# Structural identifiability analysis of epidemic models based on differential
equations: A Primer

Gerardo Chowell  Sushma Dahal  School of Public Health, Georgia State
University, Atlanta, Georgia, USA  Yuganthi R. Liyanage  Department of
Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida, USA
*corresponding author<EMAIL_ADDRESS>Amna Tariq  School of Public Health,
Georgia State University, Atlanta, Georgia, USA  Necibe Tuncer  Department of
Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida, USA
*corresponding author<EMAIL_ADDRESS>
###### Abstract

The successful application of epidemic models hinges on our ability to
estimate model parameters from limited observations reliably. An often-
overlooked step before estimating model parameters consists of ensuring that
the model parameters are structurally identifiable from a given dataset.
Structural identifiability analysis uncovers any existing parameter
correlations that preclude their estimation from the observed variables. Here
we review and illustrate methods for structural identifiability analysis based
on a differential algebra approach using DAISY and Mathematica (Wolfram
Research). We demonstrate this approach through examples of compartmental
epidemic models previously employed to study transmission dynamics and
control. We show that lack of structural identifiability may be remedied by
incorporating additional observations from different model states or fixing
some parameters based on existing parameter correlations, or by reducing the
number of parameters or state variables involved in the system dynamics. We
also underscore how structural identifiability analysis can help improve
compartmental diagrams of differential-equation models by indicating the
observed variables and the results of the structural identifiability analysis.

### Keywords:

Structural identifiability analysis; epidemic models; parameter estimation;
differential algebra; compartmental model.

### Acknowledgements

G.C. is partially supported from NSF grants 2125246 and 2026797 and R01 GM
130900.

##  1 Introduction

Mathematical models based on differential equations are frequently applied to
investigate infectious disease dynamics and assess the impact of control
interventions during epidemics [1, 2, 3, 4, 5, 6, 7]. However, their
successful application hinges on our ability to reliably estimate model
parameters from observations stemming from a limited subset of the
epidemiological states of the system (e.g., time series of new cases and
deaths) [8, 9, 10, 11, 12, 12, 13, 14]. Estimating parameters for epidemic
models is often challenging since models frequently involve several unknown
parameters and incorporate epidemiological states that are often not directly
observed. In particular, an often-overlooked step before estimating parameters
from data consists in ensuring that the model parameters are structurally
identifiable [15, 16, 9, 10]. This step is crucial since ignoring the
structural identifiability properties of the model can lead to erroneous
inferences and, in turn, misguided public health policy decisions [15, 8, 16,
9].

Structural identifiability analysis uncovers existing correlations among the
model parameters that preclude their estimation from the observed variables
[15, 17]. When structural identifiability analysis reveals that two or more
parameters are correlated, it is not possible to estimate them accurately from
the observed variables because many parameter combinations can produce the
same observations, rendering the parameters unidentifiable. Structural
identifiability issues can be resolved by reformulating the model to reduce
the number of states or parameters or by collecting additional data of the
states of the system. Beyond its role in parameter estimation, structural
identifiability analysis uncovers limitations of the proposed model and helps
the researcher determine whether additional data could make the model
parameters structurally identifiable. Structurally identifiable parameters may
still be non-identifiable in practice due to data features [8]. The so-called
“Practical identifiability” considers real-world data issues: amount and noise
in the data and sampling frequency (e.g., data collection process) [8, 10, 9,
17]. In particular, we demonstrate how structural identifiability analysis can
help enrich compartmental diagrams of differential-equation models by
indicating the observed variables (e.g., new cases or new deaths) and whether
model parameters are structurally identifiable or not.

While several methods have been proposed to analyze the structural
identifiability properties of mathematical models based on differential
equations [15, 18, 19, 20, 21, 22, 9], here we review and illustrate a
methodology for structural identifiability analysis based on a differential
algebra approach using DAISY [23] and Mathematical (Wolfram Research). This
approach allows the derivation of the model system as a function of the
parameters and the observed states of the system, eliminating the unobserved
state variables. The power of this method lies in its ability to uncover
mathematical expressions describing any existing parameter correlations, which
reveal the parameters that are not structurally identifiable from the observed
states of the system. As such, structural identifiability analysis yields a
theoretical result as it deals with the perfect scenario assuming that the
observed state variables are given continuously over time and are noise-free
[15]. We demonstrate this approach through examples of compartmental epidemic
models with variable complexity, which have been previously employed to study
the transmission dynamics and control of various infectious diseases,
including pandemic influenza, Ebola, Zika, and COVID-19.

##  2 A differential algebra approach to investigate structural parameter
identifiability

In this section we describe how differential algebra is used to investigate
the structural identifiability of epidemic models using DAISY. To begin, we
will demonstrate this approach with a simple SEIR (Susceptible-Exposed-
Infected-Recovered) model given by the following system of differential
equations [24]:

$\textbf{Model 1
(M${}_{1}$):}\begin{cases}\displaystyle\frac{dS}{dt}=-\beta\frac{SI}{N},\\\\[6.45831pt]
\displaystyle\frac{dE}{dt}=\beta\frac{SI}{N}-kE,\\\\[6.45831pt]
\displaystyle\frac{dI}{dt}=kE-\gamma I,\\\\[6.45831pt]
\displaystyle\frac{dR}{dt}=\gamma I\end{cases}$ (M1)

This model keeps track of four epidemiological states over time: The number of
susceptible individuals ($S(t)$), the number of latent individuals ($E(t)$),
the number of infected individuals ($I(t)$), and the number of recovered
individuals ($R(t)$). In the model M1 new infections are modeled with a
standard incidence term, $\beta\frac{SI}{N}$. A susceptible individual becomes
exposed (class $E$), when she/he gets into contact with infectious
individuals. Exposed individuals move to the exposed class at rate $\beta$.
After the incubation period ends ($1/k$), the exposed individual is assumed to
be infectious. Infected individuals recover at rate $\gamma$. The model has
three epidemiological parameters: The transmission rate ($\beta$), the latent
period ($k$), and the recovery rate ($\gamma$). The initial conditions are
given by: $S(0)=S_{0},\;E(0)=E_{0},\;I(0)=I_{0}$ and $R(0)=R_{0}$. The total
population size ($N(t)$) corresponds to a constant parameter since the model
describes a closed population ($N=S_{0}+E_{0}+I_{0}+R_{0}$). Hence, the full
parameter set of the SEIR model M1 is given by
$\mbox{\boldmath$p$}=\left(N,\beta,k,\gamma\right)$.

Observations are the number of new infected cases: Observations are functions
of the epidemic model’s state variables. Here, we consider the case when the
observations ($y_{1}(t)$) corresponds to the number of new infected cases.
That is:

$y_{1}(t)=kE(t).$ (1)

We investigate whether the epidemic model M1 is structured to reveal its
epidemiological parameters from the observation of daily cases, $y_{1}(t)$.
The structural identifiability analysis, which is a theoretical feature of the
epidemic model and the observations, assumes that the smooth observation
curve, $y_{1}(t)$ is known for all time and determines whether the parameters
can be identified uniquely.

In order to define structural identifiability, we need to underscore that
observations vary as the model parameters vary. Suppose there was another set
of parameters, which we will denote by
$\hat{\mbox{\boldmath$p$}}=\left(\hat{N},\hat{\beta},\hat{k},\hat{\gamma}\right)$,
that produced the same observations. That is, we assume

$y_{1}(t;\mbox{\boldmath$p$})=y_{1}(t;\hat{\mbox{\boldmath$p$}})$

Structural identifiability says that this could only happen when two parameter
sets are identical, that is $\mbox{\boldmath$p$}=\hat{\mbox{\boldmath$p$}}$.
We define structural identifiability as,

###### Definition 2.1

Let $\mbox{\boldmath$p$}=\left(N,\beta,k,\gamma\right)$ and
$\hat{\mbox{\boldmath$p$}}=\left(\hat{N},\hat{\beta},\hat{k},\hat{\gamma}\right)$
be distinct parameters of the model M1 and let $y_{1}(t)=kE(t)$ be the
observations. If

$\quad
y_{1}(t;\mbox{\boldmath$p$})=y_{1}(t;\hat{\mbox{\boldmath$p$}})\quad\text{implies}\quad\mbox{\boldmath$p$}=\hat{\mbox{\boldmath$p$}},$

then we conclude that the epidemic model M1 is structured to identify its
parameters from the observations $y_{1}(t)=kE(t)$. We say that the model M1 is
structurally (globally) identifiable.

    WRITE "MODEL1-SEIR"$

    % B_ IS THE VARIABLE VECTOR

    B_:={Y1,X1,X2,X3,X4}$
    FOR EACH EL_ IN B_ DO DEPEND EL_,T$

    %B1_ IS THE UNKNOWN PARAMETER VECTOR

    B1_:={beta,k,gamma1,n}$

    %NUMBER OF STATES
    NX_:=4$
    %NUMBER OF INPUTS
    NU_:=0$
    %NUMBER OF OUTPUTS
    NY_:=1$

    %MODEL  EQUATIONS

    C_:= {df(x1,t) = - beta*x1*x3/n,
          df(x2,t) = beta*x1*x3/n- k*x2,
          df(x3,t) = k*x2 - gamma1*x3,
          df(x4,t) = gamma1*x3,
                y1 = k*x2}$
    FLAG_:=1$
    daisy()$

    %VALUES OF INITIAL CONDITIONS
    ICK_:={x1=5,x2=1,x3=0,x4=0}$
    ICUNK_:={}$
    CONDINIZ()$
    END$

Code 1:  DAISY code for model M1

Thus, the structural identifiability definition states that two different
observations are only possible with two different sets of parameters when the
model is structurally identifiable. Several methods have been devised to
investigate the structural identifiability [15, 18, 19, 20, 21, 22, 9]. Here,
we will use the differential algebra approach since it yields additional
information about existing parameter correlations. The first step in this
approach is to eliminate the unobserved state variables from the system of
equations in M1. For this example, we only observe the number of new infected
individuals, which is given by $kE(t)$, whereas the time-dependent number of
susceptible, infected, or recovered individuals are not observed. We can
eliminate the unobserved classes by deriving the characteristic set of the
equations in M1. This is a rather complex procedure that frequently requires a
computer. To that end, we use the software DAISY (Differential Algebra for
Identifiability of SYstems) [23]. We write the code as presented in Code 1 in
DAISY for model M1. DAISY runs the Code 1 and yields the characteristic set of
the system of differential equations of model M1 and the observations
$y_{1}(t)=kE(t)$. The characteristic set of the system of differential
equations is not unique and depends on the ranking of the variables. The
default ranking in DAISY is that the observations have the lowest ranking, and
the other state variables follow the order given in the system. For instance,
the ranking for model M1 is

$y_{1}<S<E<I<R.$

The first equation in the characteristic set, which is given by aa_(1) in
DAISY, involves only the lowest ranked variable, which is $y_{1}(t)$. The next
equation in the characteristic set, which is given by aa_(2) in DAISY,
involves the 2 lowest ranked variables, which are $y_{1}(t)$ and $S(t)$.
Similarly, the third equation involves the variables, $y_{1}(t),S(t)$ and
$E(t)$. In Code 2, we present the DAISY result for model M1.

    MODEL EQUATION(S)

    c_ := {df(x1,t)=( - beta*x1*x3)/n,
    df(x2,t)=(beta*x1*x3 - k*n*x2)/n,
    df(x3,t)= - gamma1*x3 + k*x2,
    df(x4,t)=gamma1*x3,
    y1=k*x2}

    CHARACTERISTIC SET

    aa_(1) := df(y1,t,3)**2*df(y1,t)*y1*n**2 + df(y1,t,3)**2*y1**2*k*n**2 - df(y1,t,3)*df(y1,t,2)**2*y1*n**2 - df(
    y1,t,3)*df(y1,t,2)*df(y1,t)**2*n**2 + df(y1,t,3)*df(y1,t,2)*df(y1,t)*y1*n**2*(2*gamma1 - k) + 2*df(y1,t,3)*df(
    y1,t,2)*y1**2*k*n**2*(gamma1 + k) - df(y1,t,3)*df(y1,t)**3*n**2*(gamma1 + k) + 4*df(y1,t,3)*df(y1,t)**2*y1**2*
    beta*n + df(y1,t,3)*df(y1,t)**2*y1*n**2*(gamma1**2 - 2*k**2) + 8*df(y1,t,3)*df(y1,t)*y1**3*beta*k*n + df(y1,t,
    3)*df(y1,t)*y1**2*gamma1*k*n**2*(2*gamma1 + k) + 4*df(y1,t,3)*y1**4*beta*k**2*n + df(y1,t,3)*y1**3*gamma1**2*k
    **2*n**2 + df(y1,t,2)**3*df(y1,t)*n**2 - df(y1,t,2)**3*y1*n**2*(gamma1 + k) + 2*df(y1,t,2)**2*df(y1,t)**2*k*n
    **2 - 3*df(y1,t,2)**2*df(y1,t)*y1**2*beta*n + df(y1,t,2)**2*df(y1,t)*y1*n**2*(gamma1**2 - gamma1*k - 2*k**2) -
     3*df(y1,t,2)**2*y1**3*beta*k*n + df(y1,t,2)**2*y1**2*k*n**2*(gamma1**2 + 2*gamma1*k + k**2) + df(y1,t,2)*df(
    y1,t)**3*n**2*( - 2*gamma1**2 - gamma1*k + 2*k**2) + 2*df(y1,t,2)*df(y1,t)**2*y1**2*beta*n*(3*gamma1 - k) + df
    (y1,t,2)*df(y1,t)**2*y1*n**2*(gamma1**3 - gamma1**2*k - 3*gamma1*k**2 - 2*k**3) + 2*df(y1,t,2)*df(y1,t)*y1**3*
    beta*k*n*(6*gamma1 + k) + df(y1,t,2)*df(y1,t)*y1**2*gamma1*k*n**2*(2*gamma1**2 + 2*gamma1*k + k**2) + 2*df(y1,
    t,2)*y1**4*beta*k**2*n*(3*gamma1 + 2*k) + df(y1,t,2)*y1**3*gamma1**2*k**2*n**2*(gamma1 + k) - df(y1,t)**5*beta
    *n + df(y1,t)**4*y1*beta*n*( - 4*gamma1 - 3*k) + df(y1,t)**4*n**2*( - gamma1**3 - 2*gamma1**2*k + k**3) + 4*df
    (y1,t)**3*y1**3*beta**2 + df(y1,t)**3*y1**2*beta*n*(gamma1**2 - 6*gamma1*k - 6*k**2) + df(y1,t)**3*y1*gamma1*k
    *n**2*( - 2*gamma1**2 - 3*gamma1*k - k**2) + 12*df(y1,t)**2*y1**4*beta**2*k + df(y1,t)**2*y1**3*beta*k*n*(3*
    gamma1**2 - 4*k**2) - df(y1,t)**2*y1**2*gamma1**2*k**2*n**2*(gamma1 + k) + 12*df(y1,t)*y1**5*beta**2*k**2 + df
    (y1,t)*y1**4*beta*gamma1*k**2*n*(3*gamma1 + 2*k) + 4*y1**6*beta**2*k**3 + y1**5*beta*gamma1**2*k**3*n

    aa_(2) :=  - df(y1,t,3)*x1*y1*k*n + df(y1,t,2)*df(y1,t)*x1*k*n - df(y1,t,2)*df(y1,t)*y1*n - df(y1,t,2)*x1*y1*k
    *n*(gamma1 + k) - df(y1,t,2)*y1**2*k*n + df(y1,t)**3*n + df(y1,t)**2*x1*k*n*(gamma1 + k) + df(y1,t)**2*y1*n*(
    gamma1 + k) - 2*df(y1,t)*x1*y1**2*beta*k + 2*df(y1,t)*y1**2*gamma1*k*n - 2*x1*y1**3*beta*k**2 + y1**3*gamma1*k
    **2*n

    aa_(3) :=  - x2*k + y1

    aa_(4) := df(y1,t,3)*y1*n - df(y1,t,2)*df(y1,t)*n + df(y1,t,2)*x3*y1*beta + df(y1,t,2)*y1*n*(gamma1 + k) - df(
    y1,t)**2*x3*beta - df(y1,t)**2*n*(gamma1 + k) - df(y1,t)*x3*y1*beta*gamma1 + 2*df(y1,t)*y1**2*beta - x3*y1**2*
    beta*gamma1*k + 2*y1**3*beta*k

    aa_(5) :=  - df(x4,t)*df(y1,t,2)*y1*beta + df(x4,t)*df(y1,t)**2*beta + df(x4,t)*df(y1,t)*y1*beta*gamma1 + df(
    x4,t)*y1**2*beta*gamma1*k - df(y1,t,3)*y1*gamma1*n + df(y1,t,2)*df(y1,t)*gamma1*n - df(y1,t,2)*y1*gamma1*n*(
    gamma1 + k) + df(y1,t)**2*gamma1*n*(gamma1 + k) - 2*df(y1,t)*y1**2*beta*gamma1 - 2*y1**3*beta*gamma1*k

    MODEL NOT ALGEBRAICALLY OBSERVABLE

    PARAMETER VALUES

    b2_ := {beta=2,k=3,gamma1=5,n=7}

    MODEL PARAMETER SOLUTION(S)

    g_ := {{n=(7*beta)/2,gamma1=5,k=3}}

    MODEL NON IDENTIFIABLE

    IDENTIFIABILITY WITH ALL INITIAL CONDITIONS (IC_)

    ic_ := {x1=5,x2=1,x3=0,x4=0}

    MODEL PARAMETER SOLUTIONS

    gi_ := {{n=(7*beta)/2,gamma1=5,k=3}}

    MODEL NON IDENTIFIABLE

Code 2:  DAISY result for model M1

Since our goal is to obtain an equation that only involves the observed
variable, $y_{1}(t)$, we are interested in the first equation of the
characteristic set, which is aa_(1). As given by DAISY, the first equation is
not a monic differential polynomial. So, we divide by the coefficient of the
highest degree, which is $N^{2}$, and obtain the following input-output
equations of the model M1. The input-output equation (2), where input is the
model parameters, and output is the observation, is a monic differential
polynomial of the observed state variable $y_{1}(t)$ and the model parameters
$\mbox{\boldmath$p$}=\left(N,\beta,k,\gamma\right)$.

$\displaystyle(y_{1}^{{}^{\prime\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}+(y_{1}^{{}^{\prime\prime\prime}})^{2}y_{1}^{2}k-y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime\prime}})^{2}y_{1}-y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}+$
(2) $\displaystyle
y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}(2\gamma-k)+2y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}y_{1}^{2}k(\gamma+k)-y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{3}(\gamma+k)+$
$\displaystyle
4y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\frac{\beta}{N}+y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}(\gamma^{2}-2k^{2})+8y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{3}\frac{\beta
k}{N}+y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{2}\gamma
k(2\gamma+k)+$ $\displaystyle
4y_{1}^{{}^{\prime\prime\prime}}y_{1}^{4}\frac{\beta
k^{2}}{N}+y_{1}^{{}^{\prime\prime\prime}}y_{1}^{3}\gamma^{2}k^{2}+(y_{1}^{{}^{\prime\prime}})^{3}y_{1}^{{}^{\prime}}-(y_{1}^{{}^{\prime\prime}})^{3}y_{1}(\gamma+k)+2(y_{1}^{{}^{\prime\prime}})^{2}(y_{1}^{{}^{\prime}})^{2}k-$
$\displaystyle
3(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}^{2}\frac{\beta}{N}+(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}(\gamma^{2}-\gamma
k-2k^{2})-3(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{3}\frac{\beta
k}{N}+(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{2}k(\gamma^{2}+2\gamma k+k^{2})+$
$\displaystyle
y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{3}(-2\gamma^{2}-\gamma
k+2k^{2})+2y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\frac{\beta(3\gamma-k)}{N}+y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}(\gamma^{3}-\gamma^{2}k-3\gamma
k^{2}-2k^{3})+$ $\displaystyle
2y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{3}\frac{\beta
k}{N}(6\gamma+k)+y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{2}\gamma
k(2\gamma^{2}+2\gamma k+k^{2})+2y_{1}^{{}^{\prime\prime}}y_{1}^{4}\frac{\beta
k^{2}}{N}(3\gamma+2k)+$ $\displaystyle
y_{1}^{{}^{\prime\prime}}y_{1}^{3}\gamma^{2}k^{2}(\gamma+k)-(y_{1}^{{}^{\prime}})^{5}\frac{\beta}{N}+(y_{1}^{{}^{\prime}})^{4}y_{1}\frac{\beta(-4\gamma-3k)}{N}+(y_{1}^{{}^{\prime}})^{4}(-\gamma^{3}-2\gamma^{2}k+k^{3})+$
$\displaystyle
4(y_{1}^{{}^{\prime}})^{3}y_{1}^{3}\frac{\beta^{2}}{N^{2}}+(y_{1}^{{}^{\prime}})^{3}y_{1}^{2}\frac{\beta(\gamma^{2}-6\gamma
k-6k^{2})}{N}+(y_{1}^{{}^{\prime}})^{3}y_{1}\gamma k(-2\gamma^{2}-3\gamma
k-k^{2})+$ $\displaystyle
12(y_{1}^{{}^{\prime}})^{2}y_{1}^{4}\frac{\beta^{2}k}{N^{2}}+(y_{1}^{{}^{\prime}})^{2}y_{1}^{3}\frac{\beta
k(3\gamma^{2}-4k^{2})}{N}-(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\gamma^{2}k^{2}(\gamma+k)+12y_{1}^{{}^{\prime}}y_{1}^{5}\frac{\beta^{2}k^{2}}{N^{2}}+$
$\displaystyle y_{1}^{{}^{\prime}}y_{1}^{4}\frac{\beta\gamma
k^{2}(3\gamma+2k)}{N}+4y_{1}^{6}\frac{\beta^{2}k^{3}}{N^{2}}+y_{1}^{5}\frac{\beta\gamma^{2}k^{3}}{N}=0$

It is worth mentioning that for a given set of parameters
$\mbox{\boldmath$p$}=\left(N,\beta,k,\gamma\right)$ solving the monic
differential polynomial (2) is equivalent to solving the model M1 in terms of
the observed states of the system and the model parameters. For studying
identifiability we analyze whether another set of parameters,
$\hat{\mbox{\boldmath$p$}}=\left(\hat{N},\hat{\beta},\hat{k},\hat{\gamma}\right)$
could produce the same output (2). That is within differential algebra
approach, we define the structural identifiability as follows:

###### Definition 2.2

Let $c(\mbox{\boldmath$p$})$ denote the coefficients of the input-output
equation (2) with $\mbox{\boldmath$p$}=\left(N,\beta,k,\gamma\right)$ denoting
the parameters of the model M1. Then we say that the model M1 is structured to
reveal its parameters from the observations $y_{1}(t)=kE(t)$ if and only if

$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})\implies\mbox{\boldmath$p$}=\hat{\mbox{\boldmath$p$}}$

where
$\hat{\mbox{\boldmath$p$}}=\left(\hat{N},\hat{\beta},\hat{k},\hat{\gamma}\right)$
is another set of parameters assumed to produce the same observations.

Here, we list some of the coefficients of the input-output equation (2) of
model M1 as,

$\displaystyle c(\mbox{\boldmath$p$})=$
$\displaystyle\left\\{k,\;2\gamma-k,\;k(\gamma+k),\;\frac{\beta}{N},\;(\gamma^{2}-2k^{2}),\;k(2\gamma+k),\;\frac{\beta
k^{2}}{N},\;(\gamma^{2}-\gamma k-2k^{2}),\;\right.$
$\displaystyle\left.(2\gamma^{2}+2\gamma k+k^{2}),\;(-2\gamma^{2}-\gamma
k+2k^{2}),\;\frac{\beta(3\gamma-k)}{N},\;(\gamma^{3}-\gamma^{2}k-3\gamma
k^{2}-2k^{3}),\;\right.$ $\displaystyle\left.\frac{\beta
k}{N}(6\gamma+k)\right\\}$

DAISY solves $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ by
assigning random integers to the parameter values. We see that DAISY assigned
the following integers for the model parameters

$\hat{\mbox{\boldmath$p$}}=\left\\{\hat{\beta}=2,\;\hat{k}=3,\;\hat{\gamma}=5,\;\hat{N}=7\right\\}.$

Then DAISY solves the system
$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ for these integers, and
gives the following result

$\left\\{N=\displaystyle\frac{7\beta}{2},\;k=3,\;\gamma=5,\right\\}.$ (3)

And declares that the model M1 is not identifiable (see Code 2). At this
point, we would like to mention that the epidemic model parameters are usually
not integers. For instance when the incubation period is $5$ days, then the
parameter $k=1/5$. Because of this, we solve
$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ for arbitrary parameter
values. We write a Mathematica code to solve
$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ and obtain the following
parameter correlations: (Mathematica code available in file named Model1.nb in
github),

$k=\hat{k},\;\gamma=\hat{\gamma},\;\displaystyle\frac{\beta}{N}=\displaystyle\frac{\hat{\beta}}{\hat{N}}.$
(4)

This means we can only identify the rate at which individuals progress the
latent period, $k$, and the recovery rate, $\gamma$, but we can not identify
the transmission rate, $\beta$, or the total population size, $N$. The
parameter correlations derived above indicate that it is only possible to
identify the ratio $\displaystyle\frac{\beta}{N}$. That is, any $\beta$ or $N$
values that produce the same ratio $\displaystyle\frac{\beta}{N}$ which is
determined by the observation will yield the same observations ($y_{1}(t)$).
We would like to point out that the parameter correlations obtained using
Mathematica (Wolfram Research) (4) are in agreement with those derived using
DAISY (3).

We summarize the structural identifiability analysis of the model M1 with the
following Proposition 2.1.

###### Proposition 2.1

The epidemic model M1 is not structured to identify all of its parameters from
the number of new cases given by $y_{1}(t)=kE(t)$. We can only identify

$k,\gamma\text{ and }\displaystyle\frac{\beta}{N}$

However, the model can become structurally identifiable if the total
population size $N$ is known. Hence, we state the following result.

###### Proposition 2.2

If the total population size $N$ is known, then the model M1 is structurally
identifiable from the observations of $y_{1}(t)=kE(t)$. We can identify the
remaining three parameters,

$k,\gamma,\beta$

Figure 1: Flow chart of model M1 showing the transition of individuals between
the classes defined by disease status. Observations are the newly infected
individuals. Results from the structural identifiability analysis are
summarized for the scenarios when the initial conditions are known and
unknown.$S$$E$$I$$R$ Observations are new cases; $y_{1}(t)=kE(t)$
$\displaystyle\frac{\beta SI}{N}$$kE$
$\gamma I$
Identifiability Results:
Without Initial Conditions
identifiable parameters: $k,\gamma$ unidentifiable parameters: $\beta,N$
paramter correlations
$\displaystyle\frac{\beta}{N}=\displaystyle\frac{\hat{\beta}}{\hat{N}}$
Identifiability Results:
With Initial Conditions
identifiable parameters: $k,\gamma,\beta,N$

Note that the total population size $N$ is known, if all the initial
conditions are known, since $N=S_{0}+E_{0}+I_{0}+R_{0}$. Hence, we also state
the following identifiability result for model M1.

###### Proposition 2.3

If all the initial conditions are given for the model M1, then the model is
structurally identifiable from the observations of $y_{1}(t)=kE(t)$. We can
identify all model parameters,

$k,\gamma,\beta,N$

The enhanced compartmental diagram of model M1 that indicates the observed
states of the system and the parameter correlations resulting from the
structural identifiability analysis is given in Figure 1.

##  3 Structural Identifiability Analysis of Epidemic Models

###  3.1 Epidemic Model M2

$\textbf{Model 2
(M${}_{2}$):}\begin{cases}\displaystyle\frac{dS}{dt}=-\beta\frac{SI}{N},\\\\[6.45831pt]
\displaystyle\frac{dE}{dt}=\beta\frac{SI}{N}-kE,\\\\[6.45831pt]
\displaystyle\frac{dI}{dt}=k\rho E-\gamma I,\\\\[6.45831pt]
\displaystyle\frac{dA}{dt}=k(1-\rho)E-\gamma A,\\\\[6.45831pt]
\displaystyle\frac{dR}{dt}=\gamma I+\gamma A,\end{cases}$ (M2)

Next, we consider an extension of the SEIR model that incorporates symptomatic
and asymptomatic infections [25]. Therefore, in this second model there are
five epidemiological states: the number of susceptible individuals ($S(t)$),
the number of latent individuals ($E(t)$), the number of symptomatic
infectious individuals ($I(t)$), the number of asymptomatic (not infectious)
individuals ($A(t)$), and the number of recovered individuals ($R(t)$).
Compared to model M1, in this model we have one additional epidemiological
state, $A(t)$. The susceptible individuals upon contact with an infected
individual move to exposed class at rate $\beta$, the fraction of exposed
individuals become symptomatic infectious cases at a rate $k\rho$, the
remaining exposed individuals move to the asymptomatic class at rate
$k(1-\rho)$. Hence the fraction $\rho$ is always less than 1. The infected
individuals recover at a rate $\gamma$.

The model is equipped with initial conditions $S(0)=S_{0}$, $E(0)=E_{0}$,
$I(0)=I_{0}$, $A(0)=A_{0}$ and $R(0)=R_{0}$. Since the model describes a
closed population, the total population remains constant throughout the
epidemic ($N=S_{0}+E_{0}+I_{0}+A_{0}+R_{0}$). Therefore, the full parameter
set of the model M2 is given by
$\mbox{\boldmath$p$}=\left(N,\beta,k,\rho,\gamma\right)$.

Observations are the number of new symptomatic cases: In this model, we
consider the case when the observations ($y_{1}(t)$) corresponds to the number
of newly infected symptomatic cases. That is,

$y_{1}(t)=k\rho E(t).$ (5)

As before, we use DAISY software to eliminate the unobserved state variables
and to obtain input-output equations, which are differential-algebraic
polynomials of observed state variables with model parameter coefficients
(DAISY input code and results available in file named Model2$\\_$inputcode and
Model2$\\_$results in github). We obtain the input-output equations (24) of
the model M2 which is a monic differential polynomial of the observed state
variables and the parameters of the model M2 (see the input-output equations
(24) in Appendix).

Upon examining the coefficients, we see that the parameter $\rho$ does not
appear in the input-output equation (24) of the model M2, indicating that
$\rho$ is not structurally identifiable. We study the structural
identifiability of the model M2 as outlined in Section 2.

Let $c(\mbox{\boldmath$p$})$ denote the coefficients of the input-output
equation (24) with $\mbox{\boldmath$p$}=\left(N,\beta,k,\gamma,\rho\right)$
denoting the parameters of the model M2. Let
$\hat{\mbox{\boldmath$p$}}=\left(\hat{N},\hat{\beta},\hat{k},\hat{\gamma},\hat{\rho}\right)$
be another set of parameters assumed to produce the same observations. Solving
$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ (See Mathematica file)
we obtain the following solution,

$k=\hat{k},\;\gamma=\hat{\gamma},\;\displaystyle\frac{\beta}{N}=\displaystyle\frac{\hat{\beta}}{\hat{N}}.$

The solution means that only the parameters $k$ and $\gamma$ can be
identifiable. Parameter $\rho$ cannot be identified. We also see that similar
to model M1, the transmission rate and total population size are correlated.
The observations can only reveal the ratio $\displaystyle\frac{\beta}{N}$. We
summarize the structural identifiability analysis of the model M2 when no
initial conditions are considered in the following Proposition 3.1.

###### Proposition 3.1

The model M2 is not structured to reveal its parameters
$\mbox{\boldmath$p$}=\left(N,\beta,k,\gamma,\rho\right)$ from the observations
of $y_{1}(t)=k\rho E(t)$.

Our structural identifiability analysis of model M2 assumed that the initial
conditions were not known. Assuming that all of the initial conditions are
known, DAISY calculates at time $0$ all the polynomials of the characteristic
set. When the initial conditions are known, these polynomials become known and
are added to the previous summary of the model. With the knowledge of initial
conditions, DAISY declares the model as globally structurally identifiable
with the following set of parameter solutions:

    IDENTIFIABILITY WITH ALL INITIAL CONDITIONS (IC_)

    ic_ := {x1=x10,x2=x20,x3=x30,x4=x40,x5=x50}

    MODEL PARAMETER SOLUTIONS
    gi_ := {{rho=5,n=(11*beta)/2,gamma1=7,k=3}}

    MODEL GLOBALLY IDENTIFIABLE

Note that, when all the initial conditions are known, the parameter $\rho$ can
be identified. We summarize the identifiability analysis with known initial
conditions in the following Proposition 3.2.

###### Proposition 3.2

The model M2 is structured to identify all its parameters,
$\mbox{\boldmath$p$}=\left(N,\beta,k,\gamma,\rho\right)$, from the newly cases
$y_{1}(t)=k\rho E(t)$ if all the initial conditions are known.

The enhanced compartmental diagram of the model M2 that includes the observed
states of the system and the parameter correlations resulting from the
structural identifiability analysis given in Figure 2.

Figure 2: Flow chart of model M2 showing the transition of individuals between
the classes defined by disease status. Observations are the newly infected
individuals. Structural identifiability analysis is summarized for when
initial conditions are known and unknown.$S$$E$$I$$A$$R$ Observations are new
cases; $y_{1}(t)=k\rho E(t)$ $\displaystyle\frac{\beta SI}{N}$$k\rho E$
$k(1-\rho)E$
$\gamma I$
$\gamma I$
Identifiability Results:
Without Initial Conditions
identifiable parameters: $k,\gamma$ unidentifiable parameters: $\beta,N,\rho$
parameter correlations
$\displaystyle\frac{\beta}{N}=\displaystyle\frac{\hat{\beta}}{\hat{N}}$
Identifiability Results:
With Initial Conditions
identifiable parameters: $k,\gamma,\beta,N,\rho$

###  3.2 Epidemic Model M3

For model M3, we consider a disease where asymptomatic individuals are
infectious and contribute to the force of infection [26]. Thus in model M3,
the newly incidences are modeled as

$\displaystyle\frac{(\beta_{A}A+\beta_{I}I)S}{N}.$

$\textbf{Model 3
(M${}_{3}$):}\begin{cases}\displaystyle\frac{dS}{dt}=-\frac{(\beta_{A}A+\beta_{I}I)S}{N}\quad
S(0)=S_{0},\\\\[6.45831pt]
\displaystyle\frac{dE}{dt}=\frac{(\beta_{A}A+\beta_{I}I)S}{N}-kE\quad
E(0)=E_{0},\\\\[6.45831pt] \displaystyle\frac{dI}{dt}=k\rho E-\gamma I\quad
I(0)=I_{0},\\\\[6.45831pt] \displaystyle\frac{dA}{dt}=k(1-\rho)E-\gamma A\quad
A(0)=A_{0},\\\\[6.45831pt] \displaystyle\frac{dR}{dt}=\gamma I+\gamma
AR(0)=R_{0},\end{cases}$ (M3)

Compared to model M2, here we distinguish the transmission rate between
symptomatic ($\beta_{I}$) and asymptomatic infectious ($\beta_{A}$)
individuals. Just like in epidemic model M2, in this model a fraction of
exposed individuals move to symptomatic class at a rate $k\rho$ and the
remaining fraction move to the asymptomatic class at the rate $k(1-\rho)$.
Both symptomatically and asymptomatically infected individuals recover at a
rate $\gamma$. The full parameter set of the model M3 is then
$\mbox{\boldmath$p$}=\left(N,\beta_{A},\beta_{I},k,\rho,\gamma\right)$.

Observations are the number of new symptomatic cases: In this model, we also
consider the case when the observations ($y_{1}(t)$) correspond to the number
of new infectious symptomatic cases. That is;

$y_{1}(t)=k\rho E(t).$ (6)

Using DAISY, we obtain the input-output equations (25) of the model M3, which
are given in the Appendix. (DAISY input code and results available in file
named Model3$\\_$inputcode and Model3$\\_$results in github).

Let’s denote the coefficients of the input-output equation (25) with
$c(\mbox{\boldmath$p$})$. To study the identifiability of the epidemic model,
we suppose that another set of parameters produced the same observations.
Since the observation is given as a monic differential polynomial in (25), the
same observation can only happen if the coefficients are identical. Let’s
denote this other set of parameters which produced the same output as
$\hat{\mbox{\boldmath$p$}}=\left(\hat{N},\hat{k},\hat{\gamma},\hat{\rho},\hat{\beta}_{A},\hat{\beta}_{I}\right)$.
We then set the coefficients equal to each other,

$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$

and solve for the parameters
$\mbox{\boldmath$p$}=\left(N,\beta_{A},\beta_{I},k,\gamma,\rho\right)$ (see
Mathematica File). We obtain the following solution.

$k=\hat{k},\;\gamma=\hat{\gamma},\;\beta_{I}\rho+\beta_{A}(1-\rho)=\hat{\beta}_{I}\hat{\rho}+\hat{\beta}_{A}(1-\hat{\rho}),\;N\rho=\hat{N}\hat{\rho}.$
(7)

###### Proposition 3.3

The epidemic model M3 is not structured to reveal its parameters
$\mbox{\boldmath$p$}=\left(N,k,\gamma,\rho,\beta_{A},\beta_{I}\right)$ from
the observations of $y_{1}(t)=kE(t)$.

We can only identify parameters $k$ and $\gamma$. Parameters $\rho$,
$\beta_{A}$ and $\beta_{I}$ cannot be identified. Indeed, we found two
parameter correlations, one involving the transmission rates ($\beta_{A}$,
$\beta_{I}$) and the fraction of symptomatic infections ($\rho$) and the
second one involving the fraction of symptomatic infections ($\rho$) and the
total population size ($N$).

We would like to compare the Mathematica result (7) with the DAISY result for
model M3 as presented in Code 3.

    PARAMETER VALUES$

    b2_ := {betaa=2,betai=3,k=5,rho1=7,gamma1=11,n=13}$

    MODEL PARAMETER SOLUTION(S)$

    g_ := {{n=( - 91*betaa*rho1 + 91*betaa + 91*betai*rho1)/(9*rho1),gamma1=11,k=5}}$

    MODEL NON IDENTIFIABLE$

Code 3:  DAISY identifiability result for model M3 without the initial
conditions.

The Mathematica result contains two parameter correlations, while DAISY has
given the ratio of the 2 correlations.

$\displaystyle\frac{\beta_{I}\rho+\beta_{A}(1-\rho)}{N\rho}=\displaystyle\frac{\hat{\beta}_{I}\hat{\rho}+\hat{\beta}_{A}(1-\hat{\rho})}{\hat{N}\hat{\rho}}$
(8)

where
$\hat{\mbox{\boldmath$p$}}=\left\\{\hat{N},\;\hat{k},\;\hat{\gamma},\;\hat{\rho},\;\hat{\beta}_{A},\;\hat{\beta}_{I}\right\\}=\left\\{13,5,11,7,2,3\right\\}$.
Again, here we would like to mention that assigning an integer value greater
than $1$ for $\rho$ is not realistic, (as DAISY assigned $\hat{\rho}=7$).
Since $\rho$ is the fraction of infected individuals moving to the symptomatic
class, and it is clearly less than $1$.

Using the parameter correlations in (7), we can obtain a structurally
identifiable model by fixing the total population size, $N$ or the fraction of
symptomatic individuals $\rho$ and by setting $\beta_{A}=\beta_{I}$. Thus we
can modify the model M3 as following

$\textbf{Model 3'
(M${}_{3}^{\prime}$):}\begin{cases}\displaystyle\frac{dS}{dt}=-\frac{\beta(A+I)S}{N}\quad
S(0)=S_{0},\\\\[6.45831pt]
\displaystyle\frac{dE}{dt}=\frac{\beta(A+I)S}{N}-kE\quad
E(0)=E_{0},\\\\[6.45831pt] \displaystyle\frac{dI}{dt}=k\rho E-\gamma I\quad
I(0)=I_{0},\\\\[6.45831pt] \displaystyle\frac{dA}{dt}=k(1-\rho)E-\gamma A\quad
A(0)=A_{0},\\\\[6.45831pt] \displaystyle\frac{dR}{dt}=\gamma I+\gamma A,\quad
R(0)=R_{0},\end{cases}$ (M${}_{3}^{\prime}$)

###### Proposition 3.4

When the total population size $N$ is known or the fraction of the susceptible
individuals, $\rho$ is known, the epidemic model M${}_{3}^{\prime}$ is
structurally identifiable from the observations of newly symptomatic
infections.

We continue the structural identifiability analysis of the model M3 with known
initial conditions. DAISY gives the characteristic set of the model M3
according to the following default ranking,

$y1<S<E<I<A<R.$

The first equation in the characteristic set, (aa_(1) in DAISY) is the input-
output equations of the model M3, and it is used in our identifiability
analysis without initial conditions. When inital conditions are known, the
rest of the differential polynomials in the charactereistic set are considered
in the identifiability analysis. DAISY calculates at time $0$ all the
polynomials of the characteristic set. When the initial conditions are known,
these polynomials become known and are added to the previous summary of the
model. With the knowledge of initial conditions, DAISY still declares the
model as structurally unidentifiable with the parameter correlations given in
Code 4. We present the parameter correlation as given by DAISY in Code 4,

$\displaystyle\frac{N}{\beta_{A}}=\displaystyle\frac{\hat{N}}{\hat{\beta}_{A}},\;\displaystyle\frac{\beta_{I}}{\beta_{A}}=\displaystyle\frac{\hat{\beta}_{I}}{\hat{\beta}_{A}},\rho=\hat{\rho},\;k=\hat{k},\;\gamma=\hat{\gamma}$

We would like to mention that since $N$ is the total population size, and M3
represents a closed population, the initial conditions determine the size of
the population. Hence, the model M3 becomes identifiable when all the initial
conditions are known. We summarize the identifiability analysis result when
the initial conditions known, in the following Proposition 3.5.

    IDENTIFIABILITY WITH ALL INITIAL CONDITIONS (IC_)$

    ic_ := {x1=x10,x2=x20,x3=x30,x4=x40,x5=x50}$

    MODEL PARAMETER SOLUTIONS$

    gi_ := {{n=(13*betaa)/2, betai=(3*betaa)/2 ,rho1=7, gamma1=11,k=5}}$

    MODEL NON IDENTIFIABLE$

Code 4:  DAISY identifiability result for model M3 with known initial
conditions.

###### Proposition 3.5

The model M3 is globally identifiable from the observations of new symptomatic
infections if all the initial conditions are known.

The enhanced compartmental diagram of model M3 that includes the observed
states of the system and the parameter correlations resulting from the
structural identifiability analysis is given in Figure 3.

Figure 3: Flow chart of model M3 showing the transition of individuals between
the classes defined by disease status. Observations are the newly infected
symptomatic individuals. Structural identifiability analysis is summarized for
when initial conditions are known and unknown.$S$$E$$I$$A$$R$ Observations are
new cases; $y_{1}(t)=k\rho E(t)$
$\displaystyle\frac{(\beta_{A}A+\beta_{I}I)S}{N}$$k\rho E$
$k(1-\rho)E$
$\gamma I$
$\gamma I$
Identifiability Results:
Without Initial Conditions
identifiable parameters: $k,\gamma$ unidentifiable parameters: $\beta,N,\rho$
parameter correlations:
$\beta_{I}\rho+\beta_{A}(1-\rho)=\hat{\beta}_{I}\hat{\rho}+\hat{\beta}_{A}(1-\hat{\rho})$,
$N\rho=\hat{N}\hat{\rho}$ Identifiability Results:
With Initial Conditions
identifiable parameters: $k,\gamma,\beta_{A},,\beta_{I},N,\rho$

###  3.3 Epidemic Model M4

$\textbf{Model 4
(M${}_{4}$):}\begin{cases}\displaystyle\frac{dS}{dt}=-\beta\frac{SI}{N},\\\\[6.45831pt]
\displaystyle\frac{dE}{dt}=\beta\frac{SI}{N}-kE,\\\\[6.45831pt]
\displaystyle\frac{dI}{dt}=kE-(\gamma+\delta)I,\\\\[6.45831pt]
\displaystyle\frac{dR}{dt}=\gamma I,\\\\[6.45831pt]
\displaystyle\frac{dD}{dt}=\delta I\end{cases}$ (M4)

In the first three epidemic models (M1, M2, and M3), we considered diseases
where infected individuals eventually recover from the infection. Next, we
consider the epidemic model M4 that includes disease-induced deaths [24].
Here, susceptible individuals move to exposed class at rate $\beta$, $k$ is
the rate at which the exposed individuals become infectious, $\gamma$ is the
recovery rate, and $\delta$ denotes the mortality rate of infectious
individuals. Since the model includes disease-induced deaths, the total
population $N(t)$ is not constant. Hence, $N$ is not a parameter to be
identified. The full parameter set of the model M4 is given by
$\mbox{\boldmath$p$}=\left(\beta,k,\gamma,\delta\right)$. For epidemics where
disease-induced deaths occur, it is possible to observe the daily number of
deaths due to the disease. For this epidemic model M4, we will consider two
different observations; (i) new infections, (ii) new infections and new
deaths.

(i) Number of new infected cases are observed: First we consider the scenario
where the observations correspond to the number of newly infected cases. That
is,

$y_{1}(t)=kE(t).$

We use the differential algebra method for the structural identifiability
analysis. Using DAISY software, we obtain the following results. The input-
output equations were too long to transfer to a Mathematics file without a
human error.(see DAISY input code and results available in file named
Model4a$\\_$inputcode and Model4a$\\_$results in github). For this model, we
present the DAISY results only. DAISY assigns

$\hat{\mbox{\boldmath$p$}}=\left\\{\hat{\beta}=2,\hat{k}=3,\hat{\gamma}=5,\hat{\delta}=7\right\\}$

and solve $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ for these
integers and yields,

$\frac{\beta}{\delta}=\frac{2}{7},\quad\gamma+\delta=12,\quad k=3.$

These results indicate that we can identify parameter $k$, the ratio
$\displaystyle\frac{\beta}{\delta}$ and the sum $\gamma+\delta.$ That is, any
combination of parameters $\beta$ and $\delta$ that yields the same ratio
$\beta/\delta$ or any combination of parameters $\delta$ and $\gamma$ that
yields the same sum $\delta+\gamma$ will yield the same number of infected
cases. We conclude that the model M4 is not structurally identifiable. It only
reveals the parameter $k$, from the newly infected cases.

###### Proposition 3.6

The model M4 is not structurally identifiable from the observation of the
number of new infected cases.

Since we know the parameter correlations, we know which parameters to fix to
obtain a structurally identifiable model. Here, we fix $\delta,$ then the
parameter correlations become,

$\beta=\frac{2}{7}\delta,\quad\gamma=12-\delta.$

Since $\delta$ is known we can identify $\gamma$ and $\beta.$

###### Proposition 3.7

If $\delta$ is fixed, then the model M4 reveals all its parameters uniquely
from the observation of the number of newly infected cases.

With a similar analysis, one can see that if $\gamma$ is fixed, then the model
M4 becomes structurally identifiable from the number of new cases.

DAISY did not produce any results for the case when all the initial conditions
are known as this process was computationally too expensive for DAISY to
derive an identifiability, likely as a result of the long characteristic set.

Figure 4: Flow chart of model M4 showing the transition of individuals between
the classes defined by disease status. Observations are the newly infected
individuals. Structural identifiability analysis is summarized for when
initial conditions are known and unknown.$S$$E$$I$$R$$D$ Observations are new
cases; $y_{1}(t)=kE(t)$ $\displaystyle\frac{\beta SI}{N}$$kE$
$\gamma I$
$\delta I$
Identifiability Results Without Initial Conditions:
identifiable parameters: $k$ unidentifiable parameters: $\beta,\gamma,\delta$
paramter correlations
$\displaystyle\frac{\beta}{\delta}=\displaystyle\frac{\hat{\beta}}{\hat{\delta}}\quad\gamma+\delta=\hat{\gamma}+\hat{\delta}$

(ii) Number of new infected cases and deaths are observed. Now, we study the
case when 2 different observations are possible for the same epidemic: The
number of new cases $y_{1}(t)=kE(t)$, and the number of new deaths
$y_{2}(t)=\delta I(t).$ We obtain the following set of input-output equations
in DAISY,

$\displaystyle 0$ $\displaystyle=y_{2}^{\prime}-\delta
y_{1}+y_{2}(\delta+\gamma)$ (9) $\displaystyle 0$
$\displaystyle=y_{1}^{\prime\prime\prime}y_{1}^{\prime}y_{2}^{2}+y_{1}^{\prime\prime\prime}y_{1}y_{2}^{2}k+(y_{1}^{\prime\prime})^{2}y_{2}^{2}\frac{(-\beta+2\delta)}{(\beta-\delta)}-y_{1}^{\prime\prime}y_{1}^{\prime}y_{1}y_{2}\frac{\delta(\beta+\delta)}{(\beta-\delta)}+$
$\displaystyle
y_{1}^{\prime\prime}y_{1}^{\prime}y_{2}^{2}\frac{(\beta\delta+\beta\gamma-\beta
k+\delta^{2}+\delta\gamma+3\delta
k)}{(\beta-\delta)}-y_{1}^{\prime\prime}y_{1}^{2}y_{2}\frac{\delta
k(\beta+\delta)}{(\beta-\delta)}+$ $\displaystyle
y_{1}^{\prime\prime}y_{1}y_{2}^{2}\frac{k(\beta\delta+\beta\gamma+\beta
k+\delta^{2}+\delta\gamma-\delta k)}{(\beta-\delta)}-y_{1}^{\prime
3}y_{2}\delta+y_{1}^{\prime
2}y_{1}^{2}\frac{\delta^{2}(2\beta-\delta)}{(\beta-\delta)}+$ $\displaystyle
y_{1}^{\prime 2}y_{1}y_{2}\frac{\delta(-3\beta\delta-3\beta\gamma-3\beta
k+\delta^{2}+\delta\gamma+\delta k)}{(\beta-\delta)}+$ $\displaystyle
y_{1}^{\prime 2}y_{2}^{2}\frac{(\beta\delta^{2}+2\beta\delta\gamma+\beta\delta
k+\beta\gamma^{2}+\beta\gamma k-\beta k^{2}+\delta^{2}k+\delta\gamma k+2\delta
k^{2})}{(\beta-\delta)}+$ $\displaystyle
2y_{1}^{\prime}y_{1}^{3}\frac{\delta^{2}k(2\beta-\delta)}{(\beta-\delta)}+2y_{1}^{\prime}y_{1}^{2}y_{2}\frac{\delta(-3\beta\delta-3\beta\gamma-\beta
k+\delta^{2}+\delta\gamma)}{(\beta-\delta)}+$ $\displaystyle
y_{1}^{\prime}y_{1}y_{2}^{2}\frac{k(2\beta\delta^{2}+4\beta\delta\gamma+\beta\delta
k+2\beta\gamma^{2}+\beta\gamma k+\delta^{2}k+\delta\gamma
k)}{(\beta-\delta)}+y_{1}^{4}\frac{\delta^{2}k^{2}(2\beta-\delta)}{(\beta-\delta)}+$
$\displaystyle y_{1}^{3}y_{2}\frac{\delta
k^{2}(-3\beta\delta-3\beta\gamma+\delta^{2}+\delta\gamma)}{(\beta-\delta)}+$
$\displaystyle y_{1}^{2}y_{2}^{2}\frac{\beta
k^{2}(\delta^{2}+2\delta\gamma+\gamma^{2})}{(\beta-\delta)}$

Since we have two sets of observations, we obtain two monic differential
polynomials involving both observations. Thus, the input-output equations (9)
of the model M4 involves both $y_{1}$ and $y_{2}$ together with the model
parameters. Following the definition of structural identifiability, we suppose
the parameter set
$\hat{\mbox{\boldmath$p$}}=(\hat{k},\hat{\beta},\hat{\gamma},\hat{\delta})$
produces the same observations $y_{1}$ and $y_{2}$. Then setting
$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$, we obtain the following
set of solutions,

$\left\\{k=\hat{k},\quad\beta=\hat{\beta},\quad\gamma=\hat{\gamma},\quad\delta=\hat{\delta}\right\\}$

This implies that the model M4 is structurally identifiable from the
observations of new cases $y_{1}(t)=kE(t)$, and new deaths $y_{2}(t)=\delta
I(t).$ Note that this result does not consider the initial conditions. Thus,
the model M4 is structurally identifiable whether the initial conditions are
known or not. We state the following Proposition 3.8:

###### Proposition 3.8

The model M4 is structurally identifiable from the observation of newly
infected cases, $y_{1}(t)=kE(t)$ and deaths, $y_{2}(t)=\delta I(t)$).

In the previous analysis, when only the new cases are observed, we could not
identify all of the parameters. But since we have now additional observations
corresponding to the number of new deaths, the model becomes structurally
identifiable. A structurally unidentifiable model could become identifiable
either by knowing the parameter correlations and fixing the appropriate
parameters or by adding more observations.

Figure 5: Flow chart of model M4 showing the transition of individuals between
the classes defined by disease status. Observations are the newly infected
cases and deaths.$S$$E$$I$$R$$D$ Observations are new deaths; $y_{2}(t)=\delta
I(t)$ Observations are new cases; $y_{1}(t)=kE(t)$ $\displaystyle\frac{\beta
SI}{N}$$kE$
$\gamma I$
$\delta I$
Identifiability Results:
identifiable parameters: $k,\gamma,\beta,\delta$

###  3.4 Epidemic Model M5

$\textbf{Model 5
(M${}_{5}$):}\begin{cases}\displaystyle\frac{dS_{\upsilon}}{dt}=\Lambda_{\upsilon}-\frac{\beta
S_{\upsilon}I}{N}-\mu_{\upsilon}S_{\upsilon}\quad S_{\upsilon}(0)=S_{\upsilon
0},\\\\[6.45831pt] \displaystyle\frac{dI_{\upsilon}}{dt}=\frac{\beta
S_{\upsilon}I}{N}-\mu_{\upsilon}I_{\upsilon}\quad I_{\upsilon}(0)=I_{\upsilon
0},\\\\[6.45831pt]
\displaystyle\frac{dS}{dt}=-\frac{\beta_{\upsilon}SI_{\upsilon}}{N}\quad
S(0)=S_{0},\\\\[6.45831pt]
\displaystyle\frac{dI}{dt}=\frac{\beta_{\upsilon}SI_{\upsilon}}{N}-\gamma
I\quad I(0)=I_{0},\\\\[6.45831pt] \displaystyle\frac{dR}{dt}=\gamma I\quad
R(0)=R_{0}\end{cases}$ (M5)

We continue identifiability analysis of epidemic models with vector-born
diseases. The next model M5 is a simple vector-borne model equipped with five
epidemiological states over time: the number of susceptible mosquitoes
($S_{\upsilon}(t)$), the number of infected mosquitoes ($I_{\upsilon}(t)$),
the number of susceptible individuals ($S(t)$), the number of infected
individuals ($I(t)$), and the number of recovered individuals ($R(t)$). The
mosquito birth and death rates are given by ${\Lambda}_{\upsilon}$ and
${\mu}_{\upsilon}$, respectively. $\beta$ is the transmission rate from an
infected individual to a susceptible mosquito, and ${\beta_{\upsilon}}$ is the
transmission rate from an infected mosquito to a susceptible individual. Like
before, an infected individual recovers at rate ${\gamma}$. In this model, the
total human population $N$ remains the same throughout the epidemic, hence $N$
= $S_{0}$ +$I_{0}$ +$R_{0}$. The full parameter set of the model is given by
$\mbox{\boldmath$p$}=\left(\Lambda_{\upsilon},\mu_{\upsilon},\beta_{\upsilon},N,\beta,\gamma\right)$.

Observations are the cumulative number of new infections:
First, we used the number of new cases as the observations for the vector-
borne model M5. But, DAISY could not finish the computations to produce the
input-output equations. Hence, we tried changing the observations to the
cumulative number of new cases, denoted by $C(t)$:

$y_{1}(t)=C(t)=\int_{0}^{t}\frac{\beta_{\upsilon}SI_{\upsilon}}{N}=S(0)-S(t)$
(10)

So, if we know the initial number of susceptibles, $S_{0}$, the cumulative
number of infections is equivalent to observing $S(t)$. Here we investigate
whether the epidemic model M5 can reveal its epidemiological parameters from
the cumulative number of new infections.

We obtained the input-output equations using DAISY. The results and the code
are available in files Model5$\\_$inputcode and Model5$\\_$results in github.
DAISY assigns the following integers to the model parameters,

$\hat{\mbox{\boldmath$p$}}=\left\\{\hat{\Lambda}_{\upsilon}=2,\;\hat{\beta}=3,\;\hat{\mu}_{\upsilon}=5,\;\hat{\beta}_{\upsilon}=7,\hat{\gamma}=11,\hat{N}=13\right\\}$

and then solves $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ and
gives the following correlations between the parameters.

$\displaystyle\frac{\beta_{\upsilon}\Lambda_{\upsilon}}{N}=\displaystyle\frac{14}{13}=\displaystyle\frac{\hat{\beta}_{\upsilon}\hat{\Lambda}_{\upsilon}}{\hat{N}},\quad\displaystyle\frac{\beta}{N}=\displaystyle\frac{3}{13}=\displaystyle\frac{\hat{\beta}}{\hat{N}},\quad\gamma=11=\hat{\gamma},\quad\text{and}\quad\mu_{\upsilon}=5=\hat{\mu}_{\upsilon}.$
(11)

As seen from the DAISY result (11), the parameters $\gamma$ and
$\mu_{\upsilon}$ are identifiable, and the parameters
$\beta,\Lambda_{\upsilon},N$ and $\beta_{\upsilon}$ are not identifiable. But
we can only identify the ratios,

$\displaystyle\frac{\beta_{\upsilon}\Lambda_{\upsilon}}{N}\quad\text{and}\quad\displaystyle\frac{\beta}{N}$

We state the first identifiability result regarding the vector-borne model M5
in the following Proposition 3.9.

###### Proposition 3.9

The model M5 is not structurally identifiable from the observation of
cumulative incidences. Only the parameters $\gamma,\mu_{\upsilon}$ and the
parameters correlations
$\displaystyle\frac{\beta_{\upsilon}\Lambda_{\upsilon}}{N}$ and
$\displaystyle\frac{\beta}{N}$ are identifiable.

One of the advantages of the differential algebra approach for structural
identifiability analysis is that we can use the parameter correlations to
derive a structurally identifiable model. As seen in (11), $\beta_{\upsilon}$
can only be identified if $N$ and $\Lambda_{\upsilon}$ are known. Moreover,
$\beta$ can only be identified when $N$ is known.

###### Proposition 3.10

When the total human population $N$ and the recruitment rate
$\Lambda_{\upsilon}$ of the vector population are known, the model M5 becomes
structurally identifiable from the observation of cumulative incidences.

Next we carry out the structural identifiability analysis of the vector-borne
model M5 when all the initial conditions are known. With known initial
conditions DAISY solves $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$
together with the characteristic set evaluated at time $t=0$ and provides the
following solution

$\Lambda_{\upsilon}=2,\;\displaystyle\frac{\beta}{N}=\frac{3}{13},\;\displaystyle\frac{\beta_{\upsilon}}{N}=\frac{7}{13},\;\gamma=11,\;\text{and}\;\mu_{\upsilon}=5.$

Since, $N$ = $S_{0}$ +$I_{0}$ +$R_{0}$, when all the initial conditions are
known, $N$ will be identifiable, therefore also $\beta$ and
$\beta_{\upsilon}$. We conclude that the model is structurally identifiable
when the initial conditions are known, and state the result in the following
proposition.

###### Proposition 3.11

The model M5 is structurally identifiable from the observation of cumulative
cases when all the initial conditions are known.

Figure 6: Flow chart of model M5 showing the transition of individuals between
the classes defined by disease status. Observations are the cumulative number
of new cases. Structural identifiability analysis is summarized for when
initial conditions are known and unknown.$S_{\upsilon}$$I_{\upsilon}$$S$$I$$R$
Observations are cumulative cases; $y_{1}(t)=S_{0}-S(t)$
$\displaystyle\frac{\beta
S_{\upsilon}I}{N}$$\displaystyle\frac{\beta_{\upsilon}SI_{\upsilon}}{N}$$\gamma
I$
$\Lambda_{\upsilon}$$\mu_{\upsilon}$$\mu_{\upsilon}$ Identifiability Results:
Without Initial Conditions
identifiable parameters: $\mu_{\upsilon},\gamma$ unidentifiable parameters:
$\beta,\beta_{\upsilon},\Lambda_{\upsilon},N$ parameter correlations
$\displaystyle\frac{\beta}{N}=\displaystyle\frac{\hat{\beta}}{\hat{N}}$ and
$\displaystyle\frac{\beta_{\upsilon}\Lambda_{\upsilon}}{N}=\displaystyle\frac{\hat{\beta}_{\upsilon}\hat{\Lambda}_{\upsilon}}{\hat{N}}$
Identifiability Results:
With Initial Conditions
identifiable parameters:
$\beta,\beta_{\upsilon},\Lambda_{\upsilon},\mu_{\upsilon},\gamma,N$

###  3.5 Epidemic Model M6

$\textbf{Model 6
(M${}_{6}$):}\begin{cases}\displaystyle\frac{dS_{\upsilon}}{dt}=\Lambda_{\upsilon}-\frac{\beta_{I}S_{\upsilon}I+\beta_{A}S_{\upsilon}A}{N}-\mu_{\upsilon}S_{\upsilon},\\\\[6.45831pt]
\displaystyle\frac{dI_{\upsilon}}{dt}=\frac{\beta_{I}S_{\upsilon}I+\beta_{A}S_{\upsilon}A}{N}-\mu_{\upsilon}I_{\upsilon},\\\\[6.45831pt]
\displaystyle\frac{dS}{dt}=-\frac{\beta_{\upsilon}I_{\upsilon}S}{N},\\\\[6.45831pt]
\displaystyle\frac{dI}{dt}=\frac{\rho\beta_{\upsilon}I_{\upsilon}S}{N}-\gamma
I,\\\\[6.45831pt]
\displaystyle\frac{dA}{dt}=\frac{(1-\rho)\beta_{\upsilon}I_{\upsilon}S}{N}-\gamma
A,\\\\[6.45831pt] \displaystyle\frac{dR}{dt}=\gamma I+\gamma A\par\end{cases}$
(M6)

We proceed to conduct the structural identifiability analysis of the vector-
borne model in M6, which incorporates an asymptomatic class to the vector-
borne model M5. Therefore, this vector-borne model M6 has six epidemiological
states: the number of susceptible mosquitoes ($S_{\upsilon}(t)$), the number
of infected mosquitoes ($I_{\upsilon}(t)$), the number of susceptible
individuals ($S(t)$), the number of infected individuals ($I(t)$), the number
of asymptomatic individuals ($A(t)$) and the number of recovered individuals
($R(t)$). As before, the birth and death rates of mosquito population are
given by ${\Lambda}_{\upsilon}$ and ${\mu}_{\upsilon}$ respectively. Likewise,
${\gamma}$ is the recovery rate in human population. Since there are two
infectious classes, there are two routes of transmission. The first route is
when a susceptible mosquito bites a symptomatic infectious individual, and the
other route is when a susceptible mosquito bites an asymptomatic infectious
individual. We model the new incidences stemming in the mosquito population
with the following term,

$\displaystyle\frac{\beta_{I}S_{\upsilon}I+\beta_{A}S_{\upsilon}A}{N}$

${\beta_{I}}$ is the transmission rate from a symptomatic individual and
${\beta_{A}}$ is the transmission rate from an asymptomatic individual to a
susceptible mosquito. A susceptible individual becomes infected at rate
${\beta_{\upsilon}}$ when bitten by an infectious mosquito. The ${\rho}$
fraction of new infections show symptoms and move to the symptomatic class,
and the remaining fraction $1-{\rho}$ of the new infections move to the
asymptomatic class. Clearly $\rho\leq 1$.

The initial conditions of the model are given by $S_{\upsilon}(0)=S_{\upsilon
0}$, $I_{\upsilon}(0)=I_{\upsilon 0}$, $S(0)=S_{0}$, $I(0)=I_{0}$,
$A(0)=A_{0}$, and $R(0)=R_{0}$. The total human population, N remains the same
throughout the epidemic, hence $N=S_{0}+I_{0}+A_{0}+R_{0}$. The full parameter
set of the model M6 that we would like to identify given the observations is
$\mbox{\boldmath$p$}=\left(N,\Lambda_{\upsilon},\mu_{\upsilon},\beta_{\upsilon},\beta_{A},\beta_{I},\gamma,\rho\right)$.

Observations are the cumulative number of new infections: We again take the
cumulative number of incidences as the observations for this vector-borne
model M6.

$y_{1}(t)=C(t)=\rho\int_{0}^{t}\frac{SI_{\upsilon}\beta_{\upsilon}}{N}=\rho(S(0)-S(t)).$
(12)

As seen in (12), the cumulative incidence is equivalent to $\rho(S(0)-S(t))$.
Here we investigate whether the epidemic model M6 can reveal its
epidemiological parameters,
$\mbox{\boldmath$p$}=\left(N,\Lambda_{\upsilon},\mu_{\upsilon},\beta_{\upsilon},\beta_{A},\beta_{I},\gamma,\rho\right)$,
from the observation of the cumulative number of new cases.

DAISY first assigns the following integers for the model parameters (see Code
5),

$\hat{\mbox{\boldmath$p$}}=\left\\{\hat{\beta}_{I}=2,\;\hat{\beta}_{A}=3,\;\hat{\mu}_{\upsilon}=5,\;\hat{\beta}_{\upsilon}=7,\;\hat{\rho}=11,\;\hat{\gamma}=13,\;\hat{\Lambda}_{\upsilon}=17,\;\hat{N}=19\right\\}$

and then solves $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ and
gives the following results:

    PARAMETER VALUES$

    b2_ := {betai=2,betaa=3,muv=5,betav=7,rho1=11,gamma1=13,lambdaupsilon=17,n=19}$

    MODEL PARAMETER SOLUTION(S)$

    g_ := {{betaa=(209*betai + 8*n)/190,
    betav=(119*n)/(19*lambdaupsilon),
    gamma1=13,
    muv=5,
    rho1=11}}$

    MODEL NON IDENTIFIABLE$

    IDENTIFIABILITY WITH ALL INITIAL CONDITIONS (IC_)$

    ic_ := {x1=x10,x2=x20,x3=x30,x4=x40,x5=x50,x6=x60}$

    MODEL PARAMETER SOLUTIONS$

    gi_ := {{lambdaupsilon=17,
    betai=(2*n)/19,
    betaa=(3*n)/19,
    betav=(7*n)/19,
    gamma1=13,
    muv=5,
    rho1=11}}$

    MODEL NON IDENTIFIABLE$

Code 5:  DAISY identifiability result for model M6.

$209\beta_{I}+8N=190\beta_{A},\quad\displaystyle\frac{\beta_{\upsilon}\Lambda_{\upsilon}}{N}=\frac{119}{19},\quad\rho=11,\quad\gamma=13,\quad\mu_{\upsilon}=5$
(13)

Here, we would like to point out that the first correlation in (13),
$209\beta_{I}+8N=190\beta_{A}$, is the same correlation (8) as we observed in
model M3 with an asymptomatic class. Namely,

$\displaystyle\frac{\beta_{I}\rho+\beta_{A}(1-\rho)}{N\rho}=\displaystyle\frac{\hat{\beta}_{I}\hat{\rho}+\hat{\beta}_{A}(1-\hat{\rho})}{\hat{N}\hat{\rho}}$

with now $\rho=\hat{\rho}=11.$ (DAISY input code and results available in file
named Model6$\\_$inputcode and Model5$\\_$results in github) We see from the
DAISY result (13) that the parameters $\gamma$ and $\mu_{\upsilon}$ are
identifiable, and there exist the following correlations involving other
parameters:

$\displaystyle\frac{\beta_{\upsilon}\Lambda_{\upsilon}}{N}=\displaystyle\frac{\hat{\beta}_{\upsilon}\hat{\Lambda}_{\upsilon}}{\hat{N}}\quad\text{and}\quad\displaystyle\frac{\beta_{I}\hat{\rho}+\beta_{A}(1-\hat{\rho})}{N\hat{\rho}}=\displaystyle\frac{\hat{\beta}_{I}\hat{\rho}+\hat{\beta}_{A}(1-\hat{\rho})}{\hat{N}\hat{\rho}}$

This means, $\beta_{\upsilon}$ can be identified only if $N$ and
$\Lambda_{\upsilon}$ are known and $\beta_{I}$ can be identified only when
$\beta_{A}$, $N$ and $\rho$ are known.

###### Proposition 3.12

The model M6 is not structurally identifiable from the observation of
cumulative number of new infections.

We continue with the structural identifiability analysis when all the initial
conditions are known. DAISY solves
$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ together with all the
characteristic set evaluated at time $t=0$ gives the following solutions

$\Lambda_{\upsilon}=17=\hat{\Lambda}_{\upsilon},\quad\gamma=13=\hat{\gamma},\quad\rho=11=\hat{\rho},\quad\text{and}\quad\mu_{\upsilon}=5=\hat{\mu}_{\upsilon}$

together with the following correlations

$\displaystyle\frac{\beta_{\upsilon}}{N}=\frac{7}{19}=\displaystyle\frac{\hat{\beta}_{\upsilon}}{\hat{N}},\quad\displaystyle\frac{\beta_{A}}{N}=\frac{3}{19}=\displaystyle\frac{\hat{\beta}_{A}}{\hat{N}},\quad\displaystyle\frac{\beta_{I}}{N}=\frac{2}{19}=\displaystyle\frac{\hat{\beta}_{I}}{\hat{N}},$

.

This means then when we fix the total population size, $N$ the parameters
$\beta_{A}$, $\beta_{\upsilon}$ and $\beta_{I}$ will be identified. Since,
$N=S_{0}+I_{0}+A_{0}+R_{0}$, we conclude that the model is structurally
identifiable when the initial conditions are known.

###### Proposition 3.13

The model M6 is structurally identifiable from the observation of cumulative
cases when all the initial conditions are known.

Figure 7: Flow chart of model M6 showing the transition of individuals between
the classes defined by disease status. Observations are the cumulative number
of new cases. Structural identifiability analysis is summarized for the
scenario when the initial conditions are known and
unknown.$S_{\upsilon}$$I_{\upsilon}$$S$$I$$A$$R$ Observations are cumulative
cases; $y_{1}(t)=\rho(S_{0}-S(t))$
$\displaystyle\frac{\beta_{I}S_{\upsilon}I+\beta_{A}S_{\upsilon}A}{N}$$\rho\displaystyle\frac{\beta_{\upsilon}SI_{\upsilon}}{N}$$(1-\rho)\displaystyle\frac{\beta_{\upsilon}SI_{\upsilon}}{N}$$\gamma
I$
$\gamma A$
$\Lambda_{\upsilon}$$\mu_{\upsilon}$$\mu_{\upsilon}$ Identifiability Results:
Without Initial Conditions
identifiable parameters: $\mu_{\upsilon},\gamma,\rho$ unidentifiable
parameters: $\beta_{A},\beta_{I},\beta_{\upsilon},\Lambda_{\upsilon},N$
parameter correlations
$\displaystyle\frac{\beta_{\upsilon}\Lambda_{\upsilon}}{N}=\displaystyle\frac{\hat{\beta}_{\upsilon}\hat{\Lambda}_{\upsilon}}{\hat{N}}$,
$\displaystyle\frac{\beta_{I}\hat{\rho}+\beta_{A}(1-\hat{\rho})}{N\hat{\rho}}=\displaystyle\frac{\hat{\beta}_{I}\hat{\rho}+\hat{\beta}_{A}(1-\hat{\rho})}{\hat{N}\hat{\rho}}$
Identifiability Results:
With Initial Conditions
identifiable parameters:
$\beta_{A},\beta_{I}\beta_{\upsilon},\Lambda_{\upsilon},\mu_{\upsilon},\gamma,\rho,N$

###  3.6 Epidemic Model M7

$\textbf{Model 7
(M${}_{7}$):}\begin{cases}\displaystyle\frac{dS}{dt}=\displaystyle\frac{-(\beta_{I}I+\beta_{H}H+\beta_{D}D)S}{N},\\\\[6.45831pt]
\displaystyle\frac{dE}{dt}=\displaystyle\frac{(\beta_{I}I+\beta_{H}H+\beta_{D}D)S}{N}-kE,\\\\[6.45831pt]
\displaystyle\frac{dI}{dt}=kE-(\alpha+\gamma_{I}+\delta_{I})I,\\\\[6.45831pt]
\displaystyle\frac{dH}{dt}=\alpha I-(\gamma_{H}+\delta_{H})H,\\\\[6.45831pt]
\displaystyle\frac{dR}{dt}=\gamma_{I}I+\gamma_{H}H,\\\\[6.45831pt]
\displaystyle\frac{dD}{dt}=\delta_{I}I+\delta_{H}H.\\\\[6.45831pt]
\end{cases}$ (M7)

This epidemic model is motivated by the transmission dynamics of Ebola where
transmission may be amplified in health-care settings and during funerals of
Ebola deceased individuals [27]. It keeps track of six epidemiological states:
The number of susceptible individuals $(S(t))$, the number of latent
individuals $(E(t))$, the number of infected individuals in the community
$(I(t))$, the number of hospitalized individuals $(H(t))$, the number of
recovered individuals $(R(t))$ and the number of disease-induced deaths
$(D(t)).$ Here, susceptible individuals can become exposed to Ebola virus by
close contact with infected individuals in the community at the rate
$\beta_{I}$, with hospitalized individuals at the rate $\beta_{H}$ or Ebola-
deceased individuals at rate $\beta_{D}$. An exposed individual becomes
infectious with Ebola after an average latent period of $k$ days. Infectious
individuals in the community recover at a rate $\gamma_{I}$ or die from the
disease at a rate $\delta_{I}$. A hospitalized individual can either recover
at rate $\gamma_{H}$ or die at a rate $\delta_{H}.$ For this model, we would
like to identify the parameters
$\mbox{\boldmath$p$}=(\beta_{I},\beta_{H},\beta_{D},k,\alpha,\delta_{I},\delta_{H},\gamma_{H},\gamma_{I})$
given the observations.

Number of new infected cases are observed: We will first examine the case when
the observed data corresponds to the number of new infections, that is
$y_{1}(t)=kE(t).$ Using DAISY (DAISY input code and results available in file
named Model7a-inputcode and Model7a-results), we obtain the following input-
output equation.

$0=y_{1}^{\prime\prime}+y_{1}^{\prime}\frac{(\alpha\beta_{H}+\beta_{D}\delta_{I}+\beta_{I}\delta_{H}+\beta_{I}\gamma_{H})}{\beta_{I}}+y_{1}\frac{\beta_{D}}{\beta_{I}}(\alpha\delta_{H}+\delta_{H}\delta_{I}+\delta_{I}\gamma_{H})$
(14)

We note that parameters $k$ and $\gamma_{I}$ do not appear in the the input-
output equation (14). Hence, these parameters cannot be identified from the
observations of new Ebola cases alone. Following the definition of structural
identifiability, we first set
$c(\mbox{\boldmath$p$})=c(\mbox{\boldmath$\hat{p}$})$, that is

$\displaystyle\frac{(\alpha\beta_{H}+\beta_{D}\delta_{I}+\beta_{I}\delta_{H}+\beta_{I}\gamma_{H})}{\beta_{I}}=\displaystyle\frac{(\hat{\alpha}\hat{\beta}_{H}+\hat{\beta}_{D}\hat{\delta}_{I}+\hat{\beta}_{I}\hat{\delta}_{H}+\hat{\beta}_{I}\hat{\gamma}_{H})}{\hat{\beta}_{I}}$
(15)

and

$\displaystyle\frac{\beta_{D}}{\beta_{I}}(\alpha\delta_{H}+\delta_{H}\delta_{I}+\delta_{I}\gamma_{H})=\displaystyle\frac{\hat{\beta}_{D}}{\hat{\beta}_{I}}(\hat{\alpha}\hat{\delta}_{H}+\hat{\delta}_{H}\hat{\delta}_{I}+\hat{\delta_{I}}\hat{\gamma}_{H})$
(16)

Solving $c(\mbox{\boldmath$p$})=c(\mbox{\boldmath$\hat{p}$})$ for the model
parameters, we obtain the same correlations (15) and (16). Therefore, we
conclude that the model (M7) is not structured to reveal all its parameters
from the number of newly infected cases alone.

###### Proposition 3.14

The model M7 is not structurally identifiable from the observations of newly
infected cases.

We note that DAISY does not yield results when all initial conditions are
known.

Figure 8: Flow chart of model M7 showing the transition of individuals between
the classes defined disease status. Observations are new infections.
Structural identifiability analysis is summarized for the scenario when
initial conditions are known and unknown.$S$$E$$I$$R$$H$$D$ Observations are
new infections; $y_{1}(t)=kE$
$\frac{S(\beta_{I}I+\beta_{H}H+\beta_{D}D)}{N}$$kE$
$\gamma_{I}I$
$\alpha$
$\delta_{I}I$
$\delta_{H}H$
$\gamma_{H}H$
Identifiability Results:
Without Initial Conditions
identifiable parameters: None unidentifiable parameters:
$\beta_{I},\beta_{H},\beta_{D},k,\alpha,\delta_{I},\delta_{H},\gamma_{H},\gamma_{I}$
parameter correlations: See equation (15) and (16) Identifiability Results:
With Initial Conditions
identifiable parameters: DAISY did not give any results for this case.

Number of new infections and hospitalizations are observed:
Since the model parameters are not identifiable when the number of new
infections are known, we continue our analysis of model M7 when data on the
number of new hospitalizations are also available. That is, we observe both
the number of new infections $y_{1}(t)=kE(t)$ and new hospitalizations
$y_{2}(t)=\alpha I(t).$ In this scenario, DAISY yields the following input-
output equations (DAISY input code and results available in file named
Model7b-inputcode and Model7b-results).

$\displaystyle 0$
$\displaystyle=y_{1}^{\prime}-\frac{y_{1}}{\beta_{I}}(-\alpha\beta_{H}+\alpha\beta_{I}-\beta_{D}\delta_{I}-\beta_{I}\delta_{H}+\beta_{I}\delta_{I}-\beta_{I}\gamma_{H}+\beta_{I}\gamma_{I})-\frac{y_{2}}{\alpha\beta_{I}}(\alpha^{2}\beta_{H}-$
$\displaystyle\alpha^{2}\beta_{I}-\alpha\beta_{D}\delta_{H}+\alpha\beta_{D}\delta_{I}+\alpha\beta_{H}\delta_{I}+\alpha\beta_{H}\gamma_{I}+\alpha\beta_{I}\delta_{H}-2\alpha\beta_{I}\delta_{I}+\alpha\beta_{I}\gamma_{H}-2\alpha\beta_{I}\gamma_{I}-$
$\displaystyle\beta_{D}\delta_{H}\delta_{I}+\beta_{D}\delta_{I}^{2}-\beta_{D}\delta_{I}\gamma_{H}+\beta_{D}\delta_{I}\gamma_{I}+\beta_{I}\delta_{H}\delta_{I}+\beta_{I}\delta_{H}\gamma_{I}-\beta_{I}\delta_{I}^{2}+\beta_{I}\delta_{I}\gamma_{H}-2\beta_{I}\delta_{I}\gamma_{I}+$
$\displaystyle\beta_{I}\gamma_{H}\gamma_{I}-\beta_{I}\gamma_{I}^{2})$
$\displaystyle 0$
$\displaystyle=y_{2}^{\prime}-y_{1}\alpha+y_{2}(\alpha+\delta_{I}+\gamma_{I})$

We solve $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ and obtain

$\displaystyle\\{\alpha$
$\displaystyle=\hat{\alpha},\quad\gamma_{I}+\delta_{I}=+\hat{\delta}_{I}+\hat{\gamma}_{I},\quad\beta_{H}=\frac{1}{\hat{\alpha}\beta_{D}\hat{\beta_{I}}\delta_{I}}(\hat{\alpha}\beta_{D}\beta_{I}\hat{\beta_{I}}\delta_{H}-\hat{\alpha}\hat{\beta}_{D}^{2}\hat{\delta}_{H}+\hat{\alpha}\beta_{D}\hat{\beta}_{H}\beta_{I}\delta_{I}+$
(17)
$\displaystyle\beta_{D}\beta_{I}\hat{\beta_{I}}\hat{\delta}_{H}\delta_{I}-\beta_{D}^{2}\hat{\beta_{I}}\delta_{I}^{2}-\hat{\beta}_{D}\beta_{I}^{2}\hat{\delta}_{H}\hat{\delta_{I}}+\beta_{D}\hat{\beta}_{D}\beta_{I}\delta_{I}\hat{\delta_{I}}+\beta_{D}\beta_{I}\hat{\beta_{I}}\delta_{I}\hat{\gamma}_{H}-\hat{\beta_{D}}\beta_{I}^{2}\hat{\delta_{I}}\hat{\gamma}_{H}),\quad$
$\displaystyle\gamma_{H}=\frac{1}{\beta_{D}\hat{\beta_{I}}\delta_{I}}(-\hat{\alpha}\beta_{D}\hat{\beta_{I}}\delta_{H}+\hat{\alpha}\hat{\beta}_{D}\beta_{I}\hat{\delta}_{H}-\beta_{D}\hat{\beta_{I}}\delta_{H}\delta_{I}+\hat{\beta}_{D}\beta_{I}\hat{\delta}_{H}\hat{\delta_{I}}+\hat{\beta}_{D}\beta_{I}\hat{\delta_{I}}\hat{\gamma}_{H})\\}$

As seen from (17), we can only identify the parameter $\alpha$ and the sum
$\gamma_{I}+\delta_{I}$, while the rest of the parameters are involved in
complex correlations. Furthermore, the parameter $k$ does not appear in the
input output equations. We conclude that the model M7 is not identifiable from
the observations of newly infected cases and hospitalizations.

###### Proposition 3.15

The model M7 is not structurally identifiable from the number of newly
infected cases and hospitalizations.

Figure 9: Flow chart of model M7 showing the transition of individuals between
the classes defined disease status. Observations are the newly infected cases
and new hospitalizations . Structural identifiability analysis is summarized
for the scenario when initial conditions are known and
unknown.$S$$E$$I$$R$$H$$D$ Observations are new infections; $y_{1}(t)=kE(t)$
Observations are new hospitalizations; $y_{2}(t)=\alpha I$
$\frac{S(\beta_{I}I+\beta_{H}H+\beta_{D}D)}{N}$$kE$
$\gamma_{I}I$
$\alpha I$
$\delta_{I}I$
$\delta_{H}H$
$\gamma_{H}H$
Identifiability Results:
Without Initial Conditions
identifiable parameters: $\alpha$ unidentifiable parameters:
$k$,$\beta_{I}$,$\beta_{H}$,$\beta_{D}$,$k$,$\alpha$,$\delta_{I}$,$\delta_{H}$,$\gamma_{H}$,$\gamma_{I}$
parameter correlations: see (17) Identifiability Results:
With Initial Conditions
identifiable parameters: DAISY did not give any results for this case.

Number of new infections, hospitalizations and deaths are
observed:
We now consider the scenario when the number of new deaths from the disease
are available in addition to the number of new cases and hospitalizations.
Hence, here we observe three datasets: the number of newly infected cases
$y_{1}(t)=kE(t)$, the number of new hospitalizations $y_{2}(t)=\alpha I(t)$,
and the number of new deaths $y_{3}(t)=\delta_{I}I(t)+\delta_{H}H(t)$. Using
DAISY, we obtain the following input-output equations (DAISY input code and
results available in file named Model7c-inputcode and Model7c-results):

$\displaystyle 0$
$\displaystyle=y_{2}^{\prime}-y_{1}\alpha+y_{2}(\alpha+\delta_{I}+\gamma_{I})$
(18) $\displaystyle 0$
$\displaystyle=y_{3}^{\prime}-y_{1}\delta_{I}+\frac{y_{2}}{\alpha\beta_{H}}(\alpha\beta_{H}\delta_{I}-\beta_{D}\delta_{H}\delta_{I}-\beta_{H}\delta_{H}\delta_{I}+\beta_{H}\delta_{I}^{2}+\beta_{H}\delta_{I}\gamma_{I}+\beta_{I}\delta_{H}^{2})+$
$\displaystyle y_{3}\frac{\beta_{D}\delta_{H}}{\beta_{H}}$ $\displaystyle 0$
$\displaystyle=y_{1}^{\prime}-\frac{y_{3}^{\prime\prime}}{\delta_{I}}+y_{1}(\alpha-\delta_{H}+\delta_{I}+\gamma_{I})-\frac{y_{2}}{\alpha\beta_{H}\delta_{I}}(-\alpha^{2}\beta_{H}\delta_{I}-\alpha\beta_{H}\delta_{H}^{2}+\alpha\beta_{H}\delta_{H}\delta_{I}-$
$\displaystyle
2\alpha\beta_{H}\delta_{I}^{2}-2\alpha\beta_{H}\delta_{I}\gamma_{I}+\beta_{D}\delta_{H}^{2}\delta_{I}+\beta_{D}\delta_{H}\delta_{I}\gamma_{H}+\beta_{H}\delta_{H}\delta_{I}^{2}+\beta_{H}\delta_{H}\delta_{I}\gamma_{I}-\beta_{H}\delta_{I}^{3}-$
$\displaystyle
2\beta_{H}\delta_{I}^{2}\gamma_{I}-\beta_{H}\delta_{I}\gamma_{I}^{2}-\beta_{I}\delta_{H}^{3}-\beta_{I}\delta_{H}^{2}\gamma_{H})+y_{3}\frac{\beta_{D}\delta_{H}}{\beta_{H}\delta_{I}}(\delta_{H}+\gamma_{H})$

As before, we first set $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$,
and solve for the model parameters and obtain the following parameter
correlations,

$\\{\alpha=\hat{\alpha},\quad\delta_{I}=\hat{\delta}_{I},\quad\gamma_{I}=\hat{\gamma}_{I},\quad\delta_{H}=\hat{\delta}_{H},\quad\gamma_{H}=\hat{\gamma}_{H},\quad\frac{\beta_{D}}{\beta_{H}}=\frac{\hat{\beta}_{D}}{\hat{\beta}_{H}},\quad\frac{\beta_{I}}{\beta_{H}}=\frac{\hat{\beta}_{I}}{\hat{\beta}_{H}}\\}$

We can identify parameters
$\alpha,\delta_{I},\gamma_{I},\delta_{H},\gamma_{H}$ and the ratios
$\displaystyle\frac{\beta_{D}}{\beta_{H}}$,
$\displaystyle\frac{\beta_{I}}{\beta_{H}}.$ Also, the parameter $k$ is not
identifiable since it does not appear in the input-output equations. This
implies that the model M7 is not structurally identifiable given the number of
new cases, new hospitalizations and new deaths observations because we are
only able to identify parameters $\alpha,\gamma_{I},\delta_{I},\delta_{H}$ and
$\gamma_{H}$.

###### Proposition 3.16

The model M7 is not structurally identifiable from the number of newly
infected cases, new hospitalizations, and new deaths.

Figure 10: Flow chart of model M7 showing the transition of individuals
between the classes defined disease status. Observations are new infections,
new hospitalizations and new deaths. Structural identifiability analysis is
summarized for the scenario when initial conditions are known and
unknown.$S$$E$$I$$R$$H$$D$ Observations are new infections; $y_{1}(t)=kE(t)$
Observations are new hospitalizations; $y_{2}(t)=\alpha I$ Observations are
new deaths; $y_{3}(t)=\delta_{I}I(t)+\delta_{H}H(t)$
$\frac{S(\beta_{I}I+\beta_{H}H+\beta_{D}D)}{N}$$kE$
$\gamma_{I}I$
$\alpha I$
$\delta_{I}I$
$\delta_{H}H$
$\gamma_{H}H$
Identifiability Results:
Without Initial Conditions
identifiable parameters: $\alpha,\delta_{I},\gamma_{I},\delta_{H},\gamma_{H}$
unidentifiable parameters: $k,\beta_{I},\beta_{H},\beta_{D}$ parameter
correlations:
$\displaystyle\frac{\beta_{D}}{\beta_{H}}=\displaystyle\frac{\hat{\beta}_{D}}{\hat{\beta}_{H}}$,
$\displaystyle\frac{\beta_{I}}{\beta_{H}}=\displaystyle\frac{\hat{\beta_{I}}}{\hat{\beta_{H}}}.$
Identifiability Results:
With Initial Conditions
identifiable parameters: DAISY did not give any results for this case.

If we assume that transmission stemming from the death compartment is
negligible, we can eliminate the term $\beta_{D}DS$ and obtain a simpler
version of the model (M7):

$\textbf{Model 7'
(M${}_{7}^{\prime}$):}\begin{cases}\displaystyle\frac{dS}{dt}=\displaystyle\frac{-(\beta_{I}I+\beta_{H}H)S}{N},\\\\[6.45831pt]
\displaystyle\frac{dE}{dt}=\displaystyle\frac{(\beta_{I}I+\beta_{H}H)S}{N}-kE,\\\\[6.45831pt]
\displaystyle\frac{dI}{dt}=kE-(\alpha+\gamma_{I}+\delta_{I})I,\\\\[6.45831pt]
\displaystyle\frac{dH}{dt}=\alpha I-(\gamma_{H}+\delta_{H})H,\\\\[6.45831pt]
\displaystyle\frac{dR}{dt}=\gamma_{I}I+\gamma_{H}H,\\\\[6.45831pt]
\displaystyle\frac{dD}{dt}=\delta_{I}I+\delta_{H}H.\\\\[6.45831pt]
\end{cases}$ (M${}_{7}^{\prime}$)

Using DAISY (DAISY input code and results available in file named
Model7’-inputcode and Model7’-results), we obtain the following result,

$\\{\beta_{I}=\hat{\beta_{I}},\quad\beta_{H}=\hat{\beta}_{H},\quad
k=\hat{k},\quad\alpha=\hat{\alpha},\quad\gamma_{I}=\hat{\gamma_{I}},\quad\delta_{I}=\hat{\delta_{I}}\quad\gamma_{H}=\hat{\gamma}_{H},\quad\delta_{H}=\hat{\delta}_{H}\\}$

###### Proposition 3.17

The model M${}_{7}^{\prime}$ is structurally identifiable from the number of
newly infected cases, new hospitalizations, and new deaths.

Also, when all initial conditions are known, DAISY does not yield any results
for any of these cases.

Figure 11: Flow chart of model M${}_{7}^{\prime}$ showing the transition of
individuals between the classes defined disease status. Observations are new
infections, new hospitalizations and new deaths. Structural identifiability
analysis is summarized for the scenario when initial conditions are known and
unknown.$S$$E$$I$$R$$H$$D$ Observations are new infections; $y_{1}(t)=kE(t)$
Observations are new hospitalizations ; $y_{2}(t)=\alpha I$ Observations are
new deaths.; $y_{3}(t)=\delta_{I}I(t)+\delta_{H}H(t)$
$\frac{S(\beta_{I}I+\beta_{H}H)}{N}$$kE$
$\gamma_{I}I$
$\alpha I$
$\delta_{I}I$
$\delta_{H}H$
$\gamma_{H}H$
Identifiability Results:
Without Initial Conditions
identifiable parameters:
$\alpha,\delta_{I},\gamma_{I},\delta_{H},\gamma_{H},k,\beta_{I},\beta_{H},\beta_{D}$
unidentifiable parameters: None parameter correlations: None Identifiability
Results:
With Initial Conditions
identifiable parameters: DAISY did not give any results for this case.

###  3.7 Epidemic Model M8

$\textbf{Model 8
(M${}_{8}$):}\begin{cases}\displaystyle\frac{dS}{dt}=-\frac{(\beta_{\rho}I_{\rho}+\beta_{I}I)S}{N},\\\\[6.45831pt]
\displaystyle\frac{dE}{dt}=\frac{(\beta_{\rho}I_{\rho}+\beta_{I}I)S}{N}-kE,\\\\[6.45831pt]
\displaystyle\frac{dI_{\rho}}{dt}=kE-
k_{\rho}I_{\rho}-\gamma_{\rho}I_{\rho},\\\\[6.45831pt]
\displaystyle\frac{dI}{dt}=k_{\rho}I_{\rho}-\gamma I-\delta I,\\\\[6.45831pt]
\displaystyle\frac{dR}{dt}=\gamma I+\gamma_{\rho}I_{\rho},\\\\[6.45831pt]
\displaystyle\frac{dD}{dt}=\delta I\end{cases}$ (M8)

This model is motivated by the transmission dynamics of COVID-19 where pre-
symptomatic and symptomatic cases contribute to the transmission process [17].
This epidemic model involves six epidemiological states: The number of
susceptible individuals $(S(t))$, the number of latent individuals $(E(t))$,
the number of pre-symptomatic individuals $(I_{\rho}(t))$, the number of
symptomatic individuals $(I(t))$, the number of recovered individuals $(R(t))$
and the number of deaths $(D(t)).$ Unlike the previous models, this model has
a pre-symptomatic infectious state. We would like to identify the parameters
$\mbox{\boldmath$p$}=(\beta_{\rho},\beta_{I},k,k_{\rho},\gamma,\gamma_{\rho},\delta)$.

Number of new symptomatic cases $k_{\rho}I_{\rho}$ is observed. We begin by
considering the scenario in which the observed data corresponds to the number
of new cases, $y_{1}(t)=k_{\rho}I_{\rho}(t)$. In this scenario, DAISY is not
able to derive a result (DAISY input code available in file named
Model8a-inputcode).

Number of new symptomatic cases $k_{\rho}I_{\rho}$ and deaths $\delta$I are
observed. Next, we analyze the scenario when the observations correspond to
both the number of new cases and deaths;

$y_{1}(t)=k_{\rho}I_{\rho}(t)\quad\text{and}\quad y_{2}(t)=\delta I(t).$

This time, DAISY derives the input-output equations (see files
Model8b-inputcode and Model8b-results in github). DAISY assigns the following
integers to the model parameters,

$\hat{\mbox{\boldmath$p$}}=\left\\{\hat{\beta}_{\rho}=2,\;\hat{\beta_{I}}=3,\;\hat{k}=5,\;\hat{k_{\rho}}=7,\;\hat{\gamma}_{\rho}=11,\hat{\gamma}=13,\hat{\delta}=17\right\\}$

and then solves $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ and
gives the following results:

$7\beta_{\rho}+2\gamma_{\rho}=36,\quad\delta=17,\quad\gamma=13,\quad
k=5,\quad\beta_{I}=3,\quad k_{\rho}+\gamma_{\rho}=18.$ (19)

and

$7\beta_{\rho}+2\gamma_{\rho}=10,\quad\delta=17,\quad\gamma=13,\quad
k=18,\quad\beta_{I}=3,\quad k_{\rho}+\gamma_{\rho}=5.$ (20)

In this case, DAISY yields two possible solution sets given in (19) and (20).
While the parameters $\gamma$, $\delta$, and $\beta_{I}$ are identifiable,
parameters $\beta_{\rho}$, $\gamma_{\rho}$ and $k_{\rho}$ are not
identifiable. Parameter $k$ may have 2 different solutions, $k=\hat{k}$ which
is the true solution or $k=\hat{k}_{\rho}+\hat{\gamma}_{\rho}$. Even in the
case when $k$ is identified correctly, which is the solution set (19), several
correlations involve parameters $\beta_{\rho},\gamma_{\rho}$ $k_{\rho}$ and
$\gamma_{\rho}$. Therefore, we reach the following proposition for the
identifiability of M8 based on the observed number of new symptomatic cases
and deaths.

###### Proposition 3.18

The model M8 is not structurally identifiable from the observation of new
cases and new deaths. Only the parameters $\gamma$, $\delta$, $\beta_{I}$ are
identifiable.

Differential algebra approach uncovers the parameter correlations and
discovering the parameter correlations enables the formulation of a
structurally identifiable model. For instance, if the recovery rate of the
pre-symptomatic individuals is known, that is if $\gamma_{\rho}$ is fixed,
then the correlation $k_{\rho}+\gamma_{\rho}=36$ will reveal $k_{\rho}$.
Similarly, the correlation $7\beta_{\rho}+2\gamma_{\rho}=36$ would reveal
$\beta_{\rho}$. Because there are two sets of solutions, in order to obtain a
structurally identifiable model, we must additionally set the incubation
period $k$ in addition to the recovery rate $\gamma_{\rho}$. To check this, we
first ran the model (see files Model8c-inputcode and Model8c-results in
github) by fixing the value of $\gamma_{\rho}$ at 11. Then DAISY assigns the
following integers to the model parameters,

$\hat{\mbox{\boldmath$p$}}=\left\\{\hat{\beta}_{\rho}=2,\;\hat{\beta_{I}}=3,\;\hat{k}=5,\;\hat{k_{\rho}}=7,\;\hat{\gamma}=11,\hat{\delta}=13\right\\}$

and then solves $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ and
gives the following results:

$\beta_{I}=3,\quad\beta_{\rho}=2,\quad\delta=13,\quad\gamma=11,\quad k=5,\quad
k_{\rho}=7.$ (21)

and

$\beta_{I}=3,\quad\beta_{\rho}=-12/7,\quad\delta=13,\quad\gamma=11,\quad
k=18,\quad k_{\rho}=-6.$ (22)

Here, we see that the parameter $k$ is alternating again and the parameters
$\beta_{\rho}$ and $k_{\rho}$ have negative values. Therefore, we ran the
model again by fixing the values of of $\gamma_{\rho}$ at 11 and $k$ at 5 (see
files Model8d-inputcode and Model8d-results in github). Now, DAISY assigns the
following integers to the model parameters,

$\hat{\mbox{\boldmath$p$}}=\left\\{\hat{\beta}_{\rho}=2,\;\hat{\beta_{I}}=3,\;\hat{k_{\rho}}=5,\;\hat{\gamma}=7,\hat{\delta}=11\right\\}$

and then solves $c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$,
declaring the model as globally identifiable with the following model
parameter solution:

$\beta_{I}=3,\quad\beta_{\rho}=2,\quad\delta=11,\quad\gamma=7,\quad
k_{\rho}=5.$ (23)

We summarize the result in the following Proposition 3.19.

###### Proposition 3.19

If the pre-symptomatic recovery rate ($\gamma_{\rho}$) and the incubation
period ($1/k$) are fixed, then the model M8 becomes structurally identifiable
from the observation of new cases and new deaths.

Next we carry out the structural identifiability analysis of the model M8 from
the observation of new cases and deaths when all the initial conditions are
known. With known initial conditions, DAISY solves
$c(\mbox{\boldmath$p$})=c(\hat{\mbox{\boldmath$p$}})$ together with the
characteristic set evaluated at time $t=0$ and provides the following solution
$\gamma=13$, $\beta_{\rho}=2$, $\gamma_{\rho}=11$, $\beta_{I}=3$, $\delta=17$,
$k=5$, and $k_{\rho}=7$. Therefore, we conclude that the model is structurally
identifiable when the initial conditions are known, and reach the following
proposition.

###### Proposition 3.20

The model M8 is structurally identifiable from the observation of new cases
and deaths when all the initial conditions are known.

Figure 12: Flow chart of model M8 showing the transition of individuals
between the classes defined by disease status. Observations are the new
infected cases and new deaths. Structural identifiability analysis is
summarized for when initial conditions are known and
unknown.$S$$E$$I_{\rho}$$I$$R$$D$ Observations are new infected cases and new
deaths; $y_{1}(t)=kE$ and $y_{2}(t)=\delta_{I}I$
$\displaystyle\frac{(\beta_{\rho}I_{\rho}+\beta_{I}I)S}{N}$$kE$
$I_{\rho}k_{\rho}$
$I_{\rho}\gamma_{\rho}$
$\gamma I$
$\delta_{I}I$
Identifiability Results:
Without Initial Conditions
identifiable parameters: $\gamma$, $\delta$, $\beta_{I}$ unidentifiable
parameters: $\beta_{\rho}$, $\gamma_{\rho}$, $k$ and $k_{\rho}$ parameter
correlations: Two sets of correlation shown in (19) and (20) Identifiability
Results:
With Initial Conditions
identifiable parameters: $\gamma$, $\delta$, $\gamma_{\rho}$, $k$, $k_{\rho}$, $\beta_{\rho}$ and $\beta_{I}$ Notation | Meaning | Used in model
---|---|---
$S(t)$ |  Number of susceptible individuals at time $t$ |  M1, M2, M3, M4, M5, M6, M7, M8
$S_{\upsilon}(t)$ |  Number of susceptible mosquitoes at time $t$ |  M5, M6
$E(t)$ | Number of exposed individuals at time t |  M1, M2, M3, M4, M7, M8
$I(t)$ | Number of infected individuals at time t |  M1, M2, M3, M4, M5, M6, M7,M8
$I_{\upsilon}(t)$ | Number of infected mosquitoes at time t |  M5, M6
$I_{\rho}(t)$ | Number of pre-symptomatic individuals at time t | M8
$A(t)$ | Number of asymptomatic individuals at time t |  M2, M3, M4, M6
$R(t)$ | Number of recovered individuals at time t |  M1, M2, M3, M5, M6, M7, M8
$H(t)$ | Number of hospitalized individuals at time t | M7
$D(t)$ | Number of dead individuals at time t |  M4, M7, M8
Table 1: State variables and their definitions of the models M1 through M8. Notation | Meaning | Used in model
---|---|---
$\beta$ | Transmission rate in human population |  M1, M2, M4, M5
$\beta_{I}$ | Transmission rate in infected human population |  M3, M6, M7, M8
$\beta_{\rho}$ | Transmission rate in pre-symptomatic human population | M8
$\beta_{A}$ | Transmission rate in asymptomatic individuals |  M3, M6, M8
$\beta_{\upsilon}$ | Transmission rate in mosquito | M5
$\beta_{H}$ | Transmission rate in hospitalized individuals | M7
$\beta_{D}$ | Transmission rate from dead individuals | M7
k | Latent period |  M1, M2, M3, M4, M7, M8
$k_{\rho}$ | The fraction of pre-symptomatic individuals who develop symptom | M8
$\rho$ | The fraction of infected individuals who show symptoms (always less than 1) |  M2, M3, M4, M6
$\gamma$ | Recovery rate |  M1, M2, M3, M4, M5, M6
$\gamma_{I}$ | Recovery rate for infected | M7
$\gamma_{H}$ | Recovery rate for hospitalized | M7
$\gamma_{\rho}$ | The fraction of pre-symptomatic individuals who recover | M8
$\delta$ | Death rate due to disease among individuals |  M4, M7
$\delta_{H}$ | Death rate among hospitalized individuals | M7
$\delta_{I}$ | Death rate among infected individuals | M7
$\Lambda_{\upsilon}$ | Rate at which mosquitoes enter the susceptible class |  M5, M6
$\mu_{\upsilon}$ | Natural death rate of mosquitoes |  M5, M6
$\alpha$ | Rate of hospitalization of infected individuals | M7
Table 2: Parameters and their definitions the models M1 through M8. Model | Observed states | Identifiable parameters | Unidentifiable parameters
---|---|---|---
M1 |  Number of new infected cases $(kE)$ |  $k$, $\gamma$ |  $\beta$, $N$
M2 |  Number of new infected symptomatic cases $(k\rho E)$ |  $k$, $\gamma$ |  $\rho$, $\beta$, $N$
M3 |  Number of new infected symptomatic cases $(k\rho E$) |  $k$, $\gamma$ |  $\rho$, $\beta_{A}$, $\beta_{I}$, $N$
M4 |  Number of new infected cases $(kE)$ | $k$ | $\beta,\delta,\gamma$
M4 |  Number of new infected cases $(kE)$ and new deaths $(\delta I)$ |  $k$, $\beta$, $\delta$, $\gamma$ | -
M5 |  Cumulative number of new incidence $(S_{0}-S_{t})$ |  $\gamma$, $\mu_{\upsilon}$ |  $\beta$, $\beta_{\upsilon}$, $\Lambda_{\upsilon}$, $N$
M6 |  Cumulative number of new incidence $\rho$ $(S_{0}-S_{t})$ |  $\gamma$,$\mu_{\upsilon}$,$\rho$ |  $\beta_{A}$, $\beta_{I}$, $\beta_{\upsilon}$, $\Lambda_{\upsilon}$, $N$
M7 |  Number of new infected cases $(kE)$ | - | $\beta_{I},\beta_{H},\beta_{D},k,\alpha,\delta_{I},\delta_{H},\gamma_{H},\gamma_{I}$
M7 |  Number of new infected cases $(kE)$ and new hospitalization $(\alpha I)$ | $\alpha$ | $k,\beta_{I},\beta_{H},\beta_{D},k,\alpha,\delta_{I},\delta_{H},\gamma_{H},\gamma_{I}$
M7 |  Number of new infected cases $(kE)$, new hospitalization $(\alpha I)$ and, new deaths ($\delta_{I}I+\delta_{H}H)$ | $\alpha,\delta_{I},\gamma_{I},\delta_{H},\gamma_{H}$ | $k,\beta_{I},\beta_{H},\beta_{D}$
M8 |  Number of new infected cases $(kE)$ | No result obtained from DAISY | $\gamma,\delta,\gamma_{\rho},k,k_{\rho},\beta_{\rho},\beta_{I}$
M8 |  Number of new infected cases $(kE)$ and new deaths ($\delta_{I}I$) | $\gamma,\delta,k,\beta_{I}$ | $\beta_{\rho},\gamma_{\rho},k_{\rho}$
Table 3: Summary of identifiability results of the models M1 through M8 with corresponding observed states and when initial conditions are not known. Model | Observed states | Identifiability result
---|---|---
M1 |  Number of new infected cases $(kE)$ |  All parameters identifiable ($k$, $\gamma$, $\beta$, $N$)
M2 |  Number of new infected symptomatic cases $(k\rho E)$ |  All parameters identifiable (k, $\gamma$, $\rho,\beta$, $N$)
M3 |  Number of new infected symptomatic cases $(k\rho E)$ |  All parameters identifiable ($k$, $\gamma$, $\rho,\beta_{A},\beta_{I}$, $N$)
M4 |  Number of new infected cases $(kE)$ | No information in description…
M4 |  Number of new infected cases $(kE)$ and new deaths ($\delta I$) |  All parameters ($k$, $\beta,\delta,\gamma$) identifiable
M5 |  Cumulative number of new incidence $(S_{0}-S_{t})$ |  All parameters identifiable ($\gamma$, $\mu_{\upsilon}$, $\beta$, $\beta_{\upsilon}$, $\Lambda_{\upsilon}$, $N$)
M6 |  Cumulative number of new incidence $\rho$ $(S_{0}-S_{t})$ |  All parameters identifiable ($\gamma$,$\rho$,$\mu_{\upsilon}$,$\beta_{A}$,$\beta_{I}$, $\beta_{\upsilon}$, $\Lambda_{\upsilon}$,$N$)
M7 |  Number of new infected cases ($kE$) | No result obtained from DAISY
M7 |  Number of new infected cases $(kE)$ and new hospitalization $(\alpha I)$ | No result obtained from DAISY
M7 |  Number of new infected cases $(kE)$, new hospitalization $(\alpha I)$ and, new deaths $(\delta_{I}I+\delta_{H}H)$ | No result obtained from DAISY
M8 |  Number of new infected cases $(kE)$ | No result obtained from DAISY
M8 |  Number of new infected cases $(kE)$ and new deaths ($\delta_{I}I$) |  All parameters identifiable ($\gamma$, $\delta$, k, $\beta_{I}$, $\beta_{\rho}$, $\gamma_{\rho}$, $k_{\rho})$
Table 4: Identifiability results of the models M1 through M8 with
corresponding observed states and when initial conditions are known.

##  4 Discussion

In this paper, we have reviewed and illustrated a differential algebra
approach to carry out structural identifiability analyses of epidemic models
based on systems of differential equations using DAISY and Mathematica
(Wolfram Research) when a limited set of state variables are observed. The
power of this method lies in its ability to uncover parameter correlations
that make a model structurally identifiable. We demonstrate how uncovering
parameter correlations via structural identifiability analysis can help
resolve parameter identifiability roadblocks through various examples. In
particular, a structurally identifiable model can be achieved by revising the
structure of the model, obtaining additional information of specific
parameters involved in the parameter correlations, or collecting additional
data of certain states of the system. Our results underscore the need to
ensure that a model is structurally identifiable before fitting it to actual
data. Indeed, achieving a good fit to the data does not imply that the model
parameters are structurally identifiable.

Our results indicate that lack of structural identifiability in models cannot
necessarily be remedied by blindly fixing parameters based on existing
literature or collecting more data about other system states. It is important
to gain insight on how the structure of the model contributes to the lack of
parameter identifiability. While several methods exist to conduct structural
identifiability analyses of dynamical systems[18, 19, 20, 21, 22, 9, 28], the
differential algebra approach employed here goes a step further by providing
information about existing parameter correlations, which can guide the
researcher to resolve parameter identifiability issues.

We have also shown how results of structural identifiability analyses help
enrich compartmental diagrams of differential-equation models by incorporating
two pieces of information: The observed variables (e.g., new cases or new
deaths) and the corresponding results of the structural identifiability
analysis (i.e. weather parameters are identifiable or not). Importantly, these
enhanced diagrams can feature different parameter identifiability
configurations for the same model but different observed state variables.

Overall, our analyses indicate that parameter identifiability issues are more
likely to arise as model complexity increases in the number of
equations/states and parameters. However, incorporating additional data about
other system states may resolve the lack of structural identifiability. A
limitation of the approach described in this paper is the difficulty of
analyzing complex models. For situations where the differential algebra
approach cannot reveal the structural identifiability properties of the model,
alternative approaches could be explored (e.g, [28, 8]), but these often do
not provide explicit answers about existing parameter correlations involved in
the dynamical system.

The input codes and results from DAISY and Mathematica used in this study are
freely available at https://github.com/sushmadahal/Identifiability. The GitHub
repository also contains tutorial videos that describe model inputs and
results for each of the models used in the paper.

##  5 Statements and Declarations

The authors report no conflicts of interests related to this article.

##  6 Appendix

###  6.1 Input-output equations of epidemic models

Input-output equation of the model M2

$\displaystyle(y_{1}^{{}^{\prime\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}+(y_{1}^{{}^{\prime\prime\prime}})^{2}y_{1}^{2}k-y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime\prime}})^{2}y_{1}-y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}+y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}(2\gamma-k)+$
(24) $\displaystyle
2y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}y_{1}^{2}k(\gamma+k)-y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{3}(\gamma+k)+4y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\frac{\beta}{N}+y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}(\gamma^{2}-2k^{2})+$
$\displaystyle
8y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{3}\frac{\beta
k}{N}+y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{2}\gamma
k(2\gamma+k)+4y_{1}^{{}^{\prime\prime\prime}}y_{1}^{4}\frac{\beta
k^{2}}{N}+y_{1}^{{}^{\prime\prime\prime}}y_{1}^{3}\gamma^{2}k^{2}+(y_{1}^{{}^{\prime\prime}})^{3}y_{1}^{{}^{\prime}}-(y_{1}^{{}^{\prime\prime}})^{3}y_{1}(\gamma+k)+$
$\displaystyle
2(y_{1}^{{}^{\prime\prime}})^{2}(y_{1}^{{}^{\prime}})^{2}k-3(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}^{2}\frac{\beta}{N}+(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}(\gamma^{2}-\gamma
k-2k^{2})-3(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{3}\frac{\beta k}{N}+$
$\displaystyle(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{2}k(\gamma^{2}+2\gamma
k+k^{2})+y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{3}(-2\gamma^{2}-\gamma
k+2k^{2})+2y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\frac{\beta(3\gamma-k)}{N}$
$\displaystyle+y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}\frac{(\gamma^{3}-\gamma^{2}k-3\gamma
k^{2}-2k^{3})}{N^{2}}+2y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{3}\frac{\beta
k(6\gamma+k)}{N}$
$\displaystyle+y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{2}\frac{\gamma
k(2\gamma^{2}+2\gamma
k+k^{2})}{N^{2}}+2y_{1}^{{}^{\prime\prime}}y_{1}^{4}\frac{\beta
k^{2}(3\gamma+2k)}{N}$
$\displaystyle+y_{1}^{{}^{\prime\prime}}y_{1}^{3}\gamma^{2}k^{2}(\gamma+k)-(y_{1}^{{}^{\prime}})^{5}\frac{\beta}{N}+(y_{1}^{{}^{\prime}})^{4}y_{1}\frac{\beta(-4\gamma-3k)}{N}$
$\displaystyle+(y_{1}^{{}^{\prime}})^{4}(-\gamma^{3}-2\gamma^{2}k+k^{3})+4(y_{1}^{{}^{\prime}})^{3}y_{1}^{3}\frac{\beta^{2}}{N^{2}}+(y_{1}^{{}^{\prime}})^{3}y_{1}^{2}\frac{\beta(\gamma^{2}-6\gamma
k-6k^{2})}{N}+$ $\displaystyle(y_{1}^{{}^{\prime}})^{3}y_{1}\gamma
k(-2\gamma^{2}-3\gamma
k-k^{2})+12(y_{1}^{{}^{\prime}})^{2}y_{1}^{4}\frac{\beta^{2}k}{N^{2}}+(y_{1}^{{}^{\prime}})^{2}y_{1}^{3}\frac{\beta
k(3\gamma^{2}-4k^{2})}{N}-$
$\displaystyle(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\gamma^{2}k^{2}(\gamma+k)+12y_{1}^{{}^{\prime}}y_{1}^{5}\frac{\beta^{2}k^{2}}{N^{2}}+$
$\displaystyle y_{1}^{{}^{\prime}}y_{1}^{4}\frac{\beta\gamma
k^{2}(3\gamma+2k)}{N}+4y_{1}^{6}\frac{\beta^{2}k^{3}}{N^{2}}+y_{1}^{5}\frac{\beta\gamma^{2}k^{3}}{N}=0$

Input-output equation of the model M3

$\displaystyle(y_{1}^{{}^{\prime\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}+(y_{1}^{{}^{\prime\prime\prime}})^{2}y_{1}^{2}k-y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime\prime}})^{2}y_{1}-y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}+y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}(2\gamma-k)+$
$\displaystyle
2y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime\prime}}y_{1}^{2}k(\gamma+k)-y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{3}(\gamma+k)+4y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\frac{(-\beta_{A}\rho+\beta_{A}+\beta_{I}\rho)}{N\rho}+$
$\displaystyle
y_{1}^{{}^{\prime\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}(\gamma^{2}-2k^{2})+8y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{3}\frac{k(-\beta_{A}\rho+\beta_{A}+\beta_{I}\rho)}{N\rho}+y_{1}^{{}^{\prime\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{2}\gamma
k(2\gamma+k)+$ $\displaystyle
4y_{1}^{{}^{\prime\prime\prime}}y_{1}^{4}\frac{k^{2}(-\beta_{A}\rho+\beta_{A}+\beta_{I}\rho)}{N\rho}+y_{1}^{{}^{\prime\prime\prime}}y_{1}^{3}\gamma^{2}k^{2}+(y_{1}^{{}^{\prime\prime}})^{3}y_{1}^{{}^{\prime}}-(y_{1}^{{}^{\prime\prime}})^{3}y_{1}(\gamma+k)+2(y_{1}^{{}^{\prime\prime}})^{2}(y_{1}^{{}^{\prime}})^{2}k+$
$\displaystyle
3(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}^{2}\frac{(\beta_{A}\rho-\beta_{A}-\beta_{I}\rho)}{N\rho}+(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{{}^{\prime}}y_{1}(\gamma^{2}-\gamma
k-2k^{2})+3(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{3}\frac{k(\beta_{A}\rho-\beta_{A}-\beta_{I}\rho)}{N\rho}+$
$\displaystyle(y_{1}^{{}^{\prime\prime}})^{2}y_{1}^{2}k(\gamma^{2}+2\gamma
k+k^{2})+y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{3}(-2\gamma^{2}-\gamma
k+2k^{2})+$ $\displaystyle
2y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\frac{(-3\beta_{A}\gamma\rho+3\beta_{A}\gamma+\beta_{A}k\rho-\beta_{A}k+3\beta_{I}\gamma\rho-\beta_{I}k\rho)}{N\rho}+$
$\displaystyle
y_{1}^{{}^{\prime\prime}}(y_{1}^{{}^{\prime}})^{2}y_{1}(\gamma^{3}-\gamma^{2}k-3\gamma
k^{2}-2k^{3})+2y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{3}\frac{k(-6\beta_{A}\gamma\rho+6\beta_{A}\gamma-\beta_{A}k\rho+\beta_{A}k+6\beta_{I}\gamma\rho+\beta_{I}k\rho)}{N\rho}+$
$\displaystyle y_{1}^{{}^{\prime\prime}}y_{1}^{{}^{\prime}}y_{1}^{2}\gamma
k(2\gamma^{2}+2\gamma
k+k^{2})+2y_{1}^{{}^{\prime\prime}}y_{1}^{4}\frac{k^{2}(-3\beta_{A}\gamma\rho+3\beta_{A}\gamma-2\beta_{A}k\rho+2\beta_{A}k+3\beta_{I}\gamma\rho+2\beta_{I}k\rho)}{N\rho}+$
$\displaystyle
y_{1}^{{}^{\prime\prime}}y_{1}^{3}\gamma^{2}k^{2}(\gamma+k)+(y_{1}^{{}^{\prime}})^{5}\frac{(\beta_{A}\rho-\beta_{A}-\beta_{I}\rho)}{N\rho}+$
$\displaystyle(y_{1}^{{}^{\prime}})^{4}y_{1}\frac{(4\beta_{A}\gamma\rho-4\beta_{A}\gamma+3\beta_{A}k\rho-3\beta_{A}k-4\beta_{I}\gamma\rho-3\beta_{I}k\rho)}{N\rho}+$
$\displaystyle(y_{1}^{{}^{\prime}})^{4}(-\gamma^{3}-2\gamma^{2}k+k^{3})+4(y_{1}^{{}^{\prime}})^{3}y_{1}^{3}\frac{(\beta_{A}^{2}\rho^{2}-2\beta_{A}^{2}\rho+\beta_{A}^{2}-2\beta_{A}\beta_{I}\rho^{2}+2\beta_{A}\beta_{I}\rho+\beta_{I}^{2}\rho^{2})}{N^{2}\rho^{2}}+$
$\displaystyle(y_{1}^{{}^{\prime}})^{3}y_{1}^{2}\frac{(-\beta_{A}\gamma^{2}\rho+\beta_{A}\gamma^{2}+6\beta_{A}\gamma
k\rho-6\beta_{A}\gamma
k+6\beta_{A}k^{2}\rho-6\beta_{A}k^{2}+\beta_{I}\gamma^{2}\rho-6\beta_{I}\gamma
k\rho-6\beta_{I}k^{2}\rho)}{N\rho}+$
$\displaystyle(y_{1}^{{}^{\prime}})^{3}y_{1}\gamma k(-2\gamma^{2}-3\gamma
k-k^{2})+12(y_{1}^{{}^{\prime}})^{2}y_{1}^{4}\frac{k(\beta_{A}^{2}\rho^{2}-2\beta_{A}^{2}\rho+\beta_{A}^{2}-2\beta_{A}\beta_{I}\rho^{2}+2\beta_{A}\beta_{I}\rho+\beta_{I}^{2}\rho^{2})}{N^{2}\rho^{2}}+$
$\displaystyle(y_{1}^{{}^{\prime}})^{2}y_{1}^{3}\frac{k(-3\beta_{A}\gamma^{2}\rho+3\beta_{A}\gamma^{2}+4\beta_{A}k^{2}\rho-4\beta_{A}k^{2}+3\beta_{I}\gamma^{2}\rho-4\beta_{I}k^{2}\rho)}{N\rho}-(y_{1}^{{}^{\prime}})^{2}y_{1}^{2}\gamma^{2}k^{2}(\gamma+k)+$
(25) $\displaystyle
12y_{1}^{{}^{\prime}}y_{1}^{5}\frac{k^{2}(\beta_{A}^{2}\rho^{2}-2\beta_{A}^{2}\rho+\beta_{A}^{2}-2\beta_{A}\beta_{I}\rho^{2}+2\beta_{A}\beta_{I}\rho+\beta_{I}^{2}\rho^{2})}{N^{2}\rho^{2}}+$
$\displaystyle y_{1}^{{}^{\prime}}y_{1}^{4}\frac{\gamma
k^{2}(-3\beta_{A}\gamma\rho+3\beta_{A}\gamma-2\beta_{A}k\rho+2\beta_{A}k+3\beta_{I}\gamma\rho+2\beta_{I}k\rho)}{N\rho}+$
$\displaystyle
4y_{1}^{6}\frac{k^{3}(\beta_{A}^{2}\rho^{2}-2\beta_{A}^{2}\rho+\beta_{A}^{2}-2\beta_{A}\beta_{I}\rho^{2}+2\beta_{A}\beta_{I}\rho+\beta_{I}^{2}\rho^{2})}{N^{2}\rho^{2}}+y_{1}^{5}\frac{\gamma^{2}k^{3}(-\beta_{A}\rho+\beta_{A}+\beta_{I}\rho)}{N\rho}$

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