# An age-of-infection model with both symptomatic and asymptomatic infections

Fan Bai<EMAIL_ADDRESS>Hausdorff Center for Mathematics, University of
Bonn, Bonn, Germany

Keywords: Mathematical epidemiology, Deterministic compartmental models, Age
of infection models, Integro-differential equations, Final epidemic size
estimation.

AMS subject classifications: 92D30, 45J05

##  1 Abstract

We formulate a general age-of-infection epidemic model with two pathways: the
symptomatic infections and the asymptomatic infections. We then calculate the
basic reproduction number $\mathcal{R}_{0}$ and establish the final size
relation. It is shown that the ratio of accumulated counts of symptomatic
patients and asymptomatic patients is determined by the symptomatic ratio $f$
which is defined as the probability of eventually becoming symptomatic after
being infected. We also formulate and study a general age-of-infection model
with disease deaths and with two infection pathways. The final size relation
is investigated, and the upper and lower bounds for final epidemic size are
given. Several numerical simulations are performed to verify the analytical
results.

##  2 Introduction

Asymptomatic infections are recorded for COVID-$19$ (Coronavirus SARS-CoV-$2$)
pandemic and have been studied from many different aspects. Evidence suggests
that asymptomaticity has significant impacts on the development of COVID-$19$
([20, 18]). Furthermore, because of its ”silently transmissible” nature, the
pathway of asymptomatic infection is considered as a key component for
understanding the mechanisms of transmission and evaluating the efficiency of
intervention measures. However, quantification of asymptomatic infections and
further inference of value of symptomatic ratio $f$ is tricky due to the
unreliable recorded numbers of symptomatic patients and asymptomatic patients
at any specific time post epidemic outbreak. For COVID-$19$, it is suggested
that the asymptomatic ratio is around $35.1\%$ ([21]) and it is highly age
dependent. Other studies based on daily incidence data from certain geographic
locations and using various parameter estimation techniques indicate very
different values of symptomatic/asymptomatic ratio (e.g., [18, 19, 17]). The
purpose of this paper is to formulate the epidemic models with two infection
pathways, by considering different formats of the symptomatic ratio $f$
(either a constant or a piece-wise function), and obtain the relation between
the ratio, the basic reproduction number and the final epidemic size. Another
focus is to compare the final sizes of symptomatic infections and asymptomatic
infections.

The Age-of-infection model was originally derived by Kermack and McKendrick in
their well-known paper [16], it is the general form of epidemic models ([5, 4,
9, 6, 25, 15]). This type of epidemic model is especially useful to study the
epidemic models in which periods are far from exponential distributed, e.g.,
the current COVID-$19$ pandemic. The formulation of the basic reproduction
number for the age-of-infection model was well studied in [25] and the final
size relation has been established in [6]. The age-of-infection model with
disease-induced mortality was also studied in [6], an inequality of final size
relation was developed. The general model was also extended for the
heterogeneous mixing population in [12]. In this paper, we formulate and study
an age-of-infection model for epidemics with both symptomatic infections and
asymptomatic infections. The derivation and comparison of the basic
reproduction number $\mathcal{R}_{0}$ and the formulation of final size
relations are the main focus of the study. We also consider the age-of-
infection model with two infection pathways and with disease-induced
mortality.

This paper consists of five parts. In section 3, we firstly review the basic
general age-of-infection model in a homogeneously mixing population. In
section 4, we formulate the age-of-infection model with two pathways, the
symptomatic infection and the asymptomatic infection. We then calculate the
basic reproduction number and obtain the corresponding final size relation. We
will focus on the ratio of total symptomatic cases and total asymptomatic
cases in this study, which entirely depends on the symptomatic ratio $f$. As
an example, we study a special case of Susceptible-Exposed-Infectious-
Asymptomatic-Removed (SEIAR) model, by assuming exponentially distributed
disease stages. In section 5, we further consider the disease-induced deaths
for the age-of-infection model with two infection pathways and investigate the
final size relations. In section 6, we perform several sets of numerical
simulations to verify our theoretical results. In section 7, we summarize the
investigation and also propose some interesting future work in the direction
of age-of-infection modeling.

##  3 The general age-of-infection model

We have $S(t)$ denote the number of susceptibles at time $t$ and have
$\phi(t)$ be the total infectivity at time $t$. $\phi(t)$ is defined as the
sum of products of the number of infected members with each infection age and
the mean infectivity for that infection age ([25]). We assume that on average
each member of the population make $a$ contacts per unit time. The population
size is $N$ and there are no demographic effects (births, deaths, migration)
in the population. We further define $B(\tau)$ the fraction of infected
members remaining infected at infection age $\tau$ and $\pi(\tau)$ the mean
infectivity at infection age $\tau$. $B(\tau)$ is a non-increasing function
and satisfies that

$B(0)=1,\quad B(\infty)=0.$

The general age-of-infection model is

$\displaystyle S^{\prime}$ $\displaystyle=-\frac{a}{N}S\phi$ (1)
$\displaystyle\phi(t)$
$\displaystyle=\phi_{0}(t)+\int_{0}^{t}\frac{a}{N}S(t-\tau)\phi(t-\tau)B(\tau)\pi(\tau)d\tau,$

where $\phi_{0}(t)$ is the total infectivity of members of the population who
were infected at the initial time, at time $t$. The following Figure 1
demonstrates the dynamics of the general model. It is worth mentioning that
the number of infectives is described as

$I(t)=I_{0}(t)+\int_{0}^{t}\frac{a}{N}S(t-\tau)\phi(t-\tau)B(\tau)d\tau.$

It is straightforward to derive the equation about the change of number of
infectives ([25])

$I^{\prime}(t)=\frac{a}{N}S\phi+\int_{0}^{t}\frac{a}{N}S(\tau)\phi(\tau)B^{\prime}(t-\tau)d\tau,$

where $B^{\prime}$ reflects the recovery of infected individuals.

Figure 1: The diagrammatic framework of the basic age-of-infection model (1).

It has been well studied in [6, 25], that the basic reproduction number for
model (1) is

$\mathcal{R}_{0}=a\int_{0}^{\infty}B(\tau)\pi(\tau)d\tau.$ (2)

The final size relation for model (1) is

$\log\frac{S_{0}}{S_{\infty}}=\mathcal{R}_{0}\left[1-\frac{S_{\infty}}{N_{0}}\right]-\frac{a}{N_{0}}\int_{0}^{\infty}[(N_{0}-S_{0})B(t)\pi(t)-\phi_{0}(t)]dt.$
(3)

The uniqueness of $S_{\infty}$ has been confirmed in [6]. If the initial
infectivity is small and neglectible, the final size relation can be
simplified as

$\log\frac{S_{0}}{S_{\infty}}=\mathcal{R}_{0}\left[1-\frac{S_{\infty}}{N_{0}}\right].$
(4)

##  4 The age-of-infection model with both symptomatic and asymptomatic
infections

###  4.1 The model formulation

We extend the general age-of-infection model (1) by considering that the
infected individuals are possibly asymptomatic or symptomatic. It is assumed
that once an individual is infected, there is a probability $0<f<1$ that this
individual eventually becomes symptomatic, and the probability of becoming
asymptomatic is $(1-f)$. We denote $\phi_{i}(t)$ the sum of products of the
symptomatic individuals with each infection age and the mean infectivity for
that infection age, $\phi_{a}(t)$ the sum of products of the asymptomatic
individuals with each infection age and the mean infectivity for that
infection age. Thus, the total infectivity is

$\phi(t)=\phi_{i}(t)+\phi_{a}(t).$ (5)

$B_{i}(\tau)$ and $B_{a}(\tau)$ represent the fraction of symptomatic and
asymptomatic individuals remaining infected at infection age $\tau$,
respectively; $\pi_{i}(\tau)$ and $\pi_{a}(\tau)$ represent the mean
infectivity of symptomatic and asymptomatic individuals at infection age
$\tau$, respectively. $A_{i}(\tau)\mathrel{\mathop{\mathchar
58\relax}}=B_{i}(\tau)\pi_{i}(\tau)$ is the mean infectivity of a symptomatic
individual at infection age $\tau$, while
$A_{a}(\tau)\mathrel{\mathop{\mathchar 58\relax}}=B_{a}(\tau)\pi_{a}(\tau)$ is
the mean infectivity of an asymptomatic individual at infection age $\tau$.
The dynamics is depicted in Figure 2. It is noticed that the recovered
symptomatic/asymptomatic patients are theoretically counted as one stage of
the infection process, but with no infectivities.

Figure 2: The diagrammatic framework of the age-of-infection model (6).

The age-of-infection model is

$\displaystyle S^{\prime}$ $\displaystyle=-a\frac{S}{N}(\phi_{i}+\phi_{a})$
(6) $\displaystyle\phi_{i}(t)$
$\displaystyle=\phi_{i,0}(t)+\int_{0}^{t}f\frac{a}{N}S(t-\tau)(\phi_{i}(t-\tau)+\phi_{a}(t-\tau))B_{i}(\tau)\pi_{i}(\tau)d\tau,$
$\displaystyle\phi_{a}(t)$
$\displaystyle=\phi_{a,0}(t)+\int_{0}^{t}(1-f)\frac{a}{N}S(t-\tau)(\phi_{i}(t-\tau)+\phi_{a}(t-\tau))B_{a}(\tau)\pi_{a}(\tau)d\tau.$

It can be proved that model (6) is an extension of the general age-of-
infection model (1). We first calculate the basic reproduction number for
model (6),

$\mathcal{R}_{0}=a\left[f\int_{0}^{\infty}A_{i}(\tau)d\tau+(1-f)\int_{0}^{\infty}A_{a}(\tau)d\tau\right].$
(7)

This form is consistent with Theorem $3.1$ in [25] that the basic reproduction
number $\mathcal{R}_{0}$ depends only on the mean period in each infective
stage, regardless of its distribution. We now generate the final size relation
for model (6). We first write

$-\frac{S^{\prime}}{S}=\frac{a}{N}\left(\phi_{i,0}(t)+\phi_{a,0}(t)\right)+\frac{a}{N}\int_{0}^{t}[-S^{\prime}(t-\tau)]\left(fA_{i}(\tau)+(1-f)A_{a}(\tau)\right)d\tau.$

Integration with respect to $t$ from $0$ to $\infty$ yields

$\displaystyle\log\frac{S_{0}}{S_{\infty}}=$
$\displaystyle\frac{a}{N}\int_{0}^{\infty}\left(\phi_{i,0}(t)+\phi_{a,0}(t)\right)dt+\frac{a}{N}\int_{0}^{\infty}\int_{0}^{t}[-S^{\prime}(t-\tau)]\left(fA_{i}(\tau)+(1-f)A_{a}(\tau)\right)d\tau
dt$ $\displaystyle=$
$\displaystyle\frac{a}{N}\int_{0}^{\infty}\left(\phi_{i,0}(t)+\phi_{a,0}(t)\right)dt+\frac{a}{N}\int_{0}^{\infty}\left(fA_{i}(\tau)+(1-f)A_{a}(\tau)\right)\int_{\tau}^{\infty}[-S^{\prime}(t-\tau)]dtd\tau$
$\displaystyle=$
$\displaystyle\frac{a}{N}\int_{0}^{\infty}\left(\phi_{i,0}(t)+\phi_{a,0}(t)\right)dt+\frac{a}{N}[S_{0}-S_{\infty}]\int_{0}^{\infty}\left(fA_{i}(\tau)+(1-f)A_{a}(\tau)\right)d\tau$
$\displaystyle=$
$\displaystyle\frac{a}{N}\int_{0}^{\infty}\left(\phi_{i,0}(t)+\phi_{a,0}(t)\right)dt+\mathcal{R}_{0}\frac{S_{0}-S_{\infty}}{N}.$

For each infection case, the probability of proceeding to the symptomatic
state is $f$ and the probability of proceeding to the asymptomatic state is
$(1-f)$. Thus, it is intuitive to conclude that the total number of
symptomatic patients is $f(S_{0}-S_{\infty})$ and the total number of
asymptomatic patients is $(1-f)(S_{0}-S_{\infty})$. We now perform some
calculations to verity this conjecture. Firstly, the numbers of symptomatic
patients $I(t)$ and asymptomatic patients $A(t)$ are

$\displaystyle I(t)$
$\displaystyle=I_{0}(t)+\int_{0}^{t}f\frac{a}{N}S(t-\tau)\phi(t-\tau)B_{i}(\tau)d\tau,$
(8) $\displaystyle A(t)$
$\displaystyle=A_{0}(t)+\int_{0}^{t}(1-f)\frac{a}{N}S(t-\tau)\phi(t-\tau)B_{a}(\tau)d\tau.$

We now focus on $I(t)$ and calculate the derivative of $I(t)$,

$I^{\prime}(t)=f\frac{a}{N}S\phi+\int_{0}^{t}f\frac{a}{N}S(\tau)\phi(\tau)B_{i}^{\prime}(t-\tau)d\tau.$
(9)

In Equation (9), the first term indicates the rate of new symptomatic
infections and the second term represents the transition from infected stage
to recovery stage for symptomatic patients. Thus, we have

$R_{i}^{\prime}(t)=-\int_{0}^{t}f\frac{a}{N}S(\tau)\phi(\tau)B_{i}^{\prime}(t-\tau)d\tau.$

We then integrate both sides of the equation with respect to $t$ from $0$ to
$\infty$,

$\displaystyle\int_{0}^{\infty}R_{i}^{\prime}(t)dt$
$\displaystyle=-\int_{0}^{\infty}\int_{0}^{t}f\frac{a}{N}S(\tau)\phi(\tau)B_{i}^{\prime}(t-\tau)d\tau
dt$ (10) $\displaystyle R_{i}(\infty)$
$\displaystyle=-\int_{0}^{\infty}\int_{\tau}^{\infty}f[-S^{\prime}(t-\tau)]B_{i}^{\prime}(\tau)dtd\tau,$
$\displaystyle=-\int_{0}^{\infty}fB_{i}^{\prime}(\tau)\int_{\tau}^{\infty}[-S^{\prime}(t-\tau)]dtd\tau$
$\displaystyle=-\int_{0}^{\infty}fB_{i}^{\prime}(\tau)[S_{0}-S_{\infty}]d\tau$
$\displaystyle=-f[S_{0}-S_{\infty}][B_{i}(\infty)-B_{i}(0)]$
$\displaystyle=f[S_{0}-S_{\infty}].$

Similarly, we have

$R_{a}(\infty)=(1-f)[S_{0}-S_{\infty}].$

We state the following theorem to summarize all calculations.

######  Theorem 4.1.

If the population size $N$ is large and the number of initial infection cases
is small, the final size of an age-of-infection model with both symptomatic
infections and asymptomatic infections is

$\log\frac{S_{0}}{S_{\infty}}=\mathcal{R}_{0}\frac{S_{0}-S_{\infty}}{N},$ (11)

with $\mathcal{R}_{0}$ being expressed in Equation (7). Moreover, the total
estimated cases of symptomatic infections are $f(S_{0}-S_{\infty})$, while the
total estimated cases of asymptomatic infections are
$(1-f)(S_{0}-S_{\infty})$.

If the initial infection cases are not neglectible, we have the following
inequality,

$\log\frac{S_{0}}{S_{\infty}}>\mathcal{R}_{0}\frac{S_{0}-S_{\infty}}{N}.$ (12)

It is possible that the ratio $f$ is not a constant, but a function about $t$.
This may be caused by the mutations of responsible viruses. The model then
becomes

$\displaystyle S^{\prime}$ $\displaystyle=-a\frac{S}{N}(\phi_{i}+\phi_{a})$
(13) $\displaystyle\phi_{i}(t)$
$\displaystyle=\phi_{i,0}(t)+\int_{0}^{t}f(\tau)\frac{a}{N}S(t-\tau)(\phi_{i}(t-\tau)+\phi_{a}(t-\tau))A_{i}(\tau)d\tau,$
$\displaystyle\phi_{a}(t)$
$\displaystyle=\phi_{a,0}(t)+\int_{0}^{t}(1-f(\tau))\frac{a}{N}S(t-\tau)(\phi_{i}(t-\tau)+\phi_{a}(t-\tau))A_{a}(\tau)d\tau.$

The basic reproduction number $\mathcal{R}_{0}$ for model (13) depends on
initial ratio $f(0)$ and is given by,

$\mathcal{R}_{0}=a\left[f(0)\int_{0}^{\infty}A_{i}(\tau)d\tau+(1-f(0))\int_{0}^{\infty}A_{a}(\tau)d\tau\right].$
(14)

$f(t)$ may take the form of a Heaviside function (with one or multiple switch
times) or its smooth approximations (generalized logistic functions). Explicit
final size relation for model (13) is impossible to obtain, but can be
approximated by Equation (3) in Theorem 4.1 if $\max(f(\tau))-\min(f(\tau))$
is small or the switch time(s) is late or close to the end of the epidemic.
Some numerical simulations will be performed to show that the final sizes can
not be accurately predicted, but be appropriately approximated.

###  4.2 The effect on final epidemic size estimation by using incorrectly
inferred ratio

In practice, the counts of symptomatic patients are relatively accurate. While
the counts of asymptomatic patients basically rely on serology reports and
surveys. Thus, it is generally a challenging task to correctly infer the value
of $f$ based on epidemiological data. This leads to an interesting problem of
the comparison of outcomes by using the actual $f$ and the inaccurately
inferred $f^{\prime}$.

First we have a similar formula for $\mathcal{R}_{0}^{\prime}$ based on
$f^{\prime}$:

$\mathcal{R}_{0}^{\prime}=a\left[f^{\prime}\int_{0}^{\infty}A_{i}(\tau)d\tau+(1-f^{\prime})\int_{0}^{\infty}A_{a}(\tau)d\tau\right].$
(15)

We denote

$\displaystyle\mathcal{R}_{i}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=a\int_{0}^{\infty}A_{i}(\tau)d\tau,$ (16)
$\displaystyle\mathcal{R}_{a}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=a\int_{0}^{\infty}A_{a}(\tau)d\tau$

to represent the average secondary infection cases caused by a symptomatic
patient or an asymptomatic patient, when being introduced into a wholly
susceptible population.The basic reproduction number is the linear combination
of $\mathcal{R}_{i}$ and $\mathcal{R}_{a}$. The difference between
$\mathcal{R}_{0}^{\prime}$ and $\mathcal{R}_{0}$ depends on the relation
between ratios $f$, $f^{\prime}$, $\mathcal{R}_{i}$ and $\mathcal{R}_{a}$,

$\mathcal{R}_{0}-\mathcal{R}_{0}^{\prime}=(f-f^{\prime})(\mathcal{R}_{i}-\mathcal{R}_{a}).$
(17)

Therefore, the relation between $f$ and $f^{\prime}$ is insufficient to
determine how incorrect fraction will affect the assumed $\mathcal{R}_{0}$. It
is critical to compare the values of $\mathcal{R}_{i}$ and $\mathcal{R}_{a}$.
In practice, it is reasonable to assume that the inferred ratio $f^{\prime}$
is larger than the real ratio $f$, due to the fact that the asymptomatic cases
are more easier to be missed and the asymptomatic infections are
underestimated. The other situation is that the asymptomatic infections are
overestimated, if the presymptomatic cases are incorrectly categorized as
asymptomatic infections. This case indicates a smaller inferred $f^{\prime}$.
However, the comparison of $\mathcal{R}_{i}$ and $\mathcal{R}_{a}$ is trivial
and difficult to obtain in real life. We next introduce the established result
(see Lemma $2.1$ and Lemma $2.2$ in [2]) to demonstrate how different values
of basic reproduction numbers will affect the final epidemic size,

######  Theorem 4.2.

The limiting susceptible population size $S_{\infty}$ is uniquely determined
by the value of the basic reproduction number $\mathcal{R}_{0}$. If
$\mathcal{R}_{0}>1$ and increases, $S_{\infty}$ decreases and the final
epidemic size increases.

###  4.3 A special case: Exponential distributions of disease stages

A special case of the general age-of-infection model (6) is the epidemic model
with exposed stages, for both infection pathways. It is noticed that the
exposed stages can be significantly different for two different types of
patients. Therefore, we have the following SEIAR type of model,

$\displaystyle S^{\prime}$ $\displaystyle=-a\frac{S}{N}(I+A)$ (18)
$\displaystyle E_{i}^{\prime}$
$\displaystyle=af\frac{S}{N}(I+A)-\kappa_{i}E_{i}$ $\displaystyle
E_{a}^{\prime}$ $\displaystyle=a(1-f)\frac{S}{N}(I+A)-\kappa_{a}E_{a}$
$\displaystyle I^{\prime}$ $\displaystyle=\kappa_{i}E_{i}-\alpha_{i}I$
$\displaystyle A^{\prime}$ $\displaystyle=\kappa_{a}E_{a}-\alpha_{a}A$
$\displaystyle R^{\prime}$ $\displaystyle=\alpha_{i}I+\alpha_{a}A.$

Now we show that model (18) can be written in the form of the general age-of-
infection model (6). We denote $\mu_{i}(\tau)$ and $\mu_{a}(\tau)$ the
fractions of symptomatic and asymptomatic patients with infection age $\tau$
who are not yet infectious, respectively, and also define $\nu_{i}(\tau)$ and
$\nu_{i}(\tau)$ the fractions of symptomatic and asymptomatic patients with
infection age $\tau$ who are infectious, respectively ([6, 9]). The basic
dynamics of SEIAR model (18) is described in Figure 3.

Figure 3: The diagrammatic framework of the SEIAR model (18).

We first focus on the upper symptomatic infection path ($S\rightarrow
E_{i}\rightarrow I\rightarrow R$) in Figure 3, $\mu_{i}(\tau)$ and
$\nu_{i}(\tau)$ satisfy the ordinary differential equation

$\displaystyle\mu_{i}^{\prime}(\tau)$
$\displaystyle=-\kappa_{i}\mu_{i}(\tau),$ (19)
$\displaystyle\nu_{i}^{\prime}(\tau)$
$\displaystyle=\kappa_{i}\mu_{i}(\tau)-\alpha_{i}\nu(\tau),$

with initial condition $\mu_{i}(0)=1$ and $\nu_{i}(0)=0$. We are able to
obtain the solution of the ODE system (19)

$\mu_{i}(\tau)=e^{-\kappa_{i}\tau},\quad\nu_{i}(\tau)=\frac{\kappa_{i}}{\kappa_{i}-\alpha_{i}}[e^{-\alpha_{i}\tau}-e^{-\kappa_{i}\tau}].$
(20)

The change of fractions $\mu_{i}(\tau)$ and $\nu_{i}(\tau)$ with a chosen set
of parameter values ($\kappa_{i}=0.07$ and $\alpha_{i}=0.2$) is visualized in
Figure 4.

Figure 4: The change of infected stages of symptomatic patients with respect
to age of infection.

The function $B_{i}(\tau)$ is implicitly expressed by two functions $\mu_{i}$
and $\nu_{i}$ in Equation (20). It is straightforward to have the averaged
infectivities for both exposed patients and symptomatic patients for
$\pi_{E}=0$ and $\pi_{I}=1$. Based on the definition of $A_{i}(\tau)$, we have

$A_{i}(\tau)=\frac{\kappa_{i}}{\kappa_{i}-\alpha_{i}}[e^{-\alpha_{i}\tau}-e^{-\kappa_{i}\tau}].$

Similarly, the lower asymptomatic infection path ($S\rightarrow
E_{a}\rightarrow A\rightarrow R$) can be analyzed in the same manner. It
implies

$A_{a}(\tau)=\frac{\kappa_{a}}{\kappa_{a}-\alpha_{a}}[e^{-\alpha_{a}\tau}-e^{-\kappa_{a}\tau}].$

From the formula for the basic reproduction number $\mathcal{R}_{0}$ in
Equation (7), we have

$\displaystyle\mathcal{R}_{0}$
$\displaystyle=a\left[f\int_{0}^{\infty}A_{i}(\tau)d\tau+(1-f)\int_{0}^{\infty}A_{a}(\tau)d\tau\right]$
(21)
$\displaystyle=a\left[f\int_{0}^{\infty}\frac{\kappa_{i}}{\kappa_{i}-\alpha_{i}}(e^{-\alpha_{i}\tau}-e^{-\kappa_{i}\tau})d\tau+(1-f)\int_{0}^{\infty}\frac{\kappa_{a}}{\kappa_{a}-\alpha_{a}}(e^{-\alpha_{a}\tau}-e^{-\kappa_{a}\tau})d\tau\right]$
$\displaystyle=a\left(\frac{f}{\alpha_{i}}+\frac{(1-f)}{\alpha_{a}}\right).$

This result can also be verified by using the next generation matrix approach
([24]). If we consider the averaged infectivity for asymptomatic patients is
$\epsilon<1$, it can be calculated that $A_{a}(\tau)=\epsilon
e^{-\kappa_{i}\tau}+\frac{\kappa_{a}}{\kappa_{a}-\alpha_{a}}[e^{-\alpha_{a}\tau}-e^{-\kappa_{a}\tau}]$
and further we have

$\mathcal{R}_{0}=a\left(\frac{f}{\alpha_{i}}+\frac{(1-f)\epsilon}{\alpha_{a}}\right).$

The final size relation for model (18) can be generated by Theorem (4.1). We
can also use the alternative method ([7]) to verify this result. Firstly, we
denote

$f(\infty)\mathrel{\mathop{\mathchar
58\relax}}=\lim\limits_{t\rightarrow\infty}f(t),\quad\hat{f}\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{\infty}f(t)dt,$

if function $f$ is a non-negative integrable function defined on $0\leq
t<\infty$. It follows the lemma (see [7]),

######  Lemma 4.1.

If $f(t)$ is a non-negative monotone nonincreasing continuously differentiable
function, then as $t\rightarrow\infty$, $f(t)\rightarrow f_{\infty}\geq 0$ and
$f^{\prime}(t)\rightarrow 0$.

Because we have

$(S+E_{i}+E_{a}+I+A)^{\prime}=-\alpha_{i}I-\alpha_{a}A\leq 0,$

this leads to $I(\infty)=0$ and $A(\infty)=0$. We further obtain the number of
all infection cases

$S(0)-S(\infty)=\alpha_{i}\hat{I}+\alpha_{a}\hat{A}.$ (22)

We then add the second equation and the fourth equation, the third equation
and the fifth equation in (18), respectively,

$\displaystyle(E_{i}+I)^{\prime}$
$\displaystyle=af\frac{S}{N}(I+A)-\alpha_{i}I,$
$\displaystyle(E_{a}+A)^{\prime}$
$\displaystyle=a(1-f)\frac{S}{N}(I+A)-\alpha_{i}I.$

To integrate the above equations with respect to $t$ from $0$ to $\infty$
implies

$\displaystyle E_{i}(\infty)+I(\infty)-E_{i}(0)-I(0)$
$\displaystyle=af\frac{S}{N}(\hat{I}+\hat{A})-\alpha_{i}\hat{I},$
$\displaystyle E_{a}(\infty)+A(\infty)-E_{a}(0)-A(0)$
$\displaystyle=a(1-f)\frac{S}{N}(\hat{I}+\hat{A})-\alpha_{a}\hat{A}.$

It implies the linear relation between $\hat{I}$ and $\hat{A}$,

$\hat{I}=\frac{f\alpha_{a}}{(1-f)\alpha_{i}}\hat{A}.$ (23)

From the first equation in model (18), we have

$\displaystyle-\frac{S^{\prime}}{S}$
$\displaystyle=\frac{a}{N}I+\frac{a}{N}A,$
$\displaystyle\log\frac{S(0)}{S(\infty)}$
$\displaystyle=\frac{a}{N}\bar{I}+\frac{a}{N}\bar{A}.$

Using equations (22) and (23), the final size relation for model (18) can be
obtained,

$\displaystyle\log\frac{S(0)}{S(\infty)}$
$\displaystyle=a\left(\frac{f}{\alpha_{i}}+\frac{(1-f)}{\alpha_{a}}\right)\frac{S(0)-S(\infty)}{N},$
(24) $\displaystyle\log\frac{S(0)}{S(\infty)}$
$\displaystyle=\mathcal{R}_{0}\frac{S(0)-S(\infty)}{N}.$

It is noticed that, in Equation (23), $\hat{I}$ and $\hat{A}$ can be
interpreted as the sum of lengths of infection periods for all symptomatic and
asymptomatic patients, respectively. If we define the length of infection
period for symptomatic patient $j$ ($j\in[1,2,\cdots,\textrm{final size of
symptomatic patients}]$) is $\frac{1}{\alpha_{i}}|_{j}$, the law of large
numbers indicates that

$\hat{I}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{j=1}\frac{1}{\alpha_{i}}\bigg{|}_{j}\approx\textrm{final size
of symptomatic patients}\times\frac{1}{\alpha_{i}}.$

Same argument can be made for asymptomatic infections. Apparently the second
part of the above equation holds when the final size of symptomatic patients
is relatively large, and $\frac{1}{\alpha_{i}}$ is the pre-defined mean length
of infection period. Therefore, we have the final size of symptomatic patients
is $\hat{I}\alpha_{i}$ and the final size of asymptomatic patients is is
$\hat{A}\alpha_{a}$. This also implies that the ratio of sizes of symptomatic
and asymptomatic patients is $\frac{f}{1-f}$. In stochastic epidemic modeling
with only symptomatic path (the Sellke stochastic model), a similar term
$\frac{a}{N}\hat{I}$ is defined to measure the total accumulative infection
pressure on a given susceptible individual during the course of the epidemic
([22, 14]).

######  Remark.

For two pathways of symptomatic infection and asymptomatic infection, the
structures (functions $B_{i}$ and $B_{a}$) can be varied significantly. The
stage of pre-symptomatic in the pathway of symptomatic infection has been
frequently included in modeling COVID-$19$ pandemic. For an individual who is
pre-symptomatic, he or she can infect other susceptible individuals. The
infectivity is likely to be even higher than the patients with symptoms ([1]).
Thus, for upper infection pathway in Figure 3, it is reasonable to add one
more compartment ”pre-symptomatic” to describe the intermediate infectious
stage. The corresponding function $B_{i}$ can be constructed
straightforwardly. It is also possible to consider that the stays in the
exposed periods (or in other compartments) follow the Gamma distributions and
assume multiple exposed sub-stages ([3, 10, 23]).

##  5 The age-of-infection model with disease deaths

If we consider there are disease deaths, the total population size does not
remain constant. We denote $N(t)$ is the population size over time and $N_{0}$
is the initial population size. We further assume that a fraction $\mu_{1}$
($\mu_{2}$) of symptomatic (asymptomatic) infectives dies of disease, and
$\mu_{1}>\mu_{2}$. The contact rate $a(N)$ is a density dependent saturating
function ([6, 9]), where $a(N)$ is a non-decreasing function about $N$ and
$\frac{a(N)}{N}$ is a non-decreasing function about $N$. The age-of-infection
epidemic model with disease deaths is

$\displaystyle S^{\prime}$ $\displaystyle=-a(N)\frac{S}{N}(\phi_{i}+\phi_{a})$
(25) $\displaystyle\phi_{i}(t)$
$\displaystyle=\phi_{i,0}(t)+\int_{0}^{t}f[-S^{\prime}(t-\tau)]A_{i}(\tau)d\tau,$
$\displaystyle\phi_{a}(t)$
$\displaystyle=\phi_{a,0}(t)+\int_{0}^{t}(1-f)[-S^{\prime}(t-\tau)]A_{a}(\tau)d\tau.$

Because of the disease deaths, the exact final epidemic size for model (25) is
impossible to obtain. As the basic reproduction number $\mathcal{R}_{0}$ only
depends on the initial states of the epidemic and the population, it can be
explicitly defined as ([11, 5, 6])

$\mathcal{R}_{0}=a(N_{0})\left[f\int_{0}^{\infty}A_{i}(\tau)d\tau+(1-f)\int_{0}^{\infty}A_{a}(\tau)d\tau\right].$
(26)

We now establish the final size relation for model (25) with respect to the
value of $\mathcal{R}_{0}$ in (26). Based on the properties of functions
$a(N)$ and $\frac{a(N)}{N}$ ([7, 6]), $\forall t\in[0,\infty)$, we have:

$\frac{a(N_{0})}{N_{0}}\leq\frac{a(N(t))}{N(t)}\leq\frac{a(N_{\infty})}{N_{\infty}}\leq\frac{a(N_{0})}{(1-\max\\{\mu_{1},\mu_{2}\\})N_{0}},$
(27)

where $N_{\infty}$ is the final size of individuals who survive after the
epidemic ends. We now integrate the first equation in model (25) for
$\frac{S^{\prime}}{S}$ and obtain

$\displaystyle\log\frac{S_{0}}{S_{\infty}}$
$\displaystyle=\int_{0}^{\infty}\frac{a(N(t))}{N(t)}(\phi_{i,0}(t)+\phi_{a,0}(t))dt$
(28)
$\displaystyle+\int_{0}^{\infty}\frac{a(N(t))}{N(t)}\left[f\int_{0}^{t}(-S^{\prime}(t-\tau))A_{i}(\tau)d\tau+(1-f)\int_{0}^{t}(-S^{\prime}(t-\tau))A_{a}(\tau)d\tau\right]dt.$

Because $\frac{a(N(t))}{N(t)}$ is bounded and continuous for $t\in[0,\infty)$,
we are able to find the constant $N_{\star}$ such that $N_{0}\geq
N_{\star}\geq N_{\infty}$, and satisfy

$\displaystyle\log\frac{S_{0}}{S_{\infty}}=$
$\displaystyle\frac{a(N_{\star})}{N_{\star}}\bigg{(}\int_{0}^{\infty}(\phi_{i,0}(t)+\phi_{a,0}(t))dt$
(29)
$\displaystyle+\int_{0}^{\infty}\left[f\int_{0}^{t}(-S^{\prime}(t-\tau))A_{i}(\tau)d\tau+(1-f)\int_{0}^{t}(-S^{\prime}(t-\tau))A_{a}(\tau)d\tau\right]dt\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{a(N_{\star})}{N_{\star}}\bigg{(}\int_{0}^{\infty}(\phi_{i,0}(t)+\phi_{a,0}(t))dt$
$\displaystyle+\int_{0}^{\infty}\left[fA_{i}(\tau)\int_{\tau}^{\infty}(-S^{\prime}(t-\tau))dt+(1-f)A_{a}(\tau)\int_{\tau}^{\infty}(-S^{\prime}(t-\tau))dt\right]d\tau\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{a(N_{\star})}{N_{\star}}\bigg{(}\int_{0}^{\infty}(\phi_{i,0}(t)+\phi_{a,0}(t))dt+(S_{0}-S_{\infty})\int_{0}^{\infty}\left(fA_{i}(\tau)+(1-f)A_{a}(\tau)\right)d\tau\bigg{)}.$

Similarly, if we consider both the initial symptomatic cases and initial
asymptomatic cases are zero or neglectible, the integral term
$\int_{0}^{\infty}(\phi_{i,0}(t)+\phi_{a,0}(t))dt$ can be omitted in equation
(29). The estimate of final epidemic size is

$\log\frac{S_{0}}{S_{\infty}}=\frac{a(N_{\star})N_{0}}{N_{\star}a(N_{0})}\mathcal{R}_{0}\frac{S_{0}-S_{\infty}}{N_{0}}.$
(30)

It is impossible to explicitly calculate the values of $N_{\star}$ and
$a(N_{\star})$, because $N_{\star}$ depends on many factors, such as the
shapes of functions $A_{i}(t)$ and $A_{a}(t)$. However, we can obtain a
relatively accurate estimates by employing the inequalities in (27),

$\mathcal{R}_{0}\frac{S_{0}-S_{\infty}}{N_{0}}\leq\log\frac{S_{0}}{S_{\infty}}\leq\frac{1}{1-\max\\{\mu_{1},\mu_{2}\\}}\mathcal{R}_{0}\frac{S_{0}-S_{\infty}}{N_{0}}.$
(31)

It has been investigated in [2] that, $S_{\infty}$ is uniquely determined by
the value of $\mathcal{R}_{0}$. Indicated by Theorem 4.2 (Lemma $2.1$ and
Lemma $2.2$ in [2]), we have the lower and upper bounds for $S_{\infty}$. The
first part of inequality in equation (31) implies that $S_{\infty}$ is less or
equal to the final susceptible population size, when the basic reproduction
number is taken as $\mathcal{R}_{0}$ and disease-induced deaths are not
considered; the second part of inequality in equation (31) shows that
$S_{\infty}$ is larger or equal to the final susceptible population size, when
the basic reproduction number is taken as
$\frac{1}{1-\max\\{\mu_{1},\mu_{2}\\}}\mathcal{R}_{0}$ and there are no
disease deaths. It is clear to see that, if
$\max\\{\mu_{1},\mu_{2}\\}\rightarrow 0$, the elementary squeeze theorem
indicates that

$\log\frac{S_{0}}{S_{\infty}}\approx\mathcal{R}_{0}\frac{S_{0}-S_{\infty}}{N_{0}}.$

We still argue that, if the ratio $f$ is a constant, the number of total
symptomatic cases is $f(S_{0}-S_{\infty})$ and the number of total
asymptomatic cases is $(1-f)(S_{0}-S_{\infty})$. We have the following theorem
to summarize the analysis.

######  Theorem 5.1.

Assume the epidemic model has both symptomatic infections and asymptomatic
infections, the disease induced death rates are $\mu_{1}$ and $\mu_{2}$,
respectively. If the basic reproduction number is $\mathcal{R}_{0}$, then the
final susceptible population size $S_{\infty}$ has the lower bound of limiting
susceptible population size with basic reproduction number
$\frac{1}{1-\max\\{\mu_{1},\mu_{2}\\}}\mathcal{R}_{0}$ and with no disease
death; the upper bound is the limiting susceptible population size with basic
reproduction number $\mathcal{R}_{0}$ and without disease deaths. The fraction
of total symptomatic cases and total asymptomatic cases is approximately
$\frac{f}{1-f}$.

This theorem is the complement of Theorem $4.1$ in [6], it provides the upper
bound for the estimate of limiting final susceptible population size with
disease deaths considered. It also indicates that when the disease induced
death rates are larger, it is more challenging to accurately predict the
outcome of the epidemic.

##  6 Numerical experiments

In this section, we perform two sets of numerical simulations to verify the
analytical results in previous sections. The size of population is $N=10^{5}$
and the assumed symptomatic ratio $f=0.6$ which indicates that the probability
for an infected individual becoming symptomatic is $60\%$. We further assume
the contact rate $a=0.5$ (/day). Regarding the infected stages, we have
$\kappa_{i}=0.07$ (/day) and $\alpha_{i}=0.2$ (/day) for symptomatic infection
path; and have $\kappa_{a}=0.03$ (/day) and $\alpha_{a}=0.1$ (/day) for
parallel asymptomatic infection path. It is therefore calculated that the
basic reproduction number $\mathcal{R}_{0}=3.5$, and $\mathcal{R}_{i}=2.5$ and
$\mathcal{R}_{a}=5$. Clearly, if the symptomatic ratio $f$ is overestimated,
the estimated $\mathcal{R}_{0}$ is less than the actual basic reproduction
number. Assumptions of standard incidence and exponentially distributed
periods are made to simplify the numerical simulations.

###  6.1 Epidemic models without disease deaths

We first consider there are no disease-induced deaths for both symptomatic
infections and asymptomatic infections. If the ratio $f$ is a constant, we are
able to obtain the exact final epidemic size. The total numbers of symptomatic
patients and asymptomatic patients during the course of the epidemic are
$5796$ and $3863$, respectively. It can be calculated that the ratio of two
numbers is $\frac{f}{1-f}=1.5$. The dynamics is shown in the following Figure
5.

Figure 5: The constant ratio $f$ and the totally predictable progression of
the epidemic.

We next consider the case that the ratio $f$ is a function about time.
Different scenarios are considered for numerical experiments:

  1. 1.

$f$ is a piecewise function and satisfies

$f(t)=\begin{cases}0.6&\text{if $t<=150$,}\\\ 0.5&\text{if
$t>150$.}\end{cases}$ (32)

  2. 2.

$f$ is a piecewise function and satisfies

$f(t)=\begin{cases}0.6&\text{if $t<=150$,}\\\ 0.1&\text{if
$t>150$.}\end{cases}$ (33)

  3. 3.

$f$ is a piece-wise function and satisfies

$f(t)=\begin{cases}0.6&\text{if $t<=150$,}\\\ 0.9&\text{if
$t>150$.}\end{cases}$ (34)

  4. 4.

$f$ is a piece-wise function and satisfies

$f(t)=\begin{cases}0.6&\text{if $t<=300$,}\\\ 0.5&\text{if
$t>300$.}\end{cases}$ (35)

The simulations are presented in Figures 6,7,8,9 and the simulated outcomes
are summarized in Table 1.

Figure 6: The time-dependent ratio $f$ in Equation (32) and the progression of
the epidemic. Figure 7: The time-dependent ratio $f$ in Equation (33) and the
progression of the epidemic. Figure 8: The time-dependent ratio $f$ in
Equation (34) and the progression of the epidemic. Figure 9: The time-
dependent ratio $f$ in Equation (35) and the progression of the epidemic.
Table 1: Summary of the final epidemic sizes. Summary of final sizes predicted
for epidemic models
---
Model type |  Number of total symptomatic cases |  Number of total asymptomatic cases |  Ratio
Model with constant ratio |  $5796$ |  $3963$ |  $1.50$
Model with ratio function (32) |  $5550$ |  $4133$ |  $1.34$
Model with ratio function (33) |  $4516$ |  $5248$ |  $0.86$
Model with ratio function (34) |  $6497$ |  $3081$ |  $2.11$
Model with ratio function (35) |  $5795$ |  $3865$ |  $1.49$

If we consider the simulation results for epidemic model with constant ratio
as the baseline. When the switch of ratio occurs near the end of the epidemic,
it is observed in Figure 9 that the final outcome can be approximated by the
baseline. However, if the switch occurs earlier, the final epidemic can be
significantly different and it is entirely unpredictable.

###  6.2 Epidemic models with disease deaths

We now consider the epidemic models with identical parameters but with a
larger population size $N=10^{6}$ and with disease induced deaths. Firstly, we
assume the death rates are $\mu_{1}=0.2\%$ and $\mu_{2}=0.1\%$ and the
dynamics of the model is simulated in Figure 10. We are also able to obtain
the number of total symptomatic cases $R_{i}(\infty)+D_{i}(\infty)=57966$ and
the number of total asymptomatic cases $R_{a}(\infty)+D_{a}(\infty)=38643$.
Therefore the final size of epidemic is $S(0)-S(\infty)=96609$. The ratio is
$1.50$. The lower bound for final size estimation can be obtained by
simulating epidemic models without disease deaths, and we have
$S(0)-S(\infty)=96597$. If we consider another epidemic model without disease
death and with the basic reproduction number be $\mathcal{R}_{0}/(1-0.2\%)$,
the final epidemic size is $96623$ and this number is the upper bound for the
estimation of final size for the original model with disease deaths. It can be
observed that, since the death rates are very low, two bounds are both
reasonable approximations.

Figure 10: The progression of the epidemic with disease death rates
$\mu_{1,2}=0.2\%,0.1\%$ and the accumulation of death cases.

As a comparison, we next consider the epidemic model with larger death rates
$\mu_{1}=2\%$ and $\mu_{2}=1\%$. The final epidemic size is $96708$, while the
upper bounds and lower bounds are $96724$ and $96597$. If the population size
is larger, it becomes more difficult to accurately predict the final epidemic
size.

##  7 Conclusion and future work

We have formulated an age-of-infection model to study an epidemic with
symptomatic infections and asymptomatic infections. Two separate pathways are
initially determined by the symptomatic ratio $f$, which is the probability
that an individual being infected and eventually becoming symptomatic. The
basic reproduction number $\mathcal{R}_{0}$ is calculated and its value
depends on the symptomatic ratio $f$, the values $\mathcal{R}_{i}$ and
$\mathcal{R}_{a}$. The incorrect inferred value of $f$ leads to the wrongly
predicted final epidemic size. It is however a challenging task to infer these
three values in practice. We then proved that, for the age-of-infection model
with two infection pathways, the final epidemic size is uniquely determined by
$\mathcal{R}_{0}$. The ratio of total numbers of symptomatic patients and
asymptomatic patients is $\frac{f}{1-f}$. For the similar age-of-infection
epidemic model with disease deaths, the calculation of basic reproduction
number $\mathcal{R}_{0}$ is not affected. And the final epidemic size can not
be explicitly generated, but can be approximated by the given lower bounds and
upper bounds. If the death rates for both symptomatic and asymptomatic
patients are smaller, the final epidemic size can be more accurately
predicted. Two sets of numerical simulations were performed for two types of
models.

The age-of-infection epidemic model is the general structure for different
types of compartmental epidemic models ([13]) and it can be extended for other
scenarios, such as the disease spread in an age-structured population with
different activity levels, the impact of vaccination in the population and the
behavioral changes in the population. With respect to the fundamental
analysis, the complete analysis of corresponding characteristic equation for
such models is still very challenging ([13, 4, 8]).

Declaration of Competing Interest: None.

Acknowledgements: The work is dedicated to the author’s mentor Dr. Fred Brauer
($1932$-$2021$). The author is grateful for Fred’s guidance. This work is
based on numerous discussions between the author and Fred. The author also
acknowledges the Post-doctoral fellowship offered by Hausdorff Center for
Mathematics and the University of Bonn.

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