable lyapunov function can be formulated as
follows

$\displaystyle\mathcal{L}=\left[(S_{u}-S_{u}^{*})+(S_{a}-S_{a}^{*})-(S_{u}^{*}+S_{a}^{*})ln\frac{S_{u}+S_{a}}{S_{u}^{*}+S_{a}^{*}}\right]+\frac{2\mu(S_{u}^{*}+S_{a}^{*})}{\alpha
S_{a}^{*}}\left(S_{a}-S_{a}^{*}-S_{a}^{*}ln\frac{S_{a}}{S_{a}^{*}}\right),$

where, $S_{u}^{*}=\frac{\Pi(\mu+\theta)}{\mu\alpha}$ and
$S_{a}^{*}=\frac{\Pi[\alpha-(\mu+\theta)]}{\mu\alpha}$.

Therefore by combining all above equations, it follows that each solution of
the model equations (2), with initial conditions $\in\Omega$ , approaches
$E_{1}$ as $t\rightarrow\infty$ for
$max\left[\frac{\beta\gamma}{(\sigma_{1}+\mu+\delta_{1})(\mu+\gamma)},\frac{\beta\nu\gamma}{m_{2}(\mu+\gamma)}\right]<1$.
∎

####  3.2.1 Existence of awareness-free endemic equilibrium

Let $E_{2}=(S_{u}^{*},0,E^{*},I^{*},J^{*},R^{*})$ be any AFEE of system (2).
Let us denote

$\displaystyle m_{1}$ $\displaystyle=\eta+\sigma_{1}+\mu+\delta_{1},$
$\displaystyle m_{2}$ $\displaystyle=\sigma_{2}+\mu+\delta_{2},$
$\displaystyle m_{4}$ $\displaystyle=\frac{(1-p)\gamma}{m_{1}},$
$\displaystyle m_{5}$
$\displaystyle=\frac{p\gamma}{m_{2}}+\frac{\eta(1-p)\gamma}{m_{1}m_{2}}.$

Further, the force of infection be

$\displaystyle\lambda_{h}^{*}=\frac{\beta[I^{*}+\nu J^{*}]}{N^{*}}$ (3.6)

By setting the right equations of system (2) equal to zero, we have

$\displaystyle S_{u}^{*}$ $\displaystyle=\frac{\Pi}{\lambda_{h}^{*}+\mu},$
$\displaystyle E^{*}$
$\displaystyle=\frac{\lambda_{h}^{*}S_{u}^{*}}{\gamma+\mu},I^{*}=m_{4}E^{*},$
$\displaystyle J^{*}$
$\displaystyle=m_{5}E^{*},R^{*}=\frac{\sigma_{1}m_{4}E^{*}+\sigma_{2}m_{5}E^{*}}{\mu},$
$\displaystyle N^{*}$
$\displaystyle=\frac{\Pi-\delta_{1}m_{4}E^{*}-\delta_{2}m_{5}E^{*}}{\mu}.$
(3.7)

After simplification, we have the expression of $E^{*}$ as follows:

$E^{*}=\frac{\Pi}{(\beta-\delta_{1})m_{4}+(\beta\nu-\delta_{2})m_{5}}\left[R_{1}-1\right]$

, where $R_{1}$ is same as the threshold quantity for the AFDFE, given by

$R_{1}=\frac{\beta\gamma}{m_{2}(\mu+\gamma)}\left[(1-p)\frac{\eta\nu+m_{2}}{m_{1}}+p\nu\right]$

.

Therefore, the AFEE will exist if $R_{1}>1$ and
$\beta>max\\{\delta_{1},\frac{\delta_{2}}{\nu}\\}$.

####  3.2.2 Existence of endemic equilibrium

Let $E_{**}=(S_{u}^{**},S_{a}^{**},E^{**},I^{**},J^{**},R^{**})$ be any
endemic equilibrium of system (2). Let us denote

$\displaystyle m_{1}$
$\displaystyle=\eta+\sigma_{1}+\mu+\delta_{1},m_{2}=\sigma_{2}+\mu+\delta_{2},$
$\displaystyle m_{3}$
$\displaystyle=\frac{1}{\alpha}\left[(\theta+\mu)+\epsilon\\{\alpha-(\theta+\mu)\\}\right],m_{4}=\frac{(1-p)\gamma}{m_{1}},$
$\displaystyle m_{5}$
$\displaystyle=\frac{p\gamma}{m_{2}}+\frac{\eta(1-p)\gamma}{m_{1}m_{2}},m_{6}=\beta(m_{4}+\nu
m_{5}),m_{7}=\frac{\mu+\theta}{\alpha\mu}.$

Further, the force of infection be

$\displaystyle\lambda_{h}^{**}=\frac{\beta[I^{**}+\nu J^{**}]}{N^{**}}$ (3.8)

By setting the right equations of system (2) equal to zero, we have

$\displaystyle S_{u}^{**}$
$\displaystyle=\frac{N^{**}}{\alpha}\left(\epsilon\lambda_{h}^{**}+\mu+\theta\right),S_{a}^{**}=\frac{\Pi\alpha-N^{**}(\lambda_{h}^{**}+\mu)(\epsilon\lambda_{h}^{**}+\mu+\theta)}{\alpha(\epsilon\lambda_{h}^{**}+\mu)},$
$\displaystyle E^{**}$
$\displaystyle=\frac{\lambda_{h}^{**}(S_{u}^{**}+\epsilon
S_{a}^{**})}{\gamma+\mu},I^{**}=m_{4}E^{**},$ $\displaystyle J^{**}$
$\displaystyle=m_{5}E^{**},R^{**}=\frac{\sigma_{1}m_{4}E^{**}+\sigma_{2}m_{5}E^{**}}{\mu},$
$\displaystyle N^{**}$
$\displaystyle=\frac{\Pi-\delta_{1}m_{4}E^{**}-\delta_{2}m_{5}E^{**}}{\mu}.$
(3.9)

From Equations (3.8) and (3.2.2), we have

$\displaystyle\lambda_{h}^{**}$ $\displaystyle=\frac{\beta(m_{4}+\nu
m_{5})E^{**}}{N^{**}}$ (3.10) $\displaystyle=\frac{\beta(m_{4}+\nu
m_{5})\lambda_{h}^{**}}{\mu+\gamma}\left(\frac{S_{u}^{**}}{N^{**}}+\epsilon\frac{S_{a}^{**}}{N^{**}}\right)$

This implies

$\displaystyle\frac{\mu+\gamma}{\beta(m_{4}+\nu
m_{5})}=\frac{1}{\alpha}\left(\epsilon\lambda_{h}^{**}+\mu+\theta\right)+\frac{\Pi\alpha\epsilon}{\alpha(\epsilon\lambda_{h}^{**}+\mu)N^{**}}-\frac{\epsilon(\lambda_{h}^{**}+\mu)(\epsilon\lambda_{h}^{**}+\mu+\theta)}{\alpha(\epsilon\lambda_{h}^{**}+\mu)}$
(3.11)

Now, using expression of $N^{**}$ and equation (3.10), we have

$\displaystyle\frac{\Pi}{N^{**}}=\mu+\frac{(\delta_{1}m_{4}+\delta_{2}m_{5})\lambda_{h}^{**}}{\beta(m_{4}+\nu
m_{5})}$ (3.12)

Putting the value of $\frac{\Pi}{N^{**}}$ in equation (3.11) and simplifying,
we obtain

$\displaystyle\lambda_{h}^{**}=\frac{\alpha\mu(\mu+\gamma)(R_{0}-1)}{\epsilon(\mu+\gamma)\\{\alpha-\mu(1-\epsilon)R_{1}\\}-\alpha\epsilon(\delta_{1}m_{4}+\delta_{2}m_{5})}$
(3.13)

This indicate that the model 2 has unique endemic equilibrium if it exists.
The existence criterion for the EE is $R_{0}>1$ and

$\alpha(\mu+\gamma)>\mu(\mu+\gamma)(1-\epsilon)R_{1}+\alpha(\delta_{1}m_{4}+\delta_{2}m_{5})$

.

The stability analysis of the equilibria for the model (2) is summarized in
Table 2.

Table 2: The local stability of equilibria for the Model (2). LAS $\equiv$ Locally Asymptotically Stable and GAS $\equiv$ Globally Asymptotically Stable. Equilibria |  Existence condition |  Stability criterion
---|---|---
$E_{0}$ |  Always exits |  LAS as well as GAS if $max\left[R_{1},\frac{\alpha}{\mu+\theta}\right]<1$
$E_{1}$ |  $\alpha>\theta+\mu$ |  LAS if $R_{0}<1$ and GAS if $max\left[R_{0},\frac{\beta\gamma}{(\sigma_{1}+\mu+\delta_{1})(\mu+\gamma)},\frac{\beta\nu\gamma}{m_{2}(\mu+\gamma)}\right]<1$
$E_{2}$ |  $\beta>max\\{\delta_{1},\frac{\delta_{2}}{\nu}\\}$ and $R_{1}>1$ |
$E_{3}$ |  $R_{0}>1$ and $\alpha(\mu+\gamma)>\mu(\mu+\gamma)(1-\epsilon)R_{1}+\alpha(\delta_{1}m_{4}+\delta_{2}m_{5})$ |

##  4 Numerical bifurcation analysis

In this section, various possibilities of forward transcritical bifurcations
are examined based on the stability analysis of the four equilibrium points of
the model (2). To do the numerical experiments, the initial conditions are
assumed to be,

$\displaystyle N$ $\displaystyle=10000,S_{u}(0)=0.9\times N,$ $\displaystyle
S_{a}(0)$ $\displaystyle=100,E(0)=100,$ $\displaystyle I(0)$
$\displaystyle=10,J(0)=10,R(0)=0.$

The fixed parameters used in this section are as follows: $\Pi=\mu\times N$,
$\nu=0.05$, $\gamma=0.1$, $\eta=0.01$, $\sigma_{1}=0.05$, $\sigma_{2}=0.01$,
$p=0.2$, $\delta_{1}=0.01$ and $\delta_{2}=0.003$.

From the stability analysis of the equilibia $E_{0}$ and $E_{1}$, it can be
inferred that the $E_{0}$ will become unstable after a certain threshold of
$\alpha$ namely $\mu+\theta$ whenever $R_{1}<1$. In this region when
$\alpha>\mu+\theta$, the DFE will become LAS whenever $R_{0}<1$. This
phenomenon is depicted in Fig. 2(a). This type of phenomenon is called forward
transcritical bifurcation where the two equilibrium points switch their
stability at a critical value.

(a) (b) (c)

Figure 2: (a) Transcritical bifurcation for the aware susceptibles at
equilibrium ($S_{a}^{*}$) of the model (2). Using the parameter values:
$\mu=0.01$, $\beta=0.05$, $\theta=0.05$, $\epsilon=0.04$ and
$0.001<\alpha<0.2$. For this parameter set, $R_{1}=0.4842$. (b) Transcritical
bifurcation for the notified infected population at equilibrium ($J^{*}$) of
the model (2). Using the parameter values: $\mu=0.03$, $\alpha=0.03$,
$\theta=0.03$, $\epsilon=0.04$ and $0.1<\beta<0.25$. (c) Transcritical
bifurcation for the notified infected population at equilibrium ($J^{*}$) of
the model (2). Using the parameter values: $\mu=0.01$, $\alpha=0.3$,
$\theta=0.01$, $\epsilon=0.1$ and $0.4<\beta<0.9$.

Further, the forward transcritical bifurcation between the equilibria $E_{0}$
and $E_{2}$ is depicted in Fig. 2(b). The reason behind this result is that
the existence of $E_{2}$ depend on the threshold quantity $R_{1}$ of $E_{0}$.
This result indicate that depending on parameter values, the awareness free,
DFE can become unstable and one of $E_{1}$ or $E_{2}$ will become stable.

Furthermore, the forward transcritical bifurcation between the equilibria
$E_{1}$ and $E_{3}$ is depicted in Fig. 2(c). The reason behind this result is
that the existence of $E_{3}$ depend on the threshold quantity $R_{0}$ of
$E_{1}$. This result indicate that depending on parameter values, the DFE can
become unstable and $E_{3}$ will become stable.

##  5 Case study on Colombia COVID-19 data

###  5.1 Data description

As of August, $28^{th}$ 2020, there were more than $590$ thousand cases and
above $18$ thousand deaths in Colombia. Daily COVID-19 notified cases and
deaths of Colombia for the time period March $19^{th}$, 2020 to August
$24^{th}$, 2020 is considered for our study. These 159 days COVID-19 notified
cases and deaths were collected from Worldometer2020 . We use the first 149
data points to calibrate the unknown model parameters. In this time period,
COVID-19 cases and deaths both display a upward trend for Colombia. This is an
alarming situation as the pandemic continue to affect the country. This is why
we chose Colombia for the case study. However, we use the remaining 10 data
points to check the accuracy of the fitted model. The demographic parameter
values and initial conditions for fitting the proposed model to data are given
in Table 3.

Table 3: Demographic parameter values for Colombia and initial conditions

.  Parameters/IC’s Description Values Reference $N$ Total population of
Colombia 50951997 Worldometer2020  $\mu$ Natural death rate or (life
expectancy)-1 0.3518 $\times$ $10^{-4}$ Worldometer2020  $\Pi$ Recruitment
rate $\mu\times N$ $S_{u}(0)$ Initial number of unaware susceptible $0.9\times
N$ – $S_{a}(0)$ Initial number of aware susceptible 100 – $J(0)$ Initial
number of notified patients 2 Data $R(0)$ Initial number of recovered patients
0 –

###  5.2 Model calibration

We fit the model (2) to daily new notified cases of COVID-19 for Colombia.
Fixed parameters of the model (2) are given in Table 1. The demographic
parameters related to Colombia and initial condition are reported in Table 3.
We estimate three unknown model parameters such as: (a) the transmission rate
of infected individuals ($\beta$), (b) ($\alpha$) and (c) ($\delta_{2}$) by
fitting the model to newly daily reported cases. Additionally, two initial
conditions of the model (2) were also estimated from the data, namely initial
number of exposed individuals $E(0)$ and initial number of un-notified
individuals $I(0)$. During the specified time period, nonlinear least square
solver $lsqnonlin$ (in MATLAB) is used to fit simulated daily data to the
reported COVID-19 cases and deaths in Colombia. We used Delayed Rejection
Adaptive Metropolis algorithm haario2006dram  to generate the 95% confidence
region. An explanation of this technique for model fitting is given in
ghosh2017mathematical . The estimated parameters are given in Table 4. The
fitting of the daily new notified COVID-19 cases and deaths of Colombia is
displayed in Fig. 3.

Figure 3: Model (2) fitting to daily notified COVID-19 cases and notified
deaths due to COVID-19 in Colombia. Daily notified cases (deaths) are depicted
in red dots and purple curve is the model simulation. The blue dots are test
data points in both panels. Grey shaded region is the 95% confidence region.
Table 4: Estimated parameter values of the model (2)

.  Parameters Mean values 95% confidence interval $\beta$ 0.2584 (0.2514 -
0.2662) $\alpha$ 0.1069 (0.1036 - 0.1099) $\delta_{2}$ 0.0032 (0.0011 -
0.0054) $E(0)$ 1919 (1175 - 4147) $I(0)$ 1668 (984 - 3136)

Using test data points (August $15^{th}$, 2020 to August $24^{th}$, 2020), we
calculate two accuracy metrics for these 10 test data points: Root mean
squared error (RMSE) and mean absolute error (MAE). For the case fitting, we
found that RMSE=2031.5 and MAE=1491.1. On the other hand for fitting COVID-19
deaths, RMSE=137.08 and MAE=125.05 are found. This indicate that the fitting
are quite good for both scenarios.

Finally, we estimate the basic reproduction number $(R_{0})$, for the proposed
model (2). We draw 1000 samples of the estimated parameters from their
posterior distribution obtained from the MCMC run and put them in the
expression of $R_{0}$. All the fixed parameters are taken from the Table 1 and
Table 3. Estimated values of $R_{0}$ is found to be 0.7815 with 95% confidence
interval (0.7633 – 0.8014).

###  5.3 Sensitivity analysis

We performed global sensitivity analysis to identify most influential
parameters with respect to the total deaths due to COVID-19 in 6 months time
frame (starting from March, $19^{th}$ 2020). Let us denote by $D_{total}$ the
total number notified and un-notified deaths due to COVID-19. Partial rank
correlation coefficients (PRCCs) are calculated and plotted in Fig. 4.
Nonlinear and monotone relationship were observed for the parameters with
respect to $D_{total}$, which is a prerequisite for performing PRCC analysis.
Following Marino et. al marino2008methodology , we calculate PRCCs for the
parameters $\beta$, $\alpha$, $\theta$, $\nu$, $\epsilon$, $\delta_{1}$,
$\delta_{2}$ and $\eta$. The following response function is used to calculate
the PRCC values

$D_{total}=\int_{0}^{T}[\delta_{1}I(t)+\delta_{2}J(t)]\textup{d}t,$

where T=180 days (chosen arbitrarily). The base values for the parameters
$\beta$, $\alpha$ and $\delta_{2}$ are taken as the mean of estimated
parameters reported in Table 4. The other base values are taken as the fixed
values displayed in Table 1. For each of the parameters, 500 Latin Hypercube
Samples were generated from the interval (0.5 $\times$ base value, 1.5
$\times$ base value).

Figure 4: Effect of uncertainty of the model (2) on the total number of deaths
due to COVID-19 in Colombia. Parameters with significant PRCC indicated as ∗
(p-value $<$ 0.05). The fixed parameters are taken from Table 1.

It is observed that the parameters $\beta$, $\nu$, $\delta_{1}$, $\theta$ and
$\delta_{2}$ have significant positive correlations with $D_{total}$. This
indicates that transmission rate of COVID-19 will increase the total number
deaths related to COVID-19. Besides, modification factor for notified patients
and death rate of un-notified patients are positively correlated with
$D_{total}$. On the other hand, the learning factor related to aware
susceptibles has significant negative correlation with the response variable.
These results reinforces the fact that $\beta$, $\nu$ and $\alpha$ are very
crucial for reduction of COVID-19 cases in Colombia.

###  5.4 Future projections and control scenarios

In this section, we focus on four controllable parameters namely learning
factor related to aware susceptibles ($\alpha$), transmission rate ($\beta$),
modification factor for notified patients or equivalently the efficacy of
notified case containment ($\nu$) and reduction in transmission co-efficient
for aware susceptibles ($\epsilon$). Transmission rate of COVID-19 can be
reduced by social distancing, face mask use, PPE kit use and through use of
alcohol based hand wash li2020early ; cdcgov2020 . The parameters $\nu$ and
$\epsilon$ can also be reduced through effective management of notified cases
and through increased behavioral changes by aware susceptible respectively. On
the other hand, $\alpha$ should be increased to reduce the burden of COVID-19
in the society. It can be increased if the aware people show more pro-social
activities. However, the results from global sensitivity analysis suggest that
$\beta$ is most effective in terms of reduction in total COVID-19 related
deaths. The parameters $\alpha$ and $\nu$ were also found significant with
respect to the response function. Now we visualize the impacts of these four
parameters on the un-notified and notified COVID-19 cases in Colombia. Using
the estimated parameters (see Table 4) we predict un-notified and notified
infections in the coming 5 months (150 days) starting from August $15^{th}$,
2020. The baseline curve is determined by simulating the model with fixed
parameters from Table 1 and mean values of estimated parameters from Table 4.
For different values of $\alpha$, $\beta$, $\nu$ and $\epsilon$, the case
reduction in COVID-19 cases is depicted in Fig. 5 and 6.

Figure 5: Effect of control parameters $\alpha$ and $\beta$ on the notified
and un-notified COVID-19 cases.

Figure 6: Effect of control parameters $\nu$ and $\epsilon$ on the notified
and un-notified COVID-19 cases.

It can be observed that all the four controllable parameters show similar
trends in case reduction. However, $\beta$ and $\epsilon$ are showing sharp
decay in cases. Note that the scales of case reductions in the Figs. 5 and 6
are similar, but we cannot quantify the effectiveness of the parameters.
Therefore, we calculate percentage reductions in un-notified and notified
COVID-19 cases in Colombia, Table 5. The fixed parameter values are taken from
Table 1 while the values of other parameters are taken from Table 4. We used
the following basic formula to compute percentage reductions in the values of
$I(t)$ and $J(t)$

$\displaystyle\textmd{Percentage reduction}=\frac{\textmd{Base
value}-\textmd{Model output}}{\textmd{Base value}}\times 100.$

Table 5: Percentage reduction in un-notified and notified COVID-19 cases for different controllable parameter values. Parameters | Values | Un-notified case reduction | Notified case reduction
---|---|---|---
$\alpha$ | 0.15 | 18.05 | 16.20
| 0.3 | 38.73 | 34.71
| 0.6 | 47.91 | 42.98
$\beta$ | 0.2 | 42.64 | 38.18
| 0.15 | 61.03 | 54.79
| 0.1 | 71.63 | 64.41
$\nu$ | 0.4 | 13.46 | 12.08
| 0.3 | 25.75 | 23.00
| 0.2 | 35.95 | 32.12
$\epsilon$ | 0.3 | 24.38 | 21.73
| 0.2 | 40.40 | 36.08
| 0.1 | 50.99 | 45.65

From Table 5, it can be argued that $\beta$ is most effective in reduction of
COVID-19 cases. Maximum reduction in $\epsilon$ (=0.1) can reduce notified
COVID-19 cases upto 45.65%. Learning factor related to aware susceptibles
($\alpha$) also has competitive potential to reduce the COVID-19 cases in the
community.

##  6 Discussion and conclusion

This paper provides a deterministic model for the transmission dynamics of
COVID-19 outbreak incorporating prosocial behaviour. The model, which adopts
standard incidence functions in a realistic way, allows COVID-19 to be
transmitted by un-notified and notified individuals. To gain insight into its
dynamic features, the model was rigorously analyzed. The findings obtained are
as follows. There are four type of equilibrium points of the proposed model:
awareness free, disease free equilibrium ($E_{0}$), disease free equilibrium
($E_{1}$), awareness free endemic equilibrium ($E_{2}$) and endemic
equilibrium point ($E_{3}$). The awareness free, disease free equilibrium is
found to be globally asymptotically stable under a parametric condition
(($max\\{R_{1},\frac{\alpha}{\theta+\mu}\\}<1$)). The basic reproduction
number ($R_{0}$) for the proposed model is calculated using the next-
generation matrix method. The model has a locally-stable disease-free
equilibrium whenever the basic reproduction number is less than unity. The
global stability condition for the DFE is also presented using Lyapunov
function. The existence and stability criterion of these four equilibria are
presented in Table 2. Further, the two threshold quantities $R_{1}$ and
$R_{0}$ are linearly dependent on each other by the relation
$R_{0}=m_{3}R_{1}$. Using a hypothetical parameter set, we show the stability
switch or transcritical bifurcation between $E_{0}$ and $E_{1}$ (see Fig.
2(a)). Also, transcritical bifurcations are observed between the equilibria
$E_{0}$ and $E_{2}$ (see Fig. 2(b)) and between $E_{1}$ and $E_{3}$ (see Fig.
2(c)).

We calibrated the proposed model parameters to fit Colombia’s daily cases and
death data during the time period of March $19^{th}$, 2020 to August
$14^{th}$, 2020. Using test data points (August $15^{th}$, 2020 to August
$14^{th}$, 2020), we found that RMSE=2031.5 and MAE=1491.1 for notified case
fitting and for fitting COVID-19 deaths, RMSE=137.08 and MAE=125.05 are found.
Thus, the fitting is pretty good for actual data in the study period. Using
estimated parameters, the basic reproduction number is found to be 0.7815 with
95% confidence interval (0.7633 – 0.8014). Which indicate the success of
Colombian government in containing this deadly disease. This is also reflected
on the current decreasing phase of the notified cases of COVID-19 in Colombia.
Global sensitivity analysis is performed with respect to the total number of
COVID-19 related deaths. Results indicate that transmission rate ($\beta$),
modification factor ($\nu$) and learning factor related to awareness of
susceptibles ($\alpha$) are very crucial for reduction in disease related
deaths. We have also investigated the impact of four controllable parameters
on the prevalence of un-notified and notified COVID-19 cases. From Fig. 5 and
6, it can be observed that different level of controls can significantly
reduce the burden of COVID-19 from community. However, to better quantify the
impacts, we calculate the percentage reduction using a simple formula. This
reveals that reduction in transmission rate is most effective in reducing un-
notified and notified COVID-19 caese (see Table 5). Increase in the learning
factor ($\alpha$) has competitive potential to flatten the curve. This finding
reinforces the need for amplified campaigns and awareness made by individual
aware susceptible persons. Prosocial behavior is needed to combat this highly
infectious pandemic disease. Along with verified control strategies, the
governments should promote prosocial awareness by aware people. This will
definitely benefit the case reduction as well as management of future COVID-19
cases.

## References

  * [1] WHO. Coronavirus disease (covid-19) outbreak.  https://www.who.int/emergencies/diseases/novel-coronavirus-2019, 2020.  Retrieved : 2020-08-15.
  * [2] Chaolin Huang, Yeming Wang, Xingwang Li, Lili Ren, Jianping Zhao, Yi Hu, Li Zhang, Guohui Fan, Jiuyang Xu, Xiaoying Gu, et al.  Clinical features of patients infected with 2019 novel coronavirus in wuhan, china.  The Lancet, 395(10223):497–506, 2020.
  * [3] Lisa E Gralinski and Vineet D Menachery.  Return of the coronavirus: 2019-ncov.  Viruses, 12(2):135, 2020.
  * [4] Centers for disease control and prevention: 2019 novel coronavirus.  https://www.cdc.gov/coronavirus/2019-ncov, 2020.  Retrieved : 2020-03-10.
  * [5] Jasper Fuk-Woo Chan, Shuofeng Yuan, Kin-Hang Kok, Kelvin Kai-Wang To, Hin Chu, Jin Yang, Fanfan Xing, Jieling Liu, Cyril Chik-Yan Yip, Rosana Wing-Shan Poon, et al.  A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-person transmission: a study of a family cluster.  The Lancet, 395(10223):514–523, 2020.
  * [6] COVID-19 coronavirus outbreak.  https://www.worldometers.info/coronavirus/#repro, 2020.  Retrieved : 2020-08-15.
  * [7] Abba B Gumel, Shigui Ruan, Troy Day, James Watmough, Fred Brauer, P Van den Driessche, Dave Gabrielson, Chris Bowman, Murray E Alexander, Sten Ardal, et al.  Modelling strategies for controlling sars outbreaks.  Proceedings of the Royal Society of London. Series B: Biological Sciences, 271(1554):2223–2232, 2004.
  * [8] Wenhui Li, Michael J Moore, Natalya Vasilieva, Jianhua Sui, Swee Kee Wong, Michael A Berne, Mohan Somasundaran, John L Sullivan, Katherine Luzuriaga, Thomas C Greenough, et al.  Angiotensin-converting enzyme 2 is a functional receptor for the sars coronavirus.  Nature, 426(6965):450–454, 2003.
  * [9] Raoul J de Groot, Susan C Baker, Ralph S Baric, Caroline S Brown, Christian Drosten, Luis Enjuanes, Ron AM Fouchier, Monica Galiano, Alexander E Gorbalenya, Ziad A Memish, et al.  Commentary: Middle east respiratory syndrome coronavirus (mers-cov): announcement of the coronavirus study group.  Journal of virology, 87(14):7790–7792, 2013.
  * [10] Tridip Sardar, Indrajit Ghosh, Xavier Rodó, and Joydev Chattopadhyay.  A realistic two-strain model for mers-cov infection uncovers the high risk for epidemic propagation.  PLoS neglected tropical diseases, 14(2):e0008065, 2020.
  * [11] Benjamin J Cowling, Minah Park, Vicky J Fang, Peng Wu, Gabriel M Leung, and Joseph T Wu.  Preliminary epidemiologic assessment of mers-cov outbreak in south korea, may–june 2015.  Euro surveillance: bulletin Europeen sur les maladies transmissibles= European communicable disease bulletin, 20(25), 2015.
  * [12] KH Kim, TE Tandi, Jae Wook Choi, JM Moon, and MS Kim.  Middle east respiratory syndrome coronavirus (mers-cov) outbreak in south korea, 2015: epidemiology, characteristics and public health implications.  Journal of Hospital Infection, 95(2):207–213, 2017.
  * [13] Kin On Kwok, Arthur Tang, Vivian WI Wei, Woo Hyun Park, Eng Kiong Yeoh, and Steven Riley.  Epidemic models of contact tracing: Systematic review of transmission studies of severe acute respiratory syndrome and middle east respiratory syndrome.  Computational and structural biotechnology journal, 2019.
  * [14] Calistus N Ngonghala, Enahoro Iboi, Steffen Eikenberry, Matthew Scotch, Chandini Raina MacIntyre, Matthew H Bonds, and Abba B Gumel.  Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel coronavirus.  Mathematical Biosciences, page 108364, 2020.
  * [15] Adam J Kucharski, Timothy W Russell, Charlie Diamond, Yang Liu, John Edmunds, Sebastian Funk, Rosalind M Eggo, Fiona Sun, Mark Jit, James D Munday, et al.  Early dynamics of transmission and control of covid-19: a mathematical modelling study.  The lancet infectious diseases, 2020.
  * [16] Tridip Sardar, Sk Shahid Nadim, Sourav Rana, and Joydev Chattopadhyay.  Assessment of lockdown effect in some states and overall india: A predictive mathematical study on covid-19 outbreak.  Chaos, Solitons & Fractals, page 110078, 2020.
  * [17] Biao Tang, Xia Wang, Qian Li, Nicola Luigi Bragazzi, Sanyi Tang, Yanni Xiao, and Jianhong Wu.  Estimation of the transmission risk of the 2019-ncov and its implication for public health interventions.  Journal of Clinical Medicine, 9(2):462, 2020.
  * [18] Shi Zhao, Qianyin Lin, Jinjun Ran, Salihu S Musa, Guangpu Yang, Weiming Wang, Yijun Lou, Daozhou Gao, Lin Yang, Daihai He, et al.  Preliminary estimation of the basic reproduction number of novel coronavirus (2019-ncov) in china, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak.  International Journal of Infectious Diseases, 92:214–217, 2020\.
  * [19] Joshua Kiddy K Asamoah, MA Owusu, Zhen Jin, FT Oduro, Afeez Abidemi, and Esther Opoku Gyasi.  Global stability and cost-effectiveness analysis of covid-19 considering the impact of the environment: using data from ghana.  Chaos, Solitons & Fractals, page 110103, 2020.
  * [20] Qinling Yan, Yingling Tang, Dingding Yan, Jiaying Wang, Linqian Yang, Xinpei Yang, and Sanyi Tang.  Impact of media reports on the early spread of covid-19 epidemic.  Journal of Theoretical Biology, page 110385, 2020.
  * [21] Weike Zhou, Aili Wang, Fan Xia, Yanni Xiao, and Sanyi Tang.  Effects of media reporting on mitigating spread of covid-19 in the early phase of the outbreak.  2020\.
  * [22] Xinghua Chang, Maoxing Liu, Zhen Jin, and Jianrong Wang.  Studying on the impact of media coverage on the spread of covid-19 in hubei province, china.  Mathematical Biosciences and Engineering, 17(4):3147, 2020.
  * [23] Subhas Khajanchi, Kankan Sarkar, Jayanta Mondal, and Matjaz Perc.  Dynamics of the covid-19 pandemic in india.  arXiv preprint arXiv:2005.06286, 2020.
  * [24] Fekadu Tadege Kobe and Purnachandra Rao Koya.  Modeling and analysis of effect of awareness programs by media on the spread of covid-19 pandemic disease.  American Journal of Applied Mathematics, 8(4):223–229, 2020.
  * [25] Fulgensia Kamugisha Mbabazi, Yahaya Gavamukulya, Richard Awichi, Peter Olupot-Olupot, Samson Rwahwire, Saphina Biira, and Livingstone S Luboobi.  A mathematical model approach for prevention and intervention measures of the covid–19 pandemic in uganda.  2020\.
  * [26] Ahmed A Mohsen, Hassan Fadhil AL-Husseiny, Xueyong Zhou, and Khalid Hattaf.  Global stability of covid-19 model involving the quarantine strategy and media coverage effects.  2020\.
  * [27] Wikipedia. Coronavirus disease (covid-19) outbreak.  https://en.wikipedia.org/wiki/COVID-19_pandemic_in_Colombia, 2020\.  Retrieved : 2020-08-20.
  * [28] LIUYONG PANG, SANHONG LIU, XINAN ZHANG, TIANHAI TIAN, and ZHONG ZHAO.  Transmission dynamics and control strategies of covid-19 in wuhan, china.  Journal of Biological Systems, pages 1–18, 2020.
  * [29] Biao Tang, Nicola Luigi Bragazzi, Qian Li, Sanyi Tang, Yanni Xiao, and Jianhong Wu.  An updated estimation of the risk of transmission of the novel coronavirus (2019-ncov).  Infectious Disease Modelling, 2020.
  * [30] TD Frank.  Covid-19 order parameters and order parameter time constants of italy and china: A modeling approach based on synergetics.  Journal of Biological Systems, 2020.
  * [31] Jorge Humberto Rojas, Marlio Paredes, Malay Banerjee, Olcay Akman, and Anuj Mubayi.  Mathematical modeling & the transmission dynamics of sars-cov-2 in cali, colombia: Implications to a 2020 outbreak & public health preparedness.  medRxiv, 2020.
  * [32] Nana Geraldine Cabo Bizet and Damián Kaloni Mayorga Peña.  Time-dependent and time-independent sir models applied to the covid-19 outbreak in argentina, brazil, colombia, mexico and south africa.  arXiv preprint arXiv:2006.12479, 2020.
  * [33] Anibal A Teheran, Gabriel Camero, Ronald Prado de la Guardia, Carolina Hernandez, Giovanny Herrera, Luis M Pombo, Albert Avila, Carolina Florez, Esther C Barros, Luis Perez Garcia, et al.  Epidemiological characterization of asymptomatic carriers of covid-19 in colombia.  medRxiv, 2020.
  * [34] Fred G Manrique-Abril, Carlos A Agudelo-Calderon, Víctor M González-Chordá, Oscar Gutiérrez-Lesmes, Cristian F Téllez-Piñerez, and Giomar Herrera-Amaya.  Sir model of the covid-19 pandemic in colombia.  Revista de Salud Pública, 22(1), 2020.
  * [35] Joshua M Epstein, Jon Parker, Derek Cummings, and Ross A Hammond.  Coupled contagion dynamics of fear and disease: mathematical and computational explorations.  PLoS One, 3(12):e3955, 2008.
  * [36] Winfried Just, Joan Saldaña, and Ying Xin.  Oscillations in epidemic models with spread of awareness.  Journal of Mathematical Biology, 76(4):1027–1057, 2018.
  * [37] Maia Martcheva.  An introduction to mathematical epidemiology, volume 61.  Springer, 2015.
  * [38] Robert M May.  Infectious diseases of humans: dynamics and control.  Oxford University Press, 1991.
  * [39] Sudip Samanta, Sourav Rana, Anupama Sharma, Arvind Kumar Misra, and Joydev Chattopadhyay.  Effect of awareness programs by media on the epidemic outbreaks: A mathematical model.  Applied Mathematics and Computation, 219(12):6965–6977, 2013.
  * [40] Qun Li, Xuhua Guan, Peng Wu, Xiaoye Wang, Lei Zhou, Yeqing Tong, Ruiqi Ren, Kathy SM Leung, Eric HY Lau, Jessica Y Wong, et al.  Early transmission dynamics in wuhan, china, of novel coronavirus–infected pneumonia.  New England Journal of Medicine, 2020.
  * [41] Natalie M Linton, Tetsuro Kobayashi, Yichi Yang, Katsuma Hayashi, Andrei R Akhmetzhanov, Sung-mok Jung, Baoyin Yuan, Ryo Kinoshita, and Hiroshi Nishiura.  Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: a statistical analysis of publicly available case data.  Journal of clinical medicine, 9(2):538, 2020.
  * [42] Zunyou Wu and Jennifer M McGoogan.  Characteristics of and important lessons from the coronavirus disease 2019 (covid-19) outbreak in china: summary of a report of 72 314 cases from the chinese center for disease control and prevention.  Jama, 323(13):1239–1242, 2020.
  * [43] Penghui Yang, Yibo Ding, Zhe Xu, Rui Pu, Ping Li, Jin Yan, Jiluo Liu, Fanping Meng, Lei Huang, Lei Shi, et al.  Epidemiological and clinical features of covid-19 patients with and without pneumonia in beijing, china.  Medrxiv, 2020.
  * [44] Roman Woelfel, Victor Max Corman, Wolfgang Guggemos, Michael Seilmaier, Sabine Zange, Marcel A Mueller, Daniela Niemeyer, Patrick Vollmar, Camilla Rothe, Michael Hoelscher, et al.  Clinical presentation and virological assessment of hospitalized cases of coronavirus disease 2019 in a travel-associated transmission cluster.  MedRxiv, 2020.
  * [45] Lauren Tindale, Michelle Coombe, Jessica E Stockdale, Emma Garlock, Wing Yin Venus Lau, Manu Saraswat, Yen-Hsiang Brian Lee, Louxin Zhang, Dongxuan Chen, Jacco Wallinga, et al.  Transmission interval estimates suggest pre-symptomatic spread of covid-19.  MedRxiv, 2020.
  * [46] Leonardo López and Xavier Rodó.  The end of social confinement and covid-19 re-emergence risk.  Nature Human Behaviour, 4(7):746–755, 2020.
  * [47] Chengjun Sun, Wei Yang, Julien Arino, and Kamran Khan.  Effect of media-induced social distancing on disease transmission in a two patch setting.  Mathematical biosciences, 230(2):87–95, 2011.
  * [48] Carlos Castillo-Chavez, Zhilan Feng, and Wenzhang Huang.  On the computation of ro and its role on.  Mathematical approaches for emerging and reemerging infectious diseases: an introduction, 1:229, 2002.
  * [49] Pauline Van den Driessche and James Watmough.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.  Mathematical biosciences, 180(1-2):29–48, 2002.
  * [50] Joseph P LaSalle.  The stability of dynamical systems, volume 25.  Siam, 1976.
  * [51] Hal L Smith and Paul Waltman.  The theory of the chemostat: dynamics of microbial competition, volume 13.  Cambridge university press, 1995.
  * [52] Cruz Vargas-De-León.  On the global stability of sis, sir and sirs epidemic models with standard incidence.  Chaos, Solitons & Fractals, 44(12):1106–1110, 2011.
  * [53] Heikki Haario, Marko Laine, Antonietta Mira, and Eero Saksman.  Dram: efficient adaptive mcmc.  Statistics and computing, 16(4):339–354, 2006.
  * [54] Indrajit Ghosh, Tridip Sardar, and Joydev Chattopadhyay.  A mathematical study to control visceral leishmaniasis: an application to south sudan.  Bulletin of mathematical biology, 79(5):1100–1134, 2017.
  * [55] Simeone Marino, Ian B Hogue, Christian J Ray, and Denise E Kirschner.  A methodology for performing global uncertainty and sensitivity analysis in systems biology.  Journal of theoretical biology, 254(1):178–196, 2008.