key: cord-0859168-mimp60y8
authors: Liu, Suli; Li, Michael Y.
title: Epidemic models with discrete state structures()
date: 2021-03-24
journal: Physica D
DOI: 10.1016/j.physd.2021.132903
sha: db54bf74d147a7f78258d1ba024e1fe5005bed62
doc_id: 859168
cord_uid: mimp60y8

The state of an infectious disease can represent the degree of infectivity of infected individuals, or susceptibility of susceptible individuals, or immunity of recovered individuals, or a combination of these measures. When the disease progression is long such as for HIV, individuals often experience switches among different states. We derive an epidemic model in which infected individuals have a discrete set of states of infectivity and can switch among different states. The model also incorporates a general incidence form in which new infections are distributed among different disease states. We discuss the importance of the transmission-transfer network for infectious diseases. Under the assumption that the tranmission-transfer network is strongly connected, we establish that the basic reproduction number [Formula: see text] is a sharp threshold parameter: if [Formula: see text] , the disease-free equilibrium is globally asymptotically stable and the disease always dies out; if [Formula: see text] , the disease-free equilibrium is unstable, the system is uniformly persistent and initial outbreaks lead to persistent disease infection. For a restricted class of incidence functions, we prove that there is a unique endemic equilibrium and it is globally asymptotically stable when [Formula: see text]. Furthermore, we discuss the impact of different state structures on [Formula: see text] , on the distribution of the disease states at the unique endemic equilibrium, and on disease control and preventions. Implications to the COVID-19 pandemic are also discussed.

HIV, an infected individual can progress forward through different stages or backward due to treatment [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] . The stages of 25 disease progression is a good example of the state of an infected individual.

The concept of state of infection is, however, more general than that of stages and can unify many different approaches to model disease heterogeneity. For instance, the concept of state or substate can be applied to susceptible hosts to represent different level of host susceptibility to infection or reinfection after 30 recovery. Such a consideration will lead to the investigation of the impact of differential susceptibility on disease transmission (see [4] ). The concept of state or substate can also be considered for vaccinated or immune hosts and used to measure individual variations as well as community distributions of level of antibodies and immunity. The concept of state can also be extended to individual 35 characteristics including spatial location, chronological age, development stage, size etc [13] . And the distributions of individual characteristics directly influence the dynamics of population which could be generated as the state space of measures [14, 15, 16] . Interpretation of the traditional epidemic models in a state-space framework was used to track the evolution of Flu in the United 40 States [17] and predict intervention effect for COVID-19 in Japan [18] .

The concept of state has also been widely employed to diffusion models in the social and life sciences. For example, the Bass model for diffusion of innovation [19] and its generalizations and variations [20, 21] , models for the diffusion of information on social media [22] , models for the spread of opinions in modeling studies. From a transmission viewpoint, the disease state should 55 also include an individual's sexual preferences and a scoring of risky behaviours.

This approach can stimulate a more comprehensive approach to modeling and better inform the prevention and control measures of the HIV epidemic (see e.g.

[30, 31] ).

The transmission dynamics of the SARS-CoV-2 virus, which is responsi- 60 ble for the current COVID-19 pandemic, provided another good example of the importance of the state structure for epidemics. The significant number of asymptomatic infections is believed to be a key reason why the COVID-19 epidemics have proven to be more difficult to control than the SARS epidemics in 2003 [32, 33] . The substate of asymptomatic infections can be studied in 65 state-structured models. The current vaccines for COVID-19 are known to be highly effective to protect against serious symptoms and diseases, but their efficacy against transmission is still largely unknown [34] . While the vaccines can prevent serious diseases and deaths among seniors and people with underlying health, the vaccination of younger population may produce more asymptomatic 70 disease carriers, whose existence was already known for COVID-19 even before the vaccines were available [33] , if the efficacy of the vaccines against transmission is not sufficiently high. The impact of subgroup of asymptomatic carriers has been studied in the literature (e.g. [35] ) and more recently for COVID-19 in the class of A-SIR models [36, 37] . Other subgroups that capture the many char-75 acteristics of the COVID-19 were also considered in recent COVID-19 modeling, including undetected infections, infected but not infectious (latent), quarantined or isolated individuals, and hospitalized individuals (see e.g. [36, 38, 39] and references therein). Subgroups of infected individuals are special cases of the infection state and can be investigated under the framework of state-structures. 80 In our paper, we restrict our consideration to the state of infection among infected individuals and allow infected individuals switch their state of infection among finitely many different substates, hence creating a finite discrete state-structure. We formulate and investigate a class of epidemic models with a discrete state structure among infected individuals. Switch among different 85 4 J o u r n a l P r e -p r o o f Journal Pre-proof sub-states is given by a state-transfer matrix, which helps to capture the heterogeneity among the substates. We note that sub-state structures given by different state-transfer matrices may alter the probability distribution of the duration of infectiousness, and we refer the reader to the approach of the linear chain trick and its generalizations (see e.g. [40] ) for this linkage. For model-90 ing dynamical processes on networks, we refer the reader to the approximation frameworks of binary-state dynamical processes on networks [41] and its generalization [42] .

In addition to carrying out a standard stability analysis of a relatively comprehensive model, the consideration of state has inspired us to investigate a Results in our paper include as special cases earlier results on multi-staged epidemic models (see e.g. [9] and reference therein). A continuous approach to modeling the state structure was considered in [45] , and the resulting model is 105 a system of differential-integral equations with nonlocal effects. In our model (2.1) with a discrete state structure in Section 2, mathematical results on global stability are obtained under more relaxed conditions than those for infinite dimensional systems as in [45] .

The paper is organized as follows. We formulate our model in the next 110 section. In Section 3, we derive the basic reproduction number and give its biological interpretations. In Section 4, we carry out mathematical analysis of the global dynamics of the model for the case f (N ) ≡ 1: the disease free equilibrium P 0 is globally asymptotically stable when R 0 ≤ 1; and a unique endemic equilibrium P * exists and is globally asymptotically stable when R 0 > 115 1. In Section 5, we carry out numerical investigations of the impacts of the state 5 J o u r n a l P r e -p r o o f Journal Pre-proof structure on disease dynamics and outcomes.

We assume that the state of an infected individual is a finite set with a scalar index i, i = 1, 2, · · · , n. For 1 ≤ i ≤ n, let I i (t) denote the number of S(t) + I 1 (t) + I 2 (t) + · · · + I n (t) to represent the total number of individuals who are active in the infection process.

In the absence of the disease, we assume that the dynamics of population are described by a nonlinear differential equation: S = θ(S), where θ(S) is a growth function with a carrying capacity. Common forms of θ(S) are θ(S) = Λ−dS (see [6, 46, 47] ) and θ(S) = Λ+rS(1− S K )−dS (see [9, 48] ). We also assume that the disease transmission is horizontal and has a general nonlinear incidence function 135 f (N ) n j=1 g j (S, I j ), where f (N ) denotes the density dependence. Out of the total new infection f (N ) n j=1 g j (S, I j ), we assume that a fraction α i ≥ 0 will enter the i-th state, and n i=1 α i = 1. Let ζ i (I i ) denote the removal rate from I i , which may include natural death, disease-caused death, and out-migration.

A common form of ζ i is the exponential removal ζ i (I i ) = d i I i . To describe Based on these assumptions and using the transfer diagram, we derive an epidemic model with a discrete state structure described by the following system of nonlinear differential equations:

(2.1)

Since the removed population is no longer part of the transmission process and the variable R does not appear in the equations of S and I i , we consider the following equation for the removed population R separately:

The general functions in model (2.1), f (N ), g j (S, I j ), φ ij (I j ), γ i (I i ) and ζ i (I i ) are assumed to be sufficiently smooth such that existence and uniqueness of solutions are satisfied. We further make the following biologically motivated assumptions:

(H 1 ) There existsS > 0 such that θ(S) = 0 and θ(S)(S −S) < 0, ∀S ≥ 150 0, S =S;

(H 2 ) f (N ) : R + → R + is a nonincreasing positive function;

(H 3 ) For 1 ≤ j ≤ n, g j (S, I j ) ≥ 0 for all S, I j ≥ 0; and if g j ≡ 0 then g j (S, I j ) = 0 iff S = 0 or I j = 0;

It can be verified by examining direction of the vector fields on the boundary of R n+1 + that solutions to system (2.1) with nonnegative initial conditions remain nonnegative for t ≥ 0 and that the model is well defined. From the first equation of system (2.1) and assumptions (H 2 ), (H 3 ), we know that S ≤ θ(S).

Assumption (H 1 ) implies that lim sup t→∞ S(t) ≤S, a carrying capacity of the disease-free population. Adding all equations of (2.1) yields that

By assumption (H 6 ), we have

Let d * = min 1≤i≤n {d i } and we assume that d * > 0. Then (S + I 1 + I 2 + · · · + I n ) ≤ θ(S) + d * S − d * (S + I 1 + I 2 + · · · + I n )

≤ M − d * (S + I 1 + I 2 + · · · + I n ),

where M ≥ max S≥0 {θ(S) + d * S}, which implies lim sup t→∞ (S + I 1 + I 2 + · · · + I n ) ≤ M d * .

Therefore, omega limit sets of system (2.1) are contained in the following bounded region:

It can be verified that Γ is positively invariant with respect to (2.1).

assumption. When f (N ) is non-monotone, it was shown in [49] using a simple SIR model that complex dynamics such as backward bifurcations and Hopf Our model (2.1) include many previously studied compartmental models that have infectious, symptomatically infectious, asymptomatically infected, latent, vaccinated, quarantined, and isolated compartments. When α 1 = 1 and α i = 0 for i = 2, · · · , n, our model can be interpreted as a multi-staged model studied 180 in [9] , in which all new infections enter the first stage. In staged progression models, the stages are understood to have a linear sequential structure. The transfer terms φ ij , when i > j, denote the rate of disease progression (forward); and when i < j, φ ij denote the rate of amelioration (backward). For statestructured models, a sequential order among states is no longer essential, and 185 different states can be regarded as parallel structures. A key difference between state-structured models and staged-progression models is that a newly infected individual can manifest in any disease state while often enters into the first disease stage. 

For each 1 ≤ i ≤ n, let τ i denotes the co-factor of the i-th diagonal entry of Proposition 3.1 (Matrix Tree Theorem [52] ). Assume that n ≥ 2. Then

is strongly connected, then τ i > 0, for i = 1, 2, · · · , n. 

The identity in the next corollary was first given in [44] and follows from the 215 tree cycle identity.

be any family of functions, and m ij , τ i are given in Proposition 3.1. Then the following identity holds:

(3.3)

Set System (2.1) can be rewritten as the following:

g j (S, I j ),

(4.2)

It follows from assumptions (H 1 )-(H 6 ) that the disease-free equilibrium P 0 = (S, 0, · · · , 0) always exists. Let P * = (S * , I * 1 , · · · , I * n ) denote a possible endemic equilibrium of (4.2). Then, the coordinates S * , I * 1 , · · · , I * n are positive solutions to the following system of equilibrium equations:

We note that endemic equilibria of model (4.2) can be of two different forms: either a positive endemic equilibrium for which I * j > 0 for all 1 ≤ j ≤ n, or a mixed endemic equilibrium for which I * j > 0 for some j and I * j = 0 for the remaining j. A positive endemic equilibrium belongs to the interior of Γ and a mixed endemic equilibrium belongs to the boundary of Γ. In the case of a positive equilibrium, the disease is endemic among all states, while in the case of a mixed endemic equilibrium, the disease is only endemic among a subset of states, and the remaining states are disease free. It was known in the literature that complex disease models can have mixed endemic equilibria (see e.g. [54, 55] ). It was pointed out in [55] that mixed endemic equilibria exist when the transmission network of the disease is not strongly connected. Let

(4.4)

Using assumptions (H 7 ) and (H 8 ), we define

where In weights m ij defined in (4.5), the term α i f (S)c j comes from the transmis- We note that the weight matrix M of the transmission-transfer network G(M ) is obtained using the limiting values of weight functions w ij (S, I 1 , · · · , I n )/I j in (4.4) when point (S, I 1 , · · · , I n ) approaches P 0 . In later sections, we will also J o u r n a l P r e -p r o o f Journal Pre-proof consider weight matricesM = (w ij (S * , I * 1 , · · · , I * n )) n×n with w ij measured at In the weight m ij from I j to I i in (4.5), b ij represents the populations transfer from state j to state i. The α i f (S)c j term is less intuitive as weight from state 240 j to i. In the augmented directed graph with (n + 1) vertices in Figure 2 , we can see that this term is created indirectly through infection of S by I j as indicated by a dashed edge from I j to S, together with the corresponding incidence in I i as indicated by a solid edge from S to I i . This is also explained in Figure is strongly connected. Then the disease-free equilibrium P 0 = (S, 0, · · · , 0) is the 255 only equilibrium of (4.2) on the boundary of R n+1 .

Proof The proof is similar to that of Theorem 2.3 in [55] , in which only a transmission network is considered. It suffices to exclude the existence of mixed endemic equilibria on the boundary of Γ.

Let P * = (S * , I * 1 , · · · , I * n ) denote a nonnegative equilibrium of (4.2), where

Using assumption (H 1 ) and the first equilibrium equation of (4.3), we have S * > 0.

If an arc from j to i exists and I * j > 0, we show that I * i > 0. Assume otherwise that I * i = 0, then assumptions (H 4 )-(H 6 ) imply that ψ i (I * i ) = 0, and assumption (H 3 ) and I * j > 0 imply that g j (S * , I * j ) > 0. Using the equation for I i we obtain

This contradicts the fact that P * is an equilibrium, and shows I * i > 0. If an oriented path (j, j 1 , · · · , j k , i) from j to i exists and I * j > 0, using statement above repeatedly, we can show that I * i > 0.

Since the transmission-transfer network G(M ) is strongly connected, for

given two states i, j, there exists and oriented path from j to i and from i to j.

From the preceding arguments, we know that I * j > 0 implies I * i > 0. Therefore, I * j > 0 for some j implies I * i > 0 for all i, namely, for all 1 ≤ j ≤ n, I * j are either all zero or all positive, and thus the only equilibrium on the boundary is 270 the disease-free equilibrium P 0 .

If G(M ) is not strongly connected, then there may exist a state i that can reach other states but is not reachable from any other state. Then, when I * i = 0, there exist state j for which I * j > 0, and mixed endemic equilibria exist on the boundary. We give the following example to demonstrate this.

Example 4.1. Consider the epidemic model (4.6) illustrated by the transfer diagram in Figure 4 (a). Its transmission-transfer network is not strongly connected, and it has two boundary equilibria.

To see the claims in Example 4.1, we note that the state I 2 is not reachable from any other states either by state transfer or transmission, and the we can see that two boundary equilibria can exist: the disease-free equilibrium

(4.6)

In this paper, we only consider the case when the transmission-transfer net-290 work G(M ) is strongly connected. In the case when G(M ) is not strongly connected, the dynamics of model (4.2) can be analyzed following the approach in [55] using the condensed graph.

The basic reproduction number R 0 is the expected number of secondary 295 infections produced by a typical infectious individual during its entire infectious period in an entirely susceptible population. The mathematical definition of the basic reproduction number is by the spectral radius of the next generation matrix, see [56, 57] .

Let ψ i (I i ) be defined at the beginning of Section 4.1 and

Following [57] , the basic reproduction number for model (4.2) is the spectral

where ρ(A) denotes the spectral radius of a matrix A, and

(4.9)

Using (4.7), (H 8 ) and (H 9 ), it can be verified that F is a nonnegative matrix, 300 and V is a nonsingular M -matrix with a nonnegative inverse V −1 .

Assumptions (H 7 )-(H 9 ) are sufficiently weak to allow the nonlinear incidence function g(S, I i ) and transfer functions φ ij , γ i , and ζ i to be not differentiable at the disease-free equilibrium. For instance, a common incidence function

constants c i , b ij are given by the corresponding derivatives at I i = 0.

In the most common case when

It can verified that the definition of basic reproduction number R 0 in (4.8)

includes as special cases the corresponding definitions for simpler models in the literature (see e.g. [5, 6, 7, 8, 9, 46, 47, 58, 59] ). (i) Nonnegative matrix F V −1 has a unique positive eigenvalue ρ(F V −1 ).

(ii) The basic reproduction number satisfies

Proof The matrix F can be written as

(4.13)

By (4.11) and (4.13), we have F V −1 = (α 1 , α 2 , · · · , α n ) T (w 1 , w 2 , · · · , w n ). Then matrix F V −1 has rank 1, and thus its spectral radius is the only positive eigen-320 value and is equal to its trace α 1 ω 1 + α 2 ω 2 + · · · + α n ω n , leading to (4.12).

As pointed out in [57] , the (k, j) entry of matrix V −1 represents the average time of an infected individual originally at the j-th state spends in the k-th state, the (i, k) entry of matrix F is the incidence rate at which an infected individual As defined in (4.11), w j is the sum of the j-th column of matrix F V −1 , and 330 represents the expected number of new infections produced by an newly infected individual at the j-th state during its entire infectious period.

To interpret R 0 in (4.12) biologically, consider an newly infected individual, with probability α i the individual is at the state i, 1 ≤ i ≤ n, and will produce an expected w i secondary infections during its infectious period. Therefore, the total expected number of secondary infections during its infectious period is

Consider the special case when the proportions of incidence among states is such that α 1 = 1, α i = 0 for i = 2, · · · , n, namely, all new infections enter the progression. The resulting model agrees with the staged progression model considered in [9] . By (4.12), R 0 = w 1 , which agrees with the basic reproduction number given in [9] . Similarly, if α 1 = · · · = α n = 1/n, then

In the next result, we show that the basic reproduction number can also be interpreted using the average time spent by an infected individual at each disease state.

Proposition 4.3. Assume that assumptions (H 1 )-(H 9 ) hold. Let

(4.14)

Then the following results hold: 

(4.15)

Proof By (4.11) and (4.12), R 0 = (w 1 , · · · , w n )(α 1 , · · · , α n ) T = f (S)c 1 , · · · , f (S)c n V −1 (α 1 , · · · , α n ) T .

Then, (4.15) follows from (4.14).

Since the (i, j) entry of matrix V −1 represents the average time of an infected individual originally at the j-th state spends in the i-th state, we know that T i in (4.14) is the average time an infected individual spends in the i-th state. 

(4.16)

Biologically, R 0i represents the basic reproduction number of i-th state when the other states are absent. For instance, when f (N ) ≡ 1, g j (S,

then we arrive at the standard expression for the basic reproduction number of a single group SIR model:

(4.17)

The following proposition gives a lower and upper bound on the basic reproduction number R 0 .

Proposition 4.4. Suppose that assumptions (H 1 )-(H 9 ) hold. Then the basic reproduction number defined by (4.12) satisfies the following inequality:

Proof Let R 0j = min 1≤i≤n {R 0i } for some 1 ≤ j ≤ n. By (4.7) and (4.9) we obtain that (a 1 , · · · , a n )V −1 = (1, · · · , 1). Since matrix V −1 is nonnegative and using (4.11), we have (w 1 , · · · , w n ) = f (S) cj aj c1aj cj , · · · , cnaj cj V −1 ≥ R 0j (1, · · · , 1).

By (4.12), we have R 0 = α 1 w 1 + α 2 w 2 + · · · + α n w n ≥ n i=1 α i R 0j = R 0j . Similarly, we can show R 0 = α 1 w 1 + α 2 w 2 + · · · + α n w n ≤ max 1≤i≤n R 0i , completing the proof. We have adapted the graph-theoretic approach developed in [43, 44] to system (4.2). We first describe the Lyapunov function for the case R 0 ≤ 1.

Proposition 5.1. Let w i is defined by (4.11). If (I 1 , I 2 , · · · , I n ) T ≤ (F − V )(I 1 , I 2 , · · · , I n ) T (5.1) holds in Γ, and R 0 ≤ 1, then L(I 1 , I 2 , · · · , I n ) = (w 1 , w 2 , · · · , w n )(I 1 , I 2 , · · · , I n ) T (5.2)

is a Lyapunov function for system (4.2) in Γ, namely, d dt L(I 1 (t), I 2 (t), · · · , I n (t)) ≤ 0, for all t ≥ 0, along all solutions of (4.2) in Γ.

Proof Differentiating function L(t) = L(I 1 (t), · · · , I n (t)) with respect to t,

we haveL ≤ (w 1 , · · · , w n )(F − V )(I 1 , · · · , I n ) T = (w 1 , · · · , w n )(α 1 , · · · , α n ) T f (S)c 1 , · · · , f (S)c n (I 1 , · · · , I n ) T − (w 1 , · · · , w n )V (I 1 , · · · , I n ) T .

Using (4.11) and (4.12) we obtaiṅ We make the following assumptions:

It can be verified that these assumptions are satisfied by common incidence forms and linear transfer functions. Since biologically the incidence function 

for all (S, I 1 , · · · , I n ) ∈ Γ. Let K be the largest invariant subset of {(S, I 1 , · · · , I n ) ∈ Γ :L = 0}. Then P 0 ∈ K. Let (S(t), I 1 (t), · · · , I n (t)) be a solution in K. We (1), we obtain that I i = 0 for all i. In both cases, we have 395 shown that K = {P 0 }. By LaSalle's Invariance Principle [60] , P 0 is globally asymptotically stable in Γ when R 0 ≤ 1. This establishes part (a).

If R 0 > 1, then, by continuity,L > 0 for S sufficiently close toS except when I 1 = I 2 = · · · = I n = 0. Solutions starting sufficiently close to P 0 leave a small neighbourhood of P 0 , except for those on the invariant S-axis. Since the Corollary 5.2. Suppose that assumptions (H 1 ) -(H 9 ) hold. Assume that R 0 > 1 and that the transmission-transfer network of system (4.2) is strongly con-410 nected. Then system (4.2) has an endemic equilibrium in the interior of Γ.

The assumption that the transmission-transfer network is strongly connected is necessary for ruling out boundary equilibria other than P 0 , and in turn, crucial for the proof of uniform persistence and existence of positive endemic equilibrium P * when R 0 > 1. This condition was neglected in many earlier work We assume that f (N ) ≡ 1 in this subsection and establish that, when R 0 > 1, system (4.2) has a unique endemic equilibrium P * and it is globally asymptotically stable in the interior of Γ. [43, 44] has been successfully applied to many complex systems. An application of the approach to multi-staged models was shown in [9] .

Comparing to the multi-stage model in [9] , system (4.2) has disease incidence terms in all I i equations and are more interconnected. The graph-theoretic approach needs to be adapted to (4.2) to resolve its global-stability problem.

Let P * = (S * , I * 1 , · · · , I * n ), I * 1 , · · · , I * n > 0, be an endemic equilibrium. Using the weight function w ij (S, I 1 , · · · , I n ) in (4.5), we define a nonnegative matrix M = (m ij ) n×n , with m ij = w ij (S * , I * 1 , · · · , I * n ) = α i g j (S * , I * j ) + φ ij (I * j ), 1 ≤ i, j ≤ n. Motivated by method in [9] , we construct a candidate Lyapunov function of form V (S, I 1 , I 2 , · · · , I n ) =τ

where τ i is given in Proposition 3.1 andτ = n i=1 τ i α i . In the special case when α 1 = 1, and α i = 0 for i = 2, · · · , n, the Lyapunov function in (5.5) reduces to

Then the function V defined in (5.5) is a Lyapunov function for system (4.2)

with respect to the interior of Γ. 440 We remark that assumption (B 3 ) is automatically satisfied if transfer functions φ ij and ψ j are linear. If the incidence function is of separable form hold for all I j > 0, 1 ≤ j ≤ n. Therefore, along solutions that stay in the largest invariant subset of {(S, I 1 , · · · , I n ) ∈ Γ :V = 0}, the following relation holds S = S * , g j (S * , I j ) = λg j (S * , I * j ).

Using the S-equation in system (4.2), we have

This equation holds only at λ = 1. Substituting λ = 1 into (5.6), we obtain

The monotonicity assumption (B 4 ) implies that I i = I * i for all i. Thus, the largest invariant set in the set {V = 0} is the singleton {P * }. By LaSalle's Invariance Principle [60] , P * is globally asymptotically stable in the interior of Γ. The global stability also implies that the endemic equilibrium is unique.

We note that in the case when all newly infected individuals go into the first 460 state, namely, α 1 = 1, α i = 0, 2 ≤ i ≤ n, Theorem 5.1 and Theorem 5.2 give the global-stability results in [9] .

We also note that, if f (N ) is a nonconstant function, the global stability of P * when R 0 > 1 remains largely open, even for the case when f (N ) is monotone.

For certain non-monotone functions f (N ), it was shown in [49] that multiple 465 endemic equilibria can exists and periodic oscillations can occur, and the global stability results are not expected to be valid.

Results from numerical simulations are provided in this section to demonstrate our theoretical results. Several observations are made regarding the im- To focus our study on the state structure, we choose a simple form for functions f , g, φ and ψ in model (4.2). We let n = 5 and consider a special star-shaped directed graph for cross-transmissions among the 5 states, as shown in Figure 6-(a) . The corresponding weight matrixM = (m ij ) 5×5 is given bȳ

is irreducible. The simplified model is described by the following systems of differential equations: 

From the results in Section 5 we know that the disease-free equilibrium P 0 is globally asymptotically stable if R 0 ≤ 1, and that, when R 0 > 1, there exists a unique positive endemic equilibrium P * = (S * , I * 1 , · · · , I * 5 ) and it is globally asymptotically stable. equilibrium, as well as important factors that impact this distribution. This is a complicated problem on its own. Using numerical simulations, we provide some interesting observations. Let P * = (S * , I * 1 , · · · , I * 5 ) be the endemic equilibrium, we are interested in factors that influence the distribution of disease prevalence 500 I * i at state i given by (I * 1 , · · · , I * 5 ). A specific question of interest is the following: what factor (or factors) determines that a particular I * k is the highest among all I * i ? or determines the order among I * i ? Important factors we have investigated include transmission coefficients β i , fraction α i of disease incidences in state i, and transfer rates φ ij among disease states.

Observation I: Varying transmission coefficients β i do not alter the order among the disease prevalences I * i , i = 1, · · · , 5. This is intuitively clear since β i directly influence the disease incidences, which is given in the following form, for the state i:

We expect that β k will influences the overall values of all I * i but will not alter the relative order among the disease prevalence I * i . In another word, the highest β k may not produce the largest I * k .

In Figure 6 , the transmission coefficients at each state are given as (β 1 , · · · , β 5 ) = (0.2140, 0.1070, 0.3210, 0.4280, 0.2675) × 10 −4 (6.6) and plotted in Figure 6 -(b). The disease prevalence at P * among states is plotted in Figure 6 -(d). We see that the second state has the highest prevalence I * 2 , while it has the lowest transmission coefficient β 2 .

Observation II. For transfers among states, the state with the highest indegree or out-degree may not have the highest disease prevalence.

The impacts of transfer rates δ ij among states on the distribution of I * i are highly complex and is an interesting topic worthy of further investigation. In the example shown Figure 6 , the state-transfer network is given in Figure 6-(a) together with the weights. It is clear from the directed graph that the first state 28 J o u r n a l P r e -p r o o f Journal Pre-proof (vertex) has the largest number of edges coming in and going out. Its in-degree 520 and out-degree in the weighted directed graph are both the highest among all states. However, the highest disease prevalence I * i occurred at the second state i = 2, even when α 1 = α 2 .

Observation III. Proportions α i of disease incidence among states have a direct influence on distribution of disease prevalence. 525 From (6.5) we see that α i has the direct influence on the number of new infections going into state i. We expect that distribution of I * i will mimic that of α i . As shown in Figure 6 -(c) and Figure 6 -(d), this is indeed the case.

Summarizing our observations, the parameter α i , which is the proportion of new infections entering the state i, has the most important impact on deciding 530 which state will have the highest disease prevalence. The biological significance of this observation is that it is important for clinical researchers to measure proportions of newly infected HIV patients who are in different HIV states. It is known that some patients whose HIV infection progresses much faster than normal patients, the so-called 'fast progressors', while some patients whose HIV 535 infection progresses much slower, the 'nonprogressors', and others can manage to control the level HIV viral load without treatment treatments, the 'elite controllers' [28, 29] . Out results show that it is important for the HIV control to estimate the proportions of these 'abnormal' patients.

The basic reproduction number R 0 determines whether an infectious disease can be effectively controlled. Many disease interventions are aiming at reducing R 0 . It can be seen in the expression of R 0 in (4.12) that reducing the transmission coefficients β i will lower the value of R 0 , which is also biologically intuitive. Simulation results are shown in Table 6 .2. We can see that an increase in 565 δ 12 , δ 31 , δ 41 , δ 51 will decease R 0 from its baseline value 6.0074; while an increase of δ 21 , δ 13 , δ 14 , δ 15 will increase R 0 from the baseline value.

These observations strongly suggest that further theoretical investigations of the impacts of state transfers on the overall dynamics are warranted.

The authors wish to thank the anonymous referees whose comments and suggestions have helped to improve the presentation of the manuscript. 2, 0.2, 0.2, 0.2, 0.3, 0.3, 0.3, 0.3 · · · αnf (S)gn(S,In)

Therefore, condition (5.1) is verified and function L is a Lyapunov function.

2. Proof of Proposition 5.2. By Proposition 3.1 and irreducibility of matrix M , we know that τ i ,τ > 0. Let (S(t), I 1 (t), · · · , I n (t)) be a positive solution of system (4.2) and V be the defined in (5.5). Set V (t) = V (S(t), I 1 (t), · · · , I n (t)).

ThenV Assumption (B 1 ) implies that

By assumption (B 3 ), together with the fact that function s(x) = 1 − x + ln x has a global maximum at x = 1, we obtain 

Then, using the definition ofm ij in (5.4), we obtaiṅ ). An additional node S is added to demonstrate that the additional weight ↵ i f (S)c ij on the edge from I j to I i is created indirectly through infection of S by I j indicated by a dashed edge from I j to S, and the corresponding incidence in I i indicated by an solid edge from S to I i .

state. Then, when I ⇤ i = 0, there exist state j for which I ⇤ j > 0, and a boundary endemic equilibrium exists.

The state-transfer network of the disease as the weighted digraph (H, (b ij )) can also be considered, where b ij is defined in (H 8 ). If (H, (b ij )) is strongly connected, then it can be shown that P 0 is the only boundary equilibrium. However this condition is not necessary even in the simplest epidemic models. To see this, consider a simple SEIR model, whose transmission-transfer network depicted in Figure ? ? is strongly connected, while the state-transfer network among the latent state E and infectious state I only contains a single arrow from E to I, and is not strongly connected. In this paper, we only consider the case when the transmission-transfer network is strongly connected. When this assumption is not satisfied, the dynamics of model (3.2) can be analyzed using the approach in [5]. Observation III. Probability distribution ↵ i of disease incidences among states has a direct influence on distribution of disease prevalence. From (5.5) we see that ↵ i has the direct influence on the number of new infections going into state i. We expect that distribution of I ⇤ i will mimic that of ↵ i . As shown in Figure 5 .2-(c) and Figure 5 .2-(d), this is indeed the case.

Summarizing our observations, the parameter ↵ i , which is the probability of a newly infected person is in state i, has the most important impact on deciding which state will have the highest disease prevalence. The biological significance of this observation is that it is important for clinical researchers to measure proportions of newly infected HIV patients who are in di↵erent HIV states. It is known that some patients whose HIV infection progresses much faster than normal patients, the socalled 'fast progressors', while some patients whose HIV infection progresses much slower, the 'nonprogressors', and others can manage to control the level HIV viral load without ART treatments, the 'elite controllers' [16, 23] . Out results show that it is important for the HIV control to estimate the proportions of these 'abnormal' patients. 

Global dynamics of a staged-progression model with amelioration for infectious diseases

Global dynamics of a general class of multistage models for infectious diseases

A note of a staged progression HIV model with imperfect vaccine

Stationary distribution of a stochastic staged progres-605 sion HIV model with imperfect vaccination

Stationary distribution of a stochastic HIV model with two infective stages

The'cumulative'formulation of (physiologically) structured population models

Discrete-time population dynamics on the state space of measures

Persistence versus extinction for a class of discrete-time structured population models

A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients

Tracking epidemics with google flu trends data and a state-space SEIR model

Predicting intervention 630 effect for covid-19 in japan: state space modeling approach

A new product growth for model consumer durables

Influence 635 of luddism on innovation diffusion

Cellular automata modeling of resistance to innovations: Effects and solutions

An information diffusion model based on retweeting mechanism for online social media

Can a few fanatics influence the opinion of a large 645 segment of a society?

Reinforcement-driven spread of innovations and fads

A general markov chain approach for disease and rumour spreading in complex networks

Lumping of degree-based mean-field and pair-approximation equations for multistate contact processes

Diffusion of competing rumours on 660 social media

Heterogeneous neutralizing antibody and antibody-dependent 665 cell cytotoxicity responses in HIV-1 elite controllers

Epidemiology and clinical characteristics of elite controllers

Sources of racial disparities in HIV prevalence in men who have sex with men in atlanta, GA, USA: a modelling study

A dynamic network model to disentangle the roles of steady and casual partners for HIV transmission 34

among MSM

Substan-680 tial undocumented infection facilitates the rapid dissemination of novel coronavirus (sars-cov2)

Asymptomatic transmission, the achilles heel of current strategies to control covid-19

Understanding covid-19 vaccine efficacy

Modeling the effects of carriers on transmission 690 dynamics of infectious diseases

Size and timescale of epidemics in the sir framework

A simple sir model with a large set of asymptomatic infectives

A simple 700 but complex enough θ-sir type model to be used with covid-19 real data. application to the case of italy

New approximations, and policy implications, from a delayed dynamic model of a fast pandemic

Generalizations of the 'linear chain trick': incorporating more flexible dwell time distributions into mean field ODE models

Binary-state dynamics on complex networks: Pair approximation and beyond

Multistate dynamical processes on networks: 715 Analysis through degree-based approximation frameworks

A graph-theoretic approach to the method of global lyapunov functions

Global-stability problem for coupled systems of differential equations on networks

Global dynamics of an infinite dimensional epidemic model with nonlocal state structures

Global dynamics of a staged-progression model for HIV/AIDS with amelioration

Modelling the HIV/AIDS 730 epidemic trends in south africa: Insights from a simple mathematical model

Modelling the impact of HIV on the populations of south africa and botswana

Density dependence in disease incidence and its impacts on transmission dynamics

Nonnegative matrices in the mathematical 740 sciences

Counting labelled trees

Ueber die auflsung der gleichungen, auf welche man bei der untersuchung der linearen vertheilung galvanischer strme gefhrt wird, Annalen der

Global lyapunov functions and a hierarchical control scheme for networks of robotic agents

Diseases in metapopulations

Impact of network connectivity on the synchronization and global dynamics of coupled systems of differential equations

On the definition and the computation of the basic reproduction ratio r0 in models for infectious diseases in heterogeneous populations

Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission

Final and peak epidemic sizes for seir models with quarantine and isolation

Global dynamics of a multistage SIR model with distributed delays and nonlinear incidence rate

The stability of dynamical systems

Uniform persistence and flows near a closed positively invariant set

Global dynamics of a seir model with varying total population size

The theory of the chemostat: dynamics of microbial competition

Dynamical systems: stability theory and applications

A mathematical modeling study of the HIV epidemics at two rural townships in the liangshan prefecture of the sichuan province of china

Technical proofs of results in the main sections are provided.