JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.

1 (1-31)
Available online at www.sciencedirect.com

ScienceDirect
J. Differential Equations ••• (••••) •••–•••
www.elsevier.com/locate/jde

Analysis of an epidemiological model structured by

time-since-last-infection
Jorge A. Alfaro-Murillo a , Zhilan Feng b,∗ , John W. Glasser c
a Center for Infectious Disease Modeling and Analysis, Yale School of Public Health, New Haven, CT, USA
b Department of Mathematics, Purdue University, West Lafayette, IN, USA
c National Center for Immunization and Respiratory Diseases, CDC, Atlanta, GA, USA

Received 24 April 2019; revised 3 June 2019; accepted 4 June 2019

Abstract
Modeling time-since-last-infection (TSLI) provides a means of formulating epidemiological models with
fewer state variables (or epidemiological classes) and more flexible descriptions of infectivity after infection
and susceptibility after recovery than usual. The model considered here has two time variables: chrono-
logical time (t) and the TSLI (τ ), and it has only two classes: never infected (N ) and infected at least
once
 (i). Unlike
 most age-structured epidemiological models, in which the i equation is formulated using
∂ + ∂ i(τ, t), ours uses a more general differential operator. This allows weaker conditions for the
∂τ ∂t
infectivity and susceptibility functions, and thus, is more generally applicable. We reformulate the model
as an age dependent population problem for analysis, so that published results for these types of problems
can be applied, including the existence and regularity of model solutions. We also show how other coupled
models having two types of time variables can be stated as age dependent population problems.
© 2019 Elsevier Inc. All rights reserved.

Keywords: Epidemiological model; Age-since-last-infection; Existence and uniqueness of solutions; Stability

* Corresponding author.
E-mail address: zfeng@math.purdue.edu (Z. Feng).

https://doi.org/10.1016/j.jde.2019.06.002
0022-0396/© 2019 Elsevier Inc. All rights reserved.
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.2 (1-31)
2 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

1. Introduction

In many diseases with temporary immunity to reinfection, the infectivity of infected individu-
als and the susceptibility of recovered ones depends on their times since last infection. Ordinary
differential equation systems can model such diseases by adding multiple state variables. Mod-
els structured by time-since-last-infection, considered in [1,2], can instead reduce the number of
variables (or compartments) by using a single time variable for everyone who has been infected
at least once. This approach differs from models structured by age or age-of-infection (see, e.g.,
[3–14]. See also the review in [2]).
The TSLI model considered by Alfaro-Murillo, et al. [2] is a two-dimensional system includ-
ing only two variables: N (t) for the number of never infected people at time t , and i(τ, t) for the
density of those who have been infected at least once, with τ representing their times since last
infection. Let D denote the differentiation operator defined as:

(τ + h, t + h) − (τ, t)
D(τ, t) = lim , (1)
h→0+ h

for any function  that is defined on a subset of R+ × R+ (where R+ is the set of non-negative
real numbers) and has its range defined in a Banach space.We show  in Section 2.1 how the
operator D(τ, t) is a generalization of the partial derivatives ∂τ + ∂t (τ, t). The model reads:
∂ ∂

⎡ ⎤
∞
d i(υ, t) ⎦
N (t) = − ⎣ T (υ) dυ N (t) − μN (t) + μP(t),
dt P(t)
0
⎡ ⎤
∞
i(υ, t) ⎦
Di(τ, t) = − ⎣ T (υ) dυ k(τ )i(τ, t) − μi(τ, t),
P(t)
0
⎡∞ ⎤⎡ ⎤ (2)
 ∞
i(υ, t)
i(0, t) = ⎣ T (υ) dυ ⎦ ⎣N (t) + k(τ )i(τ, t) dτ ⎦ ,
P(t)
0 0
∞
N (0) = N0 , i(τ, 0) = i0 (τ ), P(t) = N (t) + i(τ, t) dτ.
0

There are two time variables in System (2). The first is t , representing chronological time (or
simply time), whereas the second is τ , representing the amount of time that has elapsed since a
person’s most recent infection, referred to as time since last infection (TSLI). N (t) denotes the
total number of individuals in the never-infected class at time t and i(τ, t) denotes the density of
individuals
u2
who have been infected at least once and have TSLI τ at time t . Thus, the quantity
u1 i(τ, t) dτ is the number of individuals at time t whose last infection was between u1 and u2
units of time ago, and P(t) denotes the total population at time t . The only parameters consid-
ered in the model are the per capita natural death rate (μ) and those for the transmission rate
(T (τ )) and infectivity (k(τ )) functions, the latter of which represents a factor of reduction in the
probability of being infected as a function of TSLI.
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.3 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 3

A solution of System (2) is a pair of functions, (N , i) with N : R+ → R+ being differen-

tiable and i : R+ → L1+ (R) being continuous (where L1+ is the space of non-negative Lebesgue
integrable functions, see Definition 3), that solve the equations in System (2) for all t ≥ 0 and
almost everywhere (a.e.) for τ ∈ (0, ∞).
The analysis presented in [2] is for the case when the parameter functions T (τ ) and k(τ )
satisfy stronger conditions than here so that i(τ, t) has continuous partial derivatives and satisfies
a partial differential equation. Specifically, the following system is considered in [2]:
⎡∞ ⎤

d i(u, t)
N (t) = − ⎣ T (u) du⎦ N (t) − μN (t) + μP ,
dt P
0
⎡∞ ⎤ (3a)

 
∂ ∂ i(u, t)
+ i(τ, t) = − ⎣ T (u) du⎦ k(τ )i(τ, t) − μi(τ, t),
∂τ ∂t P
0

with conditions:
⎡∞ ⎤⎡ ⎤
 ∞
i(u, t) ⎦ ⎣
i(0, t) = ⎣ T (u) du N (t) + k(τ )i(τ, t) dτ ⎦ ,
P
0 0 (3b)
∞
N (0) = N0 , i(τ, 0) = i0 (τ ), where P = N (t) + i(τ, t) dτ.
0

In this paper, we present an analysis of the general model (System (2)) with weaker conditions
on T and k, under which the solution i(τ, t) may not be have continuous partial derivatives (see
Theorem 4). This may allow the model to have broader applications. The approach used to study
the general model is to formulate the system as an age dependent population (ADP) problem.
We use the term “ADP problem” to refer to a particular model formulation for age-dependent
populations (specified in Section 2), for which theoretical results are available, including the ex-
istence, uniqueness, positivity, and regularity of solutions. We first introduce another formulation
of general model, termed a coupled model, or a model with two time variables (see Section 2.2).
We illustrate how coupled models can be stated as ADP problems in general, so that all theory
developed for ADP problems can be applied to coupled models.
The paper is organized as follows. In Section 2, we demonstrate the link between ADP prob-
lems and models with two time variables (or coupled models). Properties of solutions to the
generic ADP problem are also discussed in this section, and the results are applied to the refor-
mulation of the general model as a coupled model. Example reformulations  of other models as
coupled models, as well as the relation between D(τ, t) and ∂τ ∂
+ ∂t∂ (τ, t), are also presented.
Application of results in Section 2 to the general model is presented in Section 3, including the
existence and regularity of model solutions. Section 4 includes a discussion of the results.

2. Links between ADP problems and coupled models

In this section, we present solutions to a generic ADP problem and formulate a coupled model
as an ADP problem. Then solution properties of the coupled model are discussed by applying
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.4 (1-31)
4 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

results for ADP problems. Example reformulations of other models as coupled models are also
presented.

2.1. The operator D and its relation to a transport equation

Many age-structured epidemic models are stated in terms of a transport partial differential
equation of the form
∂ ∂
+ (τ, t) = f (), (4)
∂τ ∂t
where f is a given function. The i equation in System (3) is also in this form. Next we explain
why we state the coupled problem in Section 2.3 with the operator D instead.
Classical solutions of a partial differential equation such as Equation (4) are C 1 functions (i.e.,
have continuous partial derivatives). If  ∈ C 1 , we can show that D(τ, t) exists and satisfies
∂ ∂
D(τ, t) = + (τ, t).
∂τ ∂t

Indeed, suppose that  : R+ × [0, t¯ ) → R2 is a C 1 function in a neighborhood of (τ, t). Let

 > 0. There exists δ > 0 such that if 0 < h < δ then
 
∂ 
 (τ, t + h) − ∂ (τ, t) <  ,
 ∂τ ∂τ  5
 
 (τ, t + h) − (τ, t) ∂  
 − (τ, t)< ,
 h ∂t  5

and ∂
∂τ (τ, t + h) exists. Given any such h > 0, there exists h > 0 such that
 
 (τ + h, t + h) (τ + h , t + h)  
 − < ,
 h h  5
 
 (τ + h , t + h) − (τ, t + h) ∂  
 − (τ, t + h)< ,
 h ∂τ  5
 
 (τ, t + h) (τ, t + h)  
 − < .
 h h  5

Therefore,


 (τ + h, t + h) − (τ, t) ∂ ∂ 
 − (τ, t) + (τ, t) 
 h ∂τ ∂t 
   
 (τ + h, t + h) (τ + h , t + h)   (τ, t + h) − (τ, t) ∂ 
≤  −  +  − (τ, t)
h h  h ∂t
   
 (τ + h , t + h) − (τ, t + h)   
+  −
∂
(τ, t + h)  +  (τ, t + h) − (τ, t + h) 
h ∂τ   h  h 
 
∂ ∂ 
+  (τ, t + h) − (τ, t)
∂τ ∂τ
< ,
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.5 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 5

∂ 
for any 0 < h < δ. It follows that D(τ, t) exists and is equal to ∂τ + ∂t∂ (τ, t). Therefore, any
solution to a transport equation such as Equation (4) will also be a solution to the same equation
with the operator D.
The solution function i(τ, t) can be C 1 if adequate conditions are imposed on T and k (see
Theorem 4). However under weaker conditions on T and k we can obtain solutions for i that
are not C 1 and still get information about the number of infected individuals with TSLI between
u
u1 and u2 , as u12 i(τ, t) dτ does not change if the i function has different values on a set with
measure zero in τ . As we do not want to impose extra conditions for T and k to leave the
application of the general model as broad as possible, we will consider the general operator D
and solution functions i to be a continuous L1 -valued function with domain in [0, ∞), that is,
for each non-negative t the function i(·, t) defined as τ → i(τ, t) is L1 .

2.2. The generic ADP problem

We define an ADP problem as described in [15, Chapter 1]. An ADP problem is described by
the following three equations:

D(τ, t) = G((·, t))(τ ), (5a)

(0, t) = F ((·, t)), (5b)

(τ, 0) = φ(τ ), (5c)

with G : L1 → L1 , F : L1 → Rn , and φ ∈ L1 . In ADP problems Equations (5a), (5b) and (5c)

are termed the Balance Law, the Birth Law, and the initial condition, respectively.
For ease of presentation, we introduce the following definition:

Definition 1. For t¯ > 0, let Lt¯ = C([0, t¯ ]; L1 ) be the Banach space of continuous L1 -valued
functions on [0, t¯ ] with the norm:


Lt¯ = sup
(t)
,
0≤t≤t¯

where  ∈ Lt¯.

In a natural way, an element of Lt¯ can be identified with an element of L1 ((0, ∞) ×(0, t¯ ); Rn )
[15, Lemma 2.1], which allows us to use the same symbol for both; i.e.,

(t)(τ ) = (·, t)τ = (τ, t),

where 0 ≤ t ≤ t¯, and a.e. τ > 0.

Definition 2. Let t¯ > 0. Let F : L1 → Rn , G : L1 → L1 , and φ ∈ L1 . We say that a function

 ∈ Lt¯ is a solution of the ADP problem for the initial distribution φ on [0, t¯ ] provided that 
satisfies the equations in System (5) for all t ∈ [0, t¯ ] and a.e. for τ ∈ (0, ∞).
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.6 (1-31)
6 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

If we assume that  is a solution of the ADP problem on [0, t¯ ] and c ∈ R, then we can define
a “cohort function”:

wc (t) = (t + c, t)

for every tc ≤ t ≤ t¯, where tc = max{−c, 0}. Using Equation (5a), we can show that the right
derivative of this function exists and satisfies

wc (t + h) − wc (t)
wc (t+) = lim = G((·, t))(t + c) (6)
h→0+ h

a.e. for t ∈ (tc , t¯). If G is Lipschitz on norm-balls of L1 , the function G((·, t))(τ ) is integrable
as a function from (0, ∞) × (0, t¯ ) to Rn [15, Lemma 2.2], and so wc (t+) is also integrable in
[0, t¯ ]. Therefore, we have that any function of the form,

t
t → C + wc (s+) ds,
tc

has a derivative equal to wc (t+) a.e. t ∈ (tc , t¯) [16, Chapter 5, Theorem 10]. So, we can integrate
Equation (6) and obtain
 t
wc (t − τ ) + t−τ G((·, s))(s + c) ds a.e. τ ∈ (0, t),
wc (t) = t
wc (0) + 0 G((·, s))(s + c) ds a.e. τ ∈ (t, ∞).

Substituting c = τ − t, and using Equation (5b), we obtain the integral equation:

 t
F ((·, t − τ )) + t−τ G((·, s))(s + τ − t) ds a.e. τ ∈ (0, t),
(τ, t) = t (7)
φ(τ − t) + 0 G((·, s))(s + τ − t) ds a.e. τ ∈ (t, ∞).

In conclusion, if G is Lipschitz on norm-balls of L1 , every solution of the ADP problem sat-

isfies Equation (7). Clearly, not every solution of Equation (7) is a solution of the ADP problem,
because the function  in Equation (7) need not be differentiable in the sense of the operator D.
The converse is true under certain conditions (see Theorem 2.9 in [15] and Theorem 2.3 in [17]),
a fact that we will use later.
If both functions F and G are Lipschitz on norm-balls of L1 , then a function  satisfies Equa-
tion (7), for t ∈ [0, t¯ ], if and only if  is a mild solution of the ADP problem (See Theorem 2.2
in [15]) according to Definitions 4–6 in the Appendix.
We define an equilibrium solution for the ADP problem in Definition 7 of the Appendix.
A very important result in the theory of ADP problems is that, if F : L1+ → Rn+ and G : L1+ → L1
are Lipschitz on norm-balls of L1 and there exists a function c3 that satisfies (ii) in the proof of
part (c) for Proposition 2, then φ is an equilibrium solution of the ADP problem if and only if
φ is absolutely continuous with the properties that φ  ∈ L1 , φ  = G(φ), and φ(0) = F (φ) [15,
Proposition 4.1]. We will make use of this result later.
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.7 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 7

2.3. General formulation of coupled models

In this section, we focus on models consisting of both equations that depend only on time t
and variables that depend on both time t and τ (System (2) is an example). For ease of refer-
ence, we refer to this type of model as a coupled model. Several other examples are provided in
Section 2.6. A general formulation for such a system is given below.
Let X(t) denote the vector of functions that depend only on t , and let y(τ, t) denote the vector
of functions that depend on both t and τ . The general coupled model has the following form:

dX(t)
= Fx (X(t), y(·, t)) + Mx (X(t), y(·, t))X(t),
dt (8a)
Dy(τ, t) = Gy (X(t), y(·, t))(τ ),

with boundary and initial conditions

y(0, t) = Fy (X(t), y(·, t)), X(0) = X0 , y(·, 0) = φy , (8b)

where Fx : Rm × L1 (Rk ) → Rm , Mx : Rm × L1 (Rk ) → B(Rm , Rm ), Gy : Rm × L1 (Rk ) →

L1 (Rk ), Fy : Rm × L1 (Rk ) → Rk , X0 ∈ Rm and φy ∈ L1 (Rk ). The operator D is defined in
Equation (1).
A solution to System (8) is a set of functions X(t) and y(τ, t) that satisfy the equations for
time t ∈ [0, t¯ ] for some t¯ > 0 and a.e. for τ ∈ (0, ∞). An equilibrium of the system is a solution
that is constant on time t .

2.4. From coupled models to ADP problems and solution properties

We can reformulate the coupled model (System (8)) as an ADP problem described in Sys-
tem (5) by defining the functions F : L1 (Rm+k ) → Rm+k and G : L1 (Rm+k ) → L1 (Rm+k ) as

  ∞ 
φx Fx φx (τ ) dτ, φy
F = 0∞  , (9a)
φy Fy 0 φx (τ ) dτ, φy

  ∞  
φ Mx φx (υ) dυ, φy φ x (τ )
G x (τ ) = 0 ∞  , (9b)
φy Gy 0 φx (υ) dυ, φy (τ )


φx ∞
where φ = with φx ∈ L1 (Rm ) and 0 φx (τ )dτ = X0 .
φy
Let π (m) and π (−k) denote the projection functions in Definition 8 of the Appendix. Then the
following result holds:

Theorem 1. Consider System (8) as an ADP problem (System (5)) with F and G being defined
as in System (9). Assume that F and G are Lipschitz on norm-balls of L1 . If the ADP problem
has a solution  ∈ Lt¯ for the functions F and G and the initial condition φ, then System (8) has
a solution X(t), y(τ, t) for t ∈ [0, t¯ ] and a.e. for τ ∈ (0, ∞), given by
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.8 (1-31)
8 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

⎛∞ ⎞

X(t) = π (m) ⎝ (τ, t) dτ ⎠ and y(·, t) = π (−k) ((·, t)) .
0

Proof. Let t¯ > 0 such that  ∈ Lt¯ is a solution of the ADP problem on [0, t¯ ] for the functions F ,
G and the initial condition φ. Define
⎛∞ ⎞

X(t) = π (m) ⎝ (τ, t) dτ ⎠ and y(·, t) = π (−k) ((·, t)) .
0

Applying π (m) to Equation (5c), we have

π (m) ((τ, 0)) = φx (τ );

integrating, we obtain Equation (8b). Applying π (−k) to Equation (5a) and using the definition of
G in Equation (9b), we obtain the y(τ, t) in Equation (8a). In the same way, from Equation (5b)
and the definition of F in Equation (9a), we obtain the y(0, t) in Equation (8b). Also, applying
π (−k) to Equation (5c) yields the y(·, 0) in Equation (8b).
It remains to show that X satisfies Equation (8a). Notice that
⎛ ⎞
∞
π (m) ⎝F ((·, t)) + G((·, t))(τ ) dτ ⎠ = Fx (X(t), y(·, t)) + Mx (X(t), y(·, t)) X(t).
0

Thus, it suffices to show that

⎛ ⎞
∞
d
X(t) = π (m) ⎝F ((·, t)) + G((·, t))(τ ) dτ ⎠ .
dt
0

Recall from Section 2.2 that, if F and G are Lipschitz on norm-balls of L1 , a solution of the
ADP problem is also a mild solution of the ADP problem. Hence, if h > 0, then
 ⎛ ⎞
 ∞ 
 −1 
h [X(t + h) − X(t)] − π (m) ⎝F ((·, t)) + G((·, t))(τ ) dτ ⎠
 
 
0
⎛∞ ⎞ ⎛ ⎞
  ∞ ∞ 
 −1 (m) 

= h π ⎝ ⎠
(τ, t + h) dτ − (τ, t) dτ − π (m) ⎝
F ((·, t)) − G((·, t))(τ ) dτ 
⎠
0 0 0
 ⎛ h ⎞
  
 −1 (m) 
≤ h π ⎝ (τ, t + h) − F ((·, t)) dτ ⎠
 
0
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.9 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 9

 ⎛∞ ⎞
  
 (m) 

+ π ⎝ h [(τ + h, t + h) − (τ, t)] − G((·, t))(τ ) dτ 
−1 ⎠
 
0

h ∞ 
−1  
≤h |(τ, t + h) − F ((·, t))| dτ + h−1 [(τ + h, t + h) − (τ, t)] − G((·, t))(τ ) dτ,
0 0

which tends to zero as h → 0+ by the limit equations in Definition 4 of the Appendix. This shows
that the right derivative of X exists and is equal to
⎛ ⎞
∞
π (m) ⎝F ((·, t)) + G((·, t))(τ ) dτ ⎠ for t ∈ [0, t¯ ].
0

For the left derivative, let h > 0. Using similar estimates as for the right derivative, we obtain
 ⎛ ⎞
 ∞ 
 −1   
h X(t) − X(t − h) − π (m) ⎝F ((·, t)) + G((·, t))(τ ) dτ ⎠
 
 
0
 
 h 
 −1 

≤ h (τ, t) dτ − F ((·, t))
 
0
 ⎡∞ ⎤ 
  ∞ ∞ 
 −1 

+ h ⎣ (τ, t) dτ − (τ, t − h) dτ − G((·, t))(τ ) dτ  .
⎦
 
h 0 0

The first factor in the last sum goes to zero as h → 0+ by the Fundamental Theorem of Calculus
and the fact that  is a solution of the ADP problem (in particular Equation (5b)):

h
−1
lim h (τ, t) dτ = (0, t) = F ((·, t)).
h→0+
0

For the second factor, recall that, if F and G are Lipschitz on norm-balls of L1 , then  is a
mild solution of the ADP problem if and only if it satisfies the integral equation of the problem,
Equation (7) [15, Theorem 2.2]. Using Equation (7), for any 0 < h < min {τ, t}, we have

t
(τ, t) − (τ − h, t − h) = G((·, s))(s + τ − t) ds,
t−h

and for h < t,

JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.10 (1-31)
10 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

 
 ∞ ∞  ∞ 
 −1  
h (τ, t) dτ − (τ, t − h) dτ − G((·, t))(τ ) dτ 

 
h 0 0
 
 ∞  ∞ 
 −1  

= h (τ, t) − (τ − h, t − h) dτ − G((·, t))(τ ) dτ 
 
h 0
 
 ∞  t  ∞ 
 −1  
= h G((·, s))(s + τ − t) ds dτ − G((·, t))(τ ) dτ 
 
h t−h 0
∞ 
  t  

=  h−1 G((·, s))(s + τ + h − t) − G((·, t))(τ ) ds dτ 
 
0 t−h
 
∞ t   
 −1 
≤ h G((·, s))(s + τ + h − t) − G((·, t))(τ ) ds  dτ

 
0 t−h

 t ∞
−1
≤h |G((·, s))(s + τ + h − t) − G((·, t))(s + τ + h − t)| dτ ds
t−h 0

 t ∞
−1
+h |G((·, t))(s + τ + h − t) − G((·, t))(τ )| dτ ds
t−h 0

∞
≤ sup
G((·, s)) − G((·, t))
+ sup |G((·, t))(s + τ + h − t) − G((·, t))(τ )| dτ.
t−h≤s≤t t−h≤s≤t
0

In the last inequality, the first factor in the sum tends to zero as h → 0+ because the function
t → G((·, t)) is continuous [15, Lemma 2.2]. The second factor tends to zero by the continuity
of the translation in L1 . 2

2.5. Equilibrium solutions of the coupled model and ADP problem

For any coupled model where Theorem 1 can be applied, an equilibrium solution of the re-
spective ADP problem translates into an equilibrium solution of the coupled model by applying
the projection π (m) and integrating to obtain the equilibrium for X or applying the projection
π (−k) to obtain the equilibrium for y. In some cases, those are the only equilibrium solutions of
the coupled model, as stated in the following theorem.

Theorem 2. Consider System (8) as an ADP problem (System (5)) by letting F and G be as
defined in System (9). Assume that F and G are Lipschitz on norm-balls of L1 . If the ADP
problem has an equilibrium solution φ, then
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.11 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 11

⎛∞ ⎞

X0 = π (m) ⎝ φ(τ ) dτ ⎠ , φy (τ ) = π (−k) (φ(τ ))
0

is an equilibrium solution of the System (8).

Conversely, suppose that X0 , φy is an equilibrium solution of System (8) such that

(i) φy is absolutely continuous,

(ii) φy ∈ L1 , and
(iii) all eigenvalues of Mx (X0 , φy ) have negative real parts.

Then,

eMx (X0 ,φy )τ Fx (X0 , φy )
φ(τ ) =
φy (t)

is an equilibrium solution of the ADP problem.

Proof. Under the assumptions of the theorem, if the ADP problem has an equilibrium solution
φ, then we can apply Theorem 1 to obtain a solution of the coupled model (System (8)). Because
the equilibrium solution of the ADP problem does not depend on t , neither will the solution of
the coupled model.
On the other hand, let (X0 , φy ) be an equilibrium solution of the coupled model (System (8))
that satisfies (i), (ii) and (iii). Define


M (X ,φ )τ 
φx (τ ) e x 0 y Fx (X0 , φy )
φ(τ ) = = .
φy (τ ) φy (τ )

Then

τ̄
φx (τ ) dτ = (Mx (X0 , φy ))−1 eMx (X0 ,φy )τ̄ Fx (X0 , φy ) − (Mx (X0 , φy ))−1 Fx (X0 , φy ).
0

The inverse (Mx (X0 , φy ))−1 exists because we are assuming that all eigenvalues of the matrix
Mx (X0 , φy ) have negative real parts. Moreover, if all eigenvalues of a square matrix A have
negative real parts, then limτ →∞ eAτ x0 = 0 for any vector x0 of the same dimension as A [18,
Chapter 1, Theorem 2]. Thus,

∞
φx (τ ) dτ = −(Mx (X0 , φy ))−1 Fx (X0 , φy ).
0

By Equation (8a) and the fact that, if (X0 , φy ) is an equilibrium solution of the coupled model,
then it satisfies X (t) = 0, we have

−(Mx (X0 , φy ))−1 Fx (X0 , φy ) = X0 .

∞
Hence, 0 φx (τ ) dτ = X0 .
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.12 (1-31)
12 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

From the definition of φx and the fact that φy is absolutely continuous, φ is absolutely con-
tinuous. Moreover,
φx (x) = Mx (X0 , φy )φx (τ ),

so φx ∈ L1 . Also, from the assumption that φy ∈ L1 , we have that φ  ∈ L1 .

Now we can show that φ is indeed a solution of the ADP problem:



Fx (X0 , φy ) Fx (X0 , φy )
φ(0) = = = F (φ)
φy (0) Fy (X0 , φy )

and

 Mx (X0 , φy )φx (τ )
Dφ(τ ) = φ (τ ) = = G(φ)(τ ),
φy (τ )

where F and G are as in System (9). 2

2.6. Other examples of coupled models

Example 1. Brauer, et al. [4] studied a model of cholera that has three epidemiological classes:
susceptible individuals (S(t), only dependent on time), infected individuals (i(t, ·), structured by
time since infection), and contaminated water (p(t, ·), structured by the time that the pathogen
has been in the water). Let


i(·, t)
X(t) = S(t), y(·, t) = .
p(·, t)

Then the functions corresponding to those in System (8) are:

Fx (X(t), y(·, t)) = A,
∞

Mx (X(t), y(·, t)) = −μ − βd k(τ ) βi q(τ ) y(·, t) dτ,
0


θ (τ ) 0
Gy (X(t), y(·, t))(τ ) = − y(·, t),
0 δ(τ )
∞

βd k(τ ) βi q(τ )
Fy (X(t), y(·, t)) = X(t) y(τ, t) dτ,
ξ(τ ) 0
0

with the initial conditions:


i0
X0 = S 0 , φy = .
p0

Example 2. Bhattacharya and Adler [19] describe an SIRS model in which the susceptible S(t)
and infected I (t) classes depend only on time, whereas the recovered class R(·, t) is structured
by time since recovery. Their model can be formulated in the form of System (8). Let


S(t)
X(t) = , y(·, t) = R(·, t).
I (t)
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.13 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 13

The corresponding functions are

⎛ ⎞
∞
⎜ ρ(τ )y(τ, t) dτ ⎟
Fx (X(t), y(·, t)) = ⎜
⎝
⎟,
⎠
0
0


−βπ2 (X(t)) 0
Mx (X(t), y(·, t)) = ,
βπ2 (X(t)) −γ
Gy (X(t), y(·, t))(τ ) = −ρ(τ )y(τ, t),
Fy (X(t), y(·, t)) = γ π2 (X(t)),

where π2 is the projection defined as πi (x1 , x2 , · · · , xn ) = xi , with the initial conditions


S0
X0 = , φy = 0.
I0

Example 3. Magal and McCluskey [20] describe a two-group SIR model in which there are two
susceptible classes (S1 and S2 ) and two recovered classes (R1 and R2 ) that depend only on time,
and two infected classes (i1 (·, t) and i2 (·, t)) that are structured by the time since infection. Their
model can be formulated in the form of System (8) by letting
⎛ ⎞
S1 (t)

⎜ S2 (t) ⎟ i1 (·, t)
X(t) = ⎜ ⎟
⎝ R1 (t) ⎠ , y(·, t) = .
i2 (·, t)
R2 (t)

The corresponding functions are

⎛ ⎞
∞
⎜  − π (2) (X(t)) • B(τ )y(τ, t) dτ ⎟
⎜ ⎟
⎜ ⎟
⎜
Fx (X(t), y(·, t)) = ⎜ 0 ⎟,
 ∞ ⎟
⎜ ⎟
⎝ M(τ )y(τ, t) dτ ⎠
0


D 0
Mx (X(t), y(·, t)) = − ,
0 D
Gy (X(t), y(·, t))(τ ) = −(M(τ ) + D)y(τ, t),
∞ ∞
Fy (X(t), y(·, t)) = π (X(t)) • B(τ )y(τ, t) dτ M(τ )y(τ, t) dτ,
(2)

0 0

where • represents the dot product of vectors, π (2) is the projection defined as

π (m) (x1 , x2 , · · · , xn ) = (x1 , x2 , · · · , xm ),

with 0 < m < n, and with the notation

JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.14 (1-31)
14 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••



λ1 0 β2 (τ )
= , B(τ ) = ,
λ2 β1 (τ ) 0



m1 (τ ) 0 d1 0
M(τ ) = , D= .
0 m2 (τ ) 0 d2

The initial conditions are


S0
X0 = , φy = i0 .
R0

3. Solution properties of the general model

We first simplify the general model (System (2)) by showing that the total population size
P(t) remains constant for all t ≥ 0 and then analyze it by reformulating it as a coupled model.

3.1. Simplification of the general model

∞
Let N0 ∈ R+ and i0 ∈ L1+ (R) be such that N0 + 0 i0 (τ ) dτ > 0. A solution of the general
model (System (2)) is a pair of functions, N (t) : R+ → R+ differentiable and i(·, t) : R+ →
L1+ (R) continuous, that solve the equations in System (8a) for all t ≥ 0 and a.e. for τ ∈ (0, ∞).

∞
Assumption 1. Let T , k : R+ → R+ be bounded functions such that T (τ )dτ > 0 and
0
∞
(i) T (τ )dτ < ∞ or
(10)
0
(ii) k(τ ) = 0 a.e for τ > 0.

Proposition 1. Let T , k satisfy Assumption 1. For any solution (N (t), i(τ, t)) of System (2), the
total population remains constant; i.e., P(t) = P , where
∞
P = N0 + i0 (τ ) dτ. (11)
0

Proof. Suppose that (N , i) is a solution of System (2). To simplify the notation, define
∞
i(υ, t)
B(t) = T (υ) dυ
P(t)
0

and wc (t) = i(t + c, t) for any c ∈ R and t ≥ tc , where tc = max {−c, 0}. Note that

wc (t + h) − wc (t) i(t + c + h, t + h) − i(t + c, t)

lim = lim
h→0+ h h→0 + h
= Di(t + c, t)
= −B(t)k(t + c)wc (t) − μwc (t).
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.15 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 15


∞ of wc (t+) exists a.e. From Assumption 1, we know that B(t)k(t + c)
Thus, the right derivative
is either bounded by 0 T (τ ) dτ × supτ {k(τ )} or is zero a.e. Therefore, wc (t+) is integrable in
[0, t¯ ] for any t¯ > 0, whenever wc (t) is integrable in [0, t¯ ]. Because i : R+ → L1+ (R) is continu-
ous, this is the case for any t¯ > 0. So, we can integrate wc (t+) to obtain that wc satisfies a.e. the
integral equation:
t  
wc (t) = − B(s)k(s + c)wc (s) + μwc (s) ds + wc (tc );
tc

that is,
⎧
⎪ t  
⎪
⎪
⎪wc (0) −
⎪ B(s)k(s + c)w (s) + μw (s) ds if c > 0,
⎪
⎨
c c

wc (t) = 0
⎪
⎪ t  
⎪
⎪
⎪
⎪w c (−c) − B(s)k(s + c)wc (s) + μwc (s) ds if c < 0.
⎩
−c

Letting τ = t + c and using the i(0, t) and i(τ, 0) equations in System (2), we obtain:
t  
i(τ, t) = i0 (τ − t) − B(s)k(τ − t + s)i(τ − t + s, s) + μi(τ − t + s, s) ds,
0

a.e. for τ < t, and

⎡ ⎤
∞
i(τ, t) = B(t − τ ) ⎣N (t − τ ) + k(υ)i(υ, t − τ ) dυ ⎦
0
t  
− B(s)k(τ − t + s)i(τ − t + s, s) + μi(τ − t + s, s) ds,
t−τ

a.e. for τ > t. Integrating, we have

⎡ ⎤
∞ t ∞
i(τ, t) dτ = B(t − τ ) ⎣N (t − τ ) + k(υ)i(υ, t − τ ) dυ ⎦ dτ
0 0 0
t t  
− B(s)k(τ − t + s)i(τ − t + s, s) + μi(τ − t + s, s) dsdτ
0 t−τ
(12)
∞
+ i0 (τ − t) dτ
t
∞ t  
− B(s)k(τ − t + s)i(τ − t + s, s) + μi(τ − t + s, s) dsdτ.
t 0
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.16 (1-31)
16 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

Changing the limits of integration and making the change of variable υ = τ − t + s yields

t t  
B(s)k(τ − t + s)i(τ − t + s, s) + μi(τ − t + s, s) dsdτ
0 t−τ
 t s  
= B(υ)k(υ)i(υ, s) + μi(υ, s) dυ ds,
0 0

and

∞ t  
B(s)k(τ − t + s)i(τ − t + s, s) + μi(τ − t + s, s) dsdτ
t 0
 t ∞ 
= B(υ)k(υ)i(υ, s) + μi(υ, s) dυ ds.
0 s

Using s = t − τ and υ = τ − t in the other two integrals of Equation (12), we get

∞ t t ∞
i(τ, t) dτ = B(s)N (s) ds − μ
i(·, s)
ds + i0 (υ) dυ. (13)
0 0 0 0

Integrating the N equation in System (2), we obtain

t t
N (t) = − B(s)N (s) ds + μ
i(·, s)
ds + N0 . (14)
0 0

Finally, adding Equation (13) and Equation (14), we complete the proof. 2

3.2. The general model as an ADP problem

Proposition 1 allows us to reduce System (2) to the following simpler system (i.e., replacing
the function P(t) by the constant P ):
⎡∞ ⎤

d i(υ, t) ⎦
N (t) = − ⎣ T (υ) dυ N (t) − μN (t) + μP ,
dt P
0
⎡∞ ⎤

i(υ, t)
Di(τ, t) = − ⎣ T (υ) dυ ⎦ k(τ )i(τ, t) − μi(τ, t),
P
0
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.17 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 17

⎡ ⎤⎡ ⎤
∞ ∞
i(υ, t) ⎦ ⎣
i(0, t) = ⎣ T (υ) dυ N (t) + k(τ )i(τ, t) dτ ⎦ , (15)
P
0 0
N (0) = N0 , i(τ, 0) = i0 (τ ),
∞
P = N0 + i0 (τ )dτ.
0

We can then rewrite System (15) as a coupled model as in System (8), which can be studied as
an ADP problem. Let

∞
Fx (X, φy ) = μX + μ φy (τ ) dτ,
0
∞
φy (τ )
Mx (X, φy ) = − dτ − μ,
T (τ )
P
0
⎡∞ ⎤ (16)

φy (τ, t)
Gy (X, φy ) = − ⎣ T (τ ) dτ ⎦ k(τ )φy (τ, t) − μφy (τ ),
P
0
⎡∞ ⎤⎡ ⎤
 ∞
φy (τ )
Fy (X, φy ) = ⎣ T (τ ) dτ ⎦ ⎣X + k(τ )φy (τ ) dτ ⎦ ,
P
0 0

with X0 = N0 , φy = i0 .
For ease of presentation, we introduce the following notation and functions:

(i) For 0 < t¯ ≤ ∞ and φ ∈ L1 ((0, t¯ ), R2 ), let

φ n = π1 ◦ φ, φ i = π2 ◦ φ,

where π1 , π2 are the projections to the first and second entries as in Definition 8. Thus, φ n
and φ i are the never-infected part of φ and the infected-at-least-once part of φ, respectively.
(ii) Let F : L1 → R denote the function

∞
φ i (τ )
F(φ) = T (τ ) dτ. (17)
P
0

(iii) Let W : L1 → R denote the weighted function

∞ 
W(φ) = φ n (τ ) + k(τ )φ i (τ ) dτ. (18)
0
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.18 (1-31)
18 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

(iv) Let W : L1 → R denote the non-weighted function

∞ 
W(φ) = φ n (τ ) + φ i (τ ) dτ. (19)
0

Notice that W(φ) =

φ
if φ ∈ L1+ .

Let
⎛ ⎞
∞ 
⎜ μ φ n (τ ) + φ i (τ ) dτ ⎟
⎜ ⎟

 ⎜ ⎟
φn ⎜ 0 ⎟
F ⎜
= ⎜⎡ ⎟
φi ⎤ ⎟
⎜ ∞  
∞
 ⎟
⎜ φ i (υ) ⎟ (20a)
⎝ ⎣ T (υ) dυ ⎦ φ n (τ ) + k(τ )φ i (τ ) dτ ⎠
P
0 0


μW(φ)
= ,
F(φ)W(φ)
⎛ ⎡∞ ⎤ ⎞
 i (υ)
φ
⎜ − ⎣ T (υ) dυ ⎦ φ n (τ ) − μφ n (τ ) ⎟
⎜ P ⎟

n ⎜ ⎟
φ ⎜ 0 ⎟
G (τ ) = ⎜ ⎟
φ i ⎜ ⎡ ⎤ ⎟
⎜ ∞ ⎟
⎜ i ⎟ (20b)
⎝ − ⎣ T (υ) φ (υ) dυ ⎦ k(τ )φ i (τ ) − μφ i (τ ) ⎠
P
0


−F (φ)φ n (τ ) − μφ n (τ )
= .
−F (φ)S(τ )φ i (τ ) − μφ i (τ )

Using the F and G functions defined in System (20), we can translate System (15) into an ADP
problem (see System (5) and System (9)). Thus, to apply the results in Section 2 to describe
solution properties of System (15), we focus in the following section on the properties of the
functions F and G given in System (20).

3.3. Basic results for the ADP version of the general model

For ease of presentation, we state in this section some preliminary results that we will use in
the next section to obtain our main results.

Proposition 2. Let P > 0, μ ≥ 0, and T , k : R+ → R+ be bounded. The following results hold:

(a) The functions F, W and W defined in Equations (17)–(19) are bounded linear operators.
Moreover,

supτ {T (τ )}

F
op ≤ ,
W
op ≤ sup{k(τ )},
W
op = 1.
P τ
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.19 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 19

(b) If φ ∈ L1 , then there exists 0 < t¯ ≤ ∞ and  ∈ Lt¯ such that  is the unique mild solution
of the ADP problem on [0, t¯ ] for the functions F, G given in System (20) and the initial
distribution φ.
(c) If φ ∈ L1+ , then the mild solution  of the ADP problem on [0, t¯φ ) for the functions F, G
given in System (20), the initial distribution φ and t¯φ is as in Definition 5, has the property
that (·, t) ∈ L1+ for 0 ≤ t < t¯φ .

The proof can be found in Appendix B.1.

Proposition 3. Let P > 0, μ ≥ 0, T , k : R+ → R+ be bounded, and φ ∈ L1 . Let  be the mild

solution of the ADP problem on [0, t¯φ ) for the functions F, G given in System (20) and the
initial condition φ, where t¯φ is as in Definition 5. Then W((·, t)) is constant for all 0 ≤ t < t¯φ .
Additionally, if φ ∈ L1+ , then
(·, t)
=
φ
for all 0 < t < t¯φ .

The proof can be found in Appendix B.2.

Proposition 4. Let P > 0, μ ≥ 0, T , k : R+ → R+ be bounded, and φ ∈ L1+ . Let  be the mild

solution of the ADP problem on [0, t¯φ ) for the functions F, G given in System (20) and the initial
condition φ, where t¯φ is as in Definition 5. Then t¯φ = ∞.

The proof can be found in Appendix B.3.

3.4. Existence and regularity of the model solution

Based on the results stated in the previous section, we describe the properties of the solu-
tions to the general model (System (15)). Definitions for some of the terms can be found in
Appendix A. For example, a function being globally Lipschitz (Definition 9) and F-differentiable
(Definition 10).

Proposition 5. Let P > 0, μ ≥ 0. Let T : R+ → R+ be a bounded function, and let k : R+ →

R+ be a bounded globally Lipschitz function. Let φ ∈ L1+ be a continuous function such that
φ(0) = F (φ). Then there exists a unique continuous function  : R+ → L1+ that is the solution
of the ADP problem for the functions F and G given in System (20) and the initial condition φ.

The proof can be found in Appendix B.4.

The results described in Proposition 5 can be translated back to our original problem to obtain
the first theorem of existence (and regularity) of solutions:

Theorem 3. Let μ ≥ 0. Let T : R+ → R+ be a function and k : R+ → R+ a globally Lipschitz

function satisfying Assumption 1. Let N0 > 0 and let i0 : R+ → R+ be a continuous function
such that i0 ∈ L1+ and
⎡ ⎤⎡ ⎤
∞ ∞
i0 (υ)
i0 (0) = ⎣ T (υ) ∞ dυ ⎦ ⎣N0 + k(τ )i0 (τ ) dτ ⎦ . (21)
N0 + 0 i0 (τ ) dτ
0 0

Then there exists a differentiable function N : R+ → R+ and a continuous function i : R+ →

L1+ (R) that solve System (15).
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.20 (1-31)
20 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

Proof. Let φ n be any continuous function from R+ to R+ such that

∞
φ (0) = μP ,
n
φ n (τ ) dτ = N0 .
0

Because we showed in Proposition 2 that F and G given in System (20) are Lipschitz on
norm-balls of L1 , we can use Theorem 1 to translate results of a solution of the ADP problem
for the functions F and G and initial condition

n
φ
φ=
i0

to results for solutions of the System (15), and by Proposition 1, of the general model (Sys-
tem (2)).
The first result is existence of a solution. We know that φ ∈ L1+ is continuous, and by the
definition of φ and Equation (21), we have



φ n (0)  μP 
φ(0) = = ∞ = F (φ).
i0 (0) F(φ) N0 + 0 k(τ )i0 (τ ) dτ

So, given the hypothesis of Proposition 5, we can conclude that there is a solution for Sys-
tem (15). Moreover, this solution is defined for all t ∈ R+ , and satisfies
⎛∞ ⎞

N (t) = π1 ⎝ (τ, t) dτ ⎠ ,
0
i(τ, t) = π2 ((τ, t))

where  is the solution of the ADP problem.

By Part (c) of Proposition 2, (·, t) ∈ L1+ for all t ∈ R+ , so N (t) ≥ 0 and i(·, t) ∈ L1+ as
required. 2

This result is not very restrictive in the conditions imposed on the initial distribution. We only
require it to be continuous, L1 and to satisfy the non-local boundary condition. However, we are
imposing an additional restriction on the susceptibility function, k, namely for it to be globally
Lipschitz. We can dispense with this so long as we impose a stronger condition on the initial
distribution. Our regularity results will then be stronger for the solution of the ADP problem. For
this, we first need to show that our functions F and G are continuously F-differentiable.

Proposition 6. Let P > 0, μ ≥ 0. Let T , k : R+ → R+ be bounded functions. Then

1. The function F : L1 → R2 defined by Equation (20a) is a continuously F-differentiable func-

tion relative to L1 . Its F-derivative is given by


 μW(φ)
F (φ0 )(φ) = .
F(φ0 )W(φ) + F(φ)W(φ0 )
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.21 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 21

2. The function G : L1 → L1 defined by Equation (20b) is a continuously F-differentiable func-

tion relative to L1 . Its F-derivative is given by


 F(φ0 )φ n (τ ) + F(φ)φ0n (τ ) + μφ n (τ )
G (φ0 )(φ)(τ ) = − .
F(φ0 )k(τ )φ i (τ ) + F(φ)k(τ )φ0i (τ ) + μφ i (τ )

The proof can be found in Appendix B.5.

Proposition 7. Let P > 0, μ ≥ 0, and let T , k : R+ → R+ be bounded. Let φ ∈ L1+ be absolutely

continuous such that φ  ∈ L1 and φ(0) = F (φ). Then there exists a unique solution, , of the ADP
problem for the F, G given in System (20) and the initial condition φ, such that

(a) (·, t) is absolutely continuous for any t ∈ R+ .

(b) For every t ∈ R+ , the function τ → (τ, t) is differentiable and its derivative is in ∈ L1 .
(c) The function t → (·, t) is
 continuously differentiable from R+ to L .
1

(d)  also satisfies ∂τ + ∂t (τ, t) = G((·, t))(τ ) for every t ∈ R+ and a.e. for τ ∈ (0, ∞).
∂ ∂

The proof can be found in Appendix B.6.

Finally, we can translate this into a result for the general model (System (2)), as stated below:

Theorem 4. Let μ ≥ 0. Let T , k : R+ → R+ be functions that satisfy Assumption 1. Let N0 > 0,

and let i0 : R+ → R+ be an absolutely continuous function such that i0 ∈ L1+ , i0 ∈ L1 , and
⎡ ⎤⎡ ⎤
∞ ∞
i0 (υ)
i0 (0) = ⎣ T (υ) ∞ dυ ⎦ ⎣N0 + k(τ )i0 (τ ) dτ ⎦ .
N0 + 0 i0 (τ ) dτ
0 0

Then there exist a continuously differentiable function N : R+ → R+ and a continuous function

i : R+ → L1+ (R) that solve System (2). Moreover, i(·, t) is absolutely continuous for any t ∈ R+
and
∂ ∂
D(i(τ, t)) = + i(τ, t)
∂τ ∂t
for every t ∈ R+ and a.e. for τ ∈ (0, ∞).

Proof.
∞ Let φx ∈ L1+ be any absolutely continuous function such that φx ∈ L1 , φx (0) = μP and
0 φx (τ ) dτ = N0 . As in the proof of Theorem 3, the result follows by applying Theorem 1 and
Proposition 7. 2

4. Discussion

We present a novel approach for epidemiological models by using two time variables, chrono-
logical time t and time-since-last infection (TSLI). One advantage of this approach is that fewer
state variables are needed; in the general model (System (2)) considered, there are only two:
N (t), the number of never infected people at time t , and i(τ, t), the density of people at time t
who were infected at least once with their last infections occurring τ units of time ago.
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.22 (1-31)
22 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

In most models with age-of-infection τ , the infected state variable, such as i(τ, t), denotes the
density of those who are either
 latently infected or infectious, and the equation is written using
partial derivatives, ∂τ + ∂t i(τ, t). This requires stronger conditions on the model parameter
∂ ∂

functions (e.g., T (τ ), k(τ ) and i0 (τ )) for the solution i(τ, t) to be in C 1 . In our model, individuals
in the i(τ, t) class include not only latently infected and infectious, but also recovered, who
may or may not have immunity; i.e., everyone except those who have never been infected. In
addition, the equation for i(τ, t) is described using the differential operator D, which allows
weaker conditions on the parameter functions. We show that if i ∈ C 1 then Di(τ, t) = ∂τ ∂
+

∂
∂t i(τ, t).
To analyze the existence and regularity of solutions to the general model (System (2)), we
apply published results for ADP problems, a term that refers to age-dependent populations as
specified in Section 2.2 (see, e.g., [15,17]). For ease of framing the general model (System (2))
as an ADP problem, we first reformulate it as a coupled model as shown in Section 2.3. We
also reformulate several published models to illustrate how readily age-structured models can be
formulated as coupled models (see Section 2.6). In turn, coupled models can be formulated as
ADP problems (System (5)), in which case results for those problems can be applied.
The general model (System (2)) can be used to study the dynamics of transmission and control
of many infectious diseases. The special feature of the class i(τ, t), together with the parame-
ter functions T (τ ) and k(τ ) for infectivity and susceptibility based on TSLI, permits multiple
scenarios, including: (i) complete immunity from natural infections (k(τ ) = 0); (ii) partial or
temporary immunity from infections (0 < k(τ ) < 1); and (iii) enhanced susceptibility due to in-
fections (sup k(τ ) > 1). For example, one might make the following assumptions on T and k:
(i) there exists a finite period during which individuals are infectious; (ii) immunity eventually
wanes (i.e., k increases); and (iii) once-infected individuals become as susceptible as they ever
would be. In other words, there exists τ0 and τ1 with 0 < τ0 < τ1 such that T (τ ) = 0 for τ > τ0 ,
k(τ ) = 0 for τ < τ0 , and k(τ ) = sup k for τ > τ1 . Applications of the general model (System (2))
under these conditions to study diseases such as tuberculosis and influenza will be presented
elsewhere.

Acknowledgment

ZF’s research is partially supported by NSF grant DMS-1814545.

Disclaimer

The findings and conclusions in this report are those of the authors and do not necessarily rep-
resent the official position of the Centers for Disease Control and Prevention or other institutions
with which they are affiliated.

Appendix A. Definitions

This appendix includes definitions and terminology mentioned in the main text.

Definition 3. L1+ = L1+ (Rn ) = {φ ∈ L1 : φ(τ ) ∈ Rn+ a.e. τ > 0}.

JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.23 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 23

Definition 4. Let t¯ > 0,  ∈ Lt¯, F : L1 → Rn , G : L1 → L1 , and φ ∈ L1 . We say that  is a mild

solution of the ADP problem on [0, t¯ ] for the initial distribution φ provided that  satisfies:

∞
lim |h−1 [(τ + h, t + h) − (τ, t)] − G((·, t))(τ )| dτ = 0, (A.1)
h→0+
0
h
−1
lim h |(τ, t + h) − F ((·, t))| dτ = 0, (A.2)
h→0+
0

and

(·, 0) = φ, 0 ≤ t ≤ t¯. (A.3)

Definition 5. For 0 < tˆ ≤ ∞, we say that  is the solution (respectively mild solution) of the ADP
problem on [0, tˆ ) for the initial distribution φ, provided that, for all t¯ < tˆ,  restricted to [0, t¯ ] is
the solution (respectively mild solution) of the ADP problem on [0, t¯ ] for the initial condition φ
restricted to [0, t¯ ].

Definition 6. If there exists a mild solution of the ADP problem on [0, t¯ ] for some t¯ > 0, we
denote by t¯φ , the maximal tˆ > 0, such that there exists a mild solution of the ADP problem in
[0, tˆ ).

Definition 7. Given φ ∈ L1 , F : L1 (Rn ) → Rn and G : L1 (Rn ) → L1 (Rn ), we define an equi-

librium solution of the ADP problem for the functions F , G and initial condition φ as a solution
of the ADP problem for the same functions on [0, ∞) such that (·, t) = φ for all t ≥ 0.

Definition 8. We define the projection function to the i-th entry πi : Rn → R as

πi (x1 , . . . , xn ) = xi ;

for m ∈ N, 0 < m < n, the projection to the first m entries π (m) : Rn → Rm as

π (m) (x1 , . . . , xn ) = (x1 , . . . , xm );

and, for k ∈ N, 0 < k < n, the projection to the last k entries π (−k) : Rn → Rk as

π (−k) (x1 , . . . , xn ) = (xn−k+1 , . . . , xn ).

Definition 9. Let X and Y be normed spaces with norms

·
X and
·
Y , respectively, and let
H : X → Y . We say that H is Lipschitz on norm-balls of X if, for all r > 0, there exists c(r) > 0
such that

H(x1 ) − H(x2 )
Y ≤ c(r)
x1 − x2
X

for all x1 , x2 ∈ X such that

x1
X ,
x2
X ≤ r. If c(r) can be chosen to be the same constant for
all r > 0, then H is said to be globally Lipschitz.
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.24 (1-31)
24 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

Definition 10. Let X and Y be normed spaces with norms

·
X and
·
Y , respectively, and let
D ⊂ X. We say that H : D → Y is F-differentiable relative to D at x0 ∈ D if there exists H (x0 ) ∈
B(X, Y ), such that, given any  > 0, there exists δ > 0 such that, if x ∈ D and
x − x0
X < δ,
then

H(x) − H(x0 ) − H (x0 )(x − x0 )
Y ≤ 
x − x0
X .

H is said to be continuously F-differentiable relative to D on A ⊂ D if it is F-differentiable

relative to D at each x ∈ A and if the map x → H (x) is continuous from A to B(X, Y ). H (x)
is called the F-derivative of H at x.

Appendix B. Proofs

B.1. Proof of Proposition 2

Proof. To simplify notation, let T̂ = supτ {T (τ )} and k̂ = supτ {k(τ )}

Part (a). The linearity follows by the definition of the functions and fact that integration is a
linear operator.
∞ ∞
For φ ∈ L1 , we have |F(φ)| ≤ 0 T (τ )|φ o (τ )|/P dτ ≤ T̂
φ
/P , |W(φ)| ≤ 0 |φ n (τ )| +
 ∞ 
k(τ )|φ i (τ )| dτ ≤ k̂
φ
, and |W(φ)| ≤ 0 |φ n (τ )| + |φ i (τ )| τ =
φ
. In addition, |W(φ)| =

φ
if φ ∈ L1+ .
Part (b). Existence and uniqueness of the mild solution of the ADP problem is guaranteed if
F and G are Lipschitz on norm-balls of L1 [15, Theorem 2.1]. In other words, we need to show
that there exist functions c1 , c2 : R+ → R+ such that |F (φ1 ) − F (φ2 )| ≤ c1 (r)
φ1 − φ2
and

G(φ1 ) − G(φ2 )
≤ c2 (r)
φ1 − φ2
for all φ1 , φ2 ∈ L1 with
φ1
,
φ2
≤ r.
If
φ1
,
φ2
≤ r, using Part (a), we have

|F (φ1 ) − F (φ2 )| = |μW(φ1 ) − μW(φ2 )| + |F(φ1 )W(φ1 ) − F(φ2 )W(φ2 )|

= μ|W(φ1 − φ2 )| + |F(φ1 )W(φ1 ) − F(φ1 )W(φ2 ) + F(φ1 )W(φ2 ) − F(φ2 )W(φ2 )|
≤ μ
φ1 − φ2
+ |F(φ1 )||W(φ1 − φ2 )| + |W(φ2 )||F(φ1 − φ2 )|
T̂
≤ μ
φ1 − φ2
+
φ1
|W(φ1 − φ2 )| + k̂
φ2
|F(φ1 − φ2 )|
P
T̂ T̂
≤ μ
φ1 − φ2
+
φ1
k̂
φ1 − φ2
+ k̂
φ2

φ1 − φ2

P P
T̂
≤ μ
φ1 − φ2
+ 2r k̂
φ1 − φ2
.
P

Thus, we can choose

2k̂ T̂ r
c1 (r) = + μ.
P
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.25 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 25

Similarly, if
φ1
,
φ2
≤ r, then

∞
 

G(φ1 ) − G(φ2 )
= −F (φ1 )φ n (τ ) + F(φ2 )φ n (τ ) − μφ n (τ ) + μφ n (τ ) dτ
1 1 2
0
∞ 
 
+ −F (φ1 )S(τ )φ1i (τ ) + F(φ2 )k(τ )φ i (τ ) − μφ1i (τ ) + μφ2i (τ ) dτ
0
∞ ∞
≤ |F(φ1 )φ1n (τ ) − F(φ2 )φ2n (τ )| dτ +μ |φ n (τ ) − φ n (τ )| dτ
0 0
∞ ∞
+ |F(φ1 )k(τ )φ1i (τ ) − F(φ2 )k(τ )φ2i (τ )| dτ +μ |φ i (τ ) − φ i (τ )| dτ
0 0
∞
≤ |F(φ1 )φ1n (τ ) − F(φ1 )φ2n (τ ) + |F(φ1 )φ2n (τ ) − F(φ2 )φ2n (τ )| dτ
0
∞
+ |F(φ1 )k(τ )φ1i (τ ) − F(φ1 )k(τ )φ2i (τ )| + |F(φ1 )k(τ )φ2i (τ )
0

− F(φ2 )k(τ )φ2i (τ )| dτ + μ

φ1 − φ2

≤ |F(φ1 )|k̂
φ1 − φ2
+ |F(φ1 − φ2 )|k̂
φ2
+ μ
φ1 − φ2

T̂
≤2 k̂r
φ1 − φ2
+ μ
φ1 − φ2
.
P

Thus, we can also take

2k̂ T̂ r
c2 (r) = + μ.
P

Part (c). We can guarantee that (·, t) ∈ L1+ if we have the following two conditions [15,
Theorem 2.4]:

(i) F (L1+ ) ⊆ R2+ , and

(ii) there exists an increasing function c3 : R+ → R+ such that

G(φ) + c3 (r)φ ∈ L1+

whenever r > 0, φ ∈ L1+ , and

φ
≤ r.

Clearly F (L1+ ) ⊆ R2+ , so we only need to show that there exists a suitable function c3 .
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.26 (1-31)
26 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

If
φ
≤ r, using Part (a), we have

  
F(φ)φ n (τ ) + μφ n (τ )
−G(φ)(τ ) = ≤ k̂F(φ) + μ φ(τ )
F(φ)k(τ )φ (τ ) + μφ (τ )
i i
   
T̂ T̂
≤ k̂
φ
+ μ φ(τ ) ≤ k̂ r + μ φ(τ ).
P P

Therefore, we can take

k̂ T̂ r
c3 (r) = + μ. 2
P
B.2. Proof of Proposition 3

Proof. For 0 < t < t¯φ and h > 0, we have

∞
−1
 
h (τ, t + h) − (τ, t) dτ
0

h ∞ ∞
−1 −1 −1
=h (τ, t + h) dτ + h (τ, t + h) dτ − h (τ, t) dτ
0 h 0

h ∞
= h−1 (τ, t + h) dτ + h−1 [(τ + h, t + h) − (τ, t)] dτ,
0 0
∞
which converges to F ((·, t)) + 0 G((·, t))(τ ) dτ as h → 0+ because of Equations (A.1) and
(A.2). ∞ ∞
Adding the entries of vectors h−1 0 (τ, t + h) − (τ, t) dτ and F ((·, t)) + 0 G((·, t))
(τ ) dτ , we obtain

W((·, t + h)) − W((·, t))

→0
h
as h → 0+ . In other words, t → W((·, t)) is differentiable from the right in (0, t¯φ ), and its right
derivative is 0.
Given 0 < t¯ < t¯φ ,  ∈ Lt¯, so the restriction of the solution  to [0, t¯ ] is a continuous function
of t from [0, t¯ ] to L1 ; therefore, W((·, t)) is also continuous in [0, t¯ ]. Any continuous function
in [0, t¯ ] that has non-negative right derivative everywhere in (0, t¯ ) is non-decreasing in [0, t¯ ]
[16, Chapter 5, Proposition 2]. Because both W((·, t)) and −W((·, t)) have non-negative right
derivatives, we can conclude that W((·, t)) is constant in [0, t¯ ] for any 0 < t¯ < t¯φ .
Finally, if φ ∈ L1+ , because of Equation (A.3),

∞
 
W((·, 0)) = W(φ) = φ n (τ ) + φ i (τ ) dτ =
φ
,
0
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.27 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 27

so W((·, t)) =
φ
for all 0 ≤ t < t¯φ . Additionally, from Part (c) of Proposition 2, we know
that, if φ ∈ L1+ , (·, t) ∈ L1+ , then W((·, t)) =
(·, t)
for all 0 ≤ t < t¯φ . 2

B.3. Proof of Proposition 4

Proof. If t¯φ < ∞, then lim supt>0

(·, t)
= ∞ [15, Theorem 2.3]. By Proposition 3, we know
that
(·, t)
remains bounded (actually it is constant) for all t ∈ [0, t¯φ ) if φ ∈ L1+ . So, we can
conclude that t¯φ = ∞. 2

B.4. Proof of Proposition 5

Proof. The existence and uniqueness of a mild solution  of the ADP problem is guaranteed
by Part (b) of Proposition 2. Also tφ = ∞ because of Proposition 4 and (·, t) ∈ L1+ for every
t ∈ R+ because of Part (c) of Proposition 2.
Note that

G(φ)(τ ) = −M(τ, φ)φ

for all τ > 0, where M : R+ × L1 → B(R2 , R2 ) is defined as


F(φ) + μ 0
M(τ, φ) = .
0 k(τ )F(φ) + μ

For G of this form, the mild solution of the ADP problem in [0, t¯φ ) is a continuous solution
of the ADP problem in [0, t¯φ ) [15, Theorem 2.9] if

(i) φ ∈ L1 is continuous and φ(0) = F (φ),

(ii) F is Lipschitz on norm-balls of L1 , and
(iii) there exist increasing functions c4 , c5 , c6 : R+ → R+ such that, for all φ1 , φ2 ∈ L1 , τ1 ,
τ2 ≥ 0:
(a)
M(τ1 , φ1 ) − M(τ2 , φ1 )
op ≤ c4 (
φ1
)|τ1 − τ2 |
(b)
M(τ1 , φ1 )
op ≤ c5 (
φ1
)
(c)
M(τ1 , φ1 ) − M(τ1 , φ2 )
op ≤ c6 (r)
φ1 − φ2
if
φ1
,
φ2
≤ r.

(i) is part of the hypothesis and we already showed (ii) in the proof of Proposition 2, so we
proceed to prove (iii).
Define T̂ = supτ {T (τ )} and k̂ = supτ {k(τ )}. Let φ1 , φ2 ∈ L1 , τ1 , τ2 ≥ 0. Using the fact that k
is globally Lipschitz, let K be a constant such that

|k(τ ) − k(τ  )| ≤ K|τ − τ  |

for all τ, τ  ≥ 0. We have

M(τ1 , φ1 ) − M(τ2 , φ1 )
op = |k(τ1 ) − k(τ2 )| |F(φ1 )|
K T̂
≤
φ1
|τ1 − τ2 |,
P
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.28 (1-31)
28 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

and can take c4 (r) = KPT̂ r .

On the other hand,

M(τ1 , φ1 )
op = sup {|F(φ1 )x1 + μx1 | + |k(τ1 )F(φ1 )x2 + μx2 |}
|x1 |+|x2 |=1

≤ sup {|F(φ1 )||x1 | + μ|x1 | + |k(τ1 )||F(φ1 )||x2 | + μ|x2 |}

|x1 |+|x2 |=1
!
≤ sup k̂|F(φ1 )| + μ
|x1 |+|x2 |=1

k̂ T̂
≤
φ1
+ μ,
P

and we can take c5 (r) = k̂ T̂ r

P + μ.
Finally,

M(τ1 , φ1 ) − M(τ1 , φ2 )
op = sup {|F(φ1 − φ2 )x1 | + |k(τ1 )F(φ1 − φ2 )x2 |}
|x1 |+|x2 |=1
!
≤ sup k̂|F(φ1 − φ2 )|(|x1 | + |x2 |)
|x1 |+|x2 |=1

k̂ T̂
≤
φ1 − φ2
,
P

and we can take c6 (r) = k̂ T̂

P . 2

B.5. Proof of Proposition 6

Proof. Let φ0 ∈ L1 . Define T̂ = supτ {T (τ )} and k̂ = supτ {k(τ )}. Note that both F  (φ0 ) and
G (φ0 ) defined above are linear operators from L1 to R2 and from L1 to L1 , respectively. They
are bounded linear operators because

|F  (φ0 )(φ)| =|μW(φ)| + |F(φ0 )W(φ) + F(φ)W(φ0 )|

 
T̂
≤ μ + |F(φ0 )|k̂ + |W(φ0 )|
φ
,
P

and
∞


G (φ0 )(φ)
= |F(φ0 )φ n (τ ) + F(φ)φ0n (τ ) + μφ n (τ )| dτ
0
∞
+ |F(φ0 )k(τ )φ i (τ ) + F(φ)k(τ )φ0i (τ ) + μφ i (τ )| dτ
0
 
T̂
≤ |F(φ0 )|k̂ + k̂
φ0
+ μ
φ
,
P
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.29 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 29

for any φ ∈ L1 .
Now, let  > 0 and φ ∈ L1 . We have

|F (φ) − F (φ0 ) − F  (φ0 )(φ − φ0 )|

= |F(φ)W(φ) − F(φ0 )W(φ0 ) − F(φ0 )W(φ − φ0 ) − F(φ − φ0 )W(φ0 )|
= |F(φ)W(φ − φ0 ) − F(φ0 )W(φ − φ0 )|
= |F(φ − φ0 )W(φ − φ0 )|
T̂
≤ k̂
φ − φ0
2
P
≤ 
φ − φ0
,
!
if T̂ = 0, k̂ = 0, or if
φ − φ0
< δ = min 1, P  when T̂ = 0 and k̂ = 0.
T̂ k̂
Likewise, we have

G(φ0 ) − G(φ) − G (φ0 )(φ − φ0 )

∞
= |F(φ − φ0 )φ n (τ ) − F(φ − φ0 )φ0n (τ )| dτ
0
∞
+ |F(φ − φ0 )k(τ )φ i (τ ) − F(φ − φ0 )k(τ )φ0i (τ )| dτ
0
∞
≤ k̂|F(φ − φ0 )| |φ n (τ ) − φ n (τ ) + |φ 0 (τ ) − φ0i (τ )| dτ
0

= k̂|F(φ − φ0 )|
φ − φ0

T̂
≤ k̂
φ − φ0
2 ,
P
!
which again is smaller than  if T̂ = 0, k̂ = 0, or if
φ − φ0
< δ = min 1, P  when T̂ = 0
T̂ k̂
and k̂ = 0.
Now, let φ1 , φ2 ∈ L1 . We have

F  (φ1 ) − F  (φ2 )
op = sup |F(φ1 − φ2 )W(φ) + F(φ)W(φ1 − φ2 )|

φ
=1
 "
T̂ T̂
≤ sup
φ1 − φ2
k̂
φ
+
φ
k̂
φ1 − φ2

φ
=1 P P

T̂
=2 k̂
φ1 − φ2
.
P
Thus, φ → F  (φ) is a continuous function from L1 to B(L1 , R2 ).
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.30 (1-31)
30 J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–•••

On the other hand,

# ∞
 

G (φ1 ) − G (φ2 )
op = sup |F(φ1 − φ2 )φ n (τ ) + F(φ)(φ1n (τ ) − φ2n (τ ))| dτ

φ
=1
0
∞ $
+ |F(φ1 − φ2 )k(τ )φ i (τ ) + F(φ)k(τ (φ1i (τ ) − φ2i (τ ))| dτ
0
!
≤ sup |F(φ1 − φ2 )|k̂
φ
+ F(φ)k̂
φ1 − φ2

φ
=1

T̂
≤2 k̂
φ1 − φ2
.
P

Thus, φ → G (φ) is also continuous as a function from L1 to B(L1 , L1 ). 2

B.6. Proof of Proposition 7

Proof. A mild solution of the ADP problem on [0, t¯φ ) is a solution of the ADP problem and
satisfies conditions (a)–(d) for any t ∈ [0, t¯φ ) as long as the following conditions hold [17, The-
orem 2.3]:

(i) The functions F and G are Lipschitz on norm-balls of L1+ ,

(ii) There exists a function c3 that satisfies (ii) in the proof of Proposition 2, Part (c).
(iii) The functions F and G are continuously F-differentiable relative to L1+ .
(iv) The initial condition φ has the properties: φ ∈ L1+ and is absolutely continuous, φ  ∈ L1 ,
and φ(0) = F (φ).

The existence and uniqueness of the mild solution of the ADP problem can be guaranteed by
Proposition 2. Conditions (i) and (ii) are shown in the proofs of Parts (b) and (c) of Proposition 2,
respectively. Condition (iii) is Proposition 6, whereas (iv) is part of the hypothesis. Finally, the
fact that t¯φ = ∞ was the result of Proposition 4. 2

References

[1] J.A. Alfaro-Murillo, An Epidemic Model Structured by the Time Since Last Infection, Ph.D. thesis, Purdue Univer-
sity, 2013.
[2] J.A. Alfaro-Murillo, Z. Feng, J.W. Glasser, A review and extension of time-since-infection models in epidemiology,
submitted for publication.
[3] F. Brauer, The Kermack-McKendrick epidemic model revisited, Mathematical Biosciences 198 (2) (2005) 119–131.
[4] F. Brauer, Z. Shuai, P. van den Driessche, Dynamics of an age-of-infection cholera model, Mathematical Biosciences
and Engineering 10 (5 & 6) (2013) 1335–1349.
[5] D. Breda, O. Diekmann, W.F. de Graaf, A. Pugliese, R. Vermiglio, On the formulation of epidemic models (an
appraisal of Kermack and McKendrick), Journal of Biological Dynamics 6 (sup2) (2012) 103–117.
[6] O. Diekmann, R. Montijn, Prelude to Hopf bifurcation in an epidemic model: analysis of a characteristic equation
associated with a nonlinear Volterra integral equation, Journal of Mathematical Biology 14 (1) (1982) 117–127.
[7] Z. Feng, M. Iannelli, F.A. Milner, A two-strain tuberculosis model with age of infection, SIAM Journal on Applied
Mathematics 62 (5) (2002) 1634–1656.
JID:YJDEQ AID:9868 /FLA [m1+; v1.300; Prn:12/06/2019; 9:46] P.31 (1-31)
J.A. Alfaro-Murillo et al. / J. Differential Equations ••• (••••) •••–••• 31

[8] Z. Feng, H. Thieme, Endemic models with arbitrarily distributed periods of infection I: fundamental properties of
the model, SIAM Journal on Applied Mathematics 61 (3) (2000) 803–833.
[9] Z. Feng, H.R. Thieme, Endemic models with arbitrarily distributed periods of infection II: fast disease dynamics
and permanent recovery, SIAM Journal on Applied Mathematics 61 (3) (2000) 983–1012.
[10] H.W. Hethcote, H.R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math-
ematical Biosciences 75 (2) (1985) 205–227.
[11] H. Inaba, Kermack and McKendrick revisited: the variable susceptibility model for infectious diseases, Japan Jour-
nal of Industrial and Applied Mathematics 18 (2) (2001) 273–292.
[12] W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the
Royal Society of London. Series A 115 (772) (1927) 700–721.
[13] H.R. Thieme, C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?,
SIAM Journal on Applied Mathematics 53 (5) (1993) 1447–1479.
[14] G.F. Webb, E.M.C. D’Agata, P. Magal, S. Ruan, S. Falkow, A model of antibiotic-resistant bacterial epidemics
in hospitals, Proceedings of the National Academy of Sciences of the United States of America 102 (37) (2005)
13343–13348.
[15] G.F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and
Applied Mathematics, vol. 89, Marcel Dekker, New York, 1985.
[16] H.L. Royden, Real Analysis, 3rd edition, Macmillan, New York, 1988.
[17] W.M. Ruess, Linearized stability and regularity for nonlinear age-dependent population models, in: Functional
Analysis and Evolution Equations, 2007, pp. 561–576, http://dx.doi.org/10.1007/978-3-7643-7794-6_34.
[18] L. Perko, Differential Equations and Dynamical Systems, 3rd edition, Springer, 2001.
[19] S. Bhattacharya, F.R. Adler, A time since recovery model with varying rates of loss of immunity, Bulletin of Math-
ematical Biology 74 (12) (2012) 2810–2819.
[20] P. Magal, C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM
Journal on Applied Mathematics 73 (2) (2013) 1058–1095, https://doi.org/10.1137/120882056, http://epubs.siam.
org/doi/abs/10.1137/120882056.