Analysis of cholera epidemics with bacterial growth andspatial movement

Xueying Wang a,∗ and Jin Wang b

aDepartment of Mathematics,Washington State University, Pullman, WA 99164, USA

∗ Email: xueying@math.wsu.edu

b Department of Mathematics,University of Tennessee at Chattanooga, Chattanooga, TN 37403

Email: jin-wang02@utc.edu

ABSTRACT

In this work, we propose novel epidemic models (named, susceptible-infected-recovered-susceptible-bacteria, or SIRS-B) for cholera dynamics by incorporating a general for-mulation of bacteria growth and spatial variation. In the first part, a generalizedordinary differential equation (ODE) model is presented and it is found that bacte-rial growth contributes to the increase of the basic reproduction number, R0 . Withthe derived basic reproduction number, we analyze the local and global dynamicsof the model. Particularly, we give a rigorous proof on the endemic global stabil-ity by employing the geometric approach. In the second part, we extend the ODEmodel to a partial differential equation (PDE) model with the inclusion of diffusionto capture the movement of human hosts and bacteria in a heterogeneous environ-ment. The disease threshold of this PDE model is studied again by using the basicreproduction number. The results on the threshold dynamics of the ODE and PDEmodels are compared, and verified through numerical simulation. Additionally, ouranalysis shows that incorporating diffusive spatial spread does not produce a Turinginstability when R0 associated with the ODE model is less than the unity.

1. Introduction

Recent years witnessed an increasing number of cholera outbreaks worldwide [42],including one of the largest cholera epidemics in modern history that took placein Haiti during 2010-2012 with more than 530,000 reported cases and over 7,000deaths, and the worst African cholera outbreak in the past 20 years that devastatedZimbabwe during 2008-2009 with nearly 100,000 reported cases and more than 4,000deaths. These outbreaks, with their increased frequency and severity, indicate thatour current knowledge in cholera dynamics and public health guidelines to controlthe disease are not adequate.

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Cholera is an acute intestinal infectious disease caused by the bacterium Vibriocholerae. It can spread rapidly and lead to death within days if left untreated. Thecomplexity of cholera dynamics stems from the fact that both direct (i.e., human-to-human) and indirect (i.e., environment-to-human) routes are involved in the diseasetransmission [21]. Thus, the dynamics of cholera involve multiple interactions amongthe human host, the pathogen, and the environment [25]. In an effort to gain deeperunderstanding of cholera dynamics, many mathematical models have recently beenproposed and analyzed. Codeco published a model [6] that added the pathogen con-centration in the water supply into a regular SIR epidemiological model. Hartley etal. [12] included a hyperinfectious state of the pathogen, representing the “explosive”infectivity of freshly shed vibrios [1]. Tien and Earn [34] and Mukandavire et al. [21]explicitly considered both human-to-human and environment-to-human transmissionpathways. Wang and Liao [38] proposed a cholera model that incorporated generalincidence and pathogen functions. Shuai et al. [30] investigated cholera dynamicswith both hyperinfectivity and temporary immunity. Other cholera models include,but not limited to, [2–5, 17, 20, 26, 31, 33, 35].

One limitation in most of these mathematical cholera studies is that the dynamicsof the pathogen (i.e., the vibrios) are poorly addressed. A standard assumptionin almost all the cholera models is that the vibrios cannot sustain themselves inthe absence of human contribution (e.g., shedding from infected individuals andinflow from contaminated sewage). This assumption allows a simple, often linear,representation of the rate of change for the bacterial density: a positive contributionfrom the infected human population, and a negative contribution due to naturaldeath of the vibrios. On the other hand, there have been strong evidences [7] thatthe vibrios can independently persist in the environment and, consequently, theirintrinsic growth and decay may play an essential role in shaping cholera epidemics.Only very few studies [3, 17] so far have taken into account the intrinsic bacterialdynamics in cholera modeling.

Another challenge in current mathematical cholera studies is that spatial hetero-geneity is rarely considered, resulting in insufficient understanding of the spatialspread of cholera infection. In the work of [21], basic reproduction numbers wereestimated for all the 10 provinces in Zimbabwe and the results were highly heteroge-neous, implying that the underlying transmission pattern varied widely throughoutthe country. Similarly, Tuite et al. [35] obtained very different reproduction numbersfor the 10 administrative departments in Haiti during the recent cholera outbreak.The findings in these studies underscore the importance of spatial heterogeneity incholera transmission and the design of control strategies, and suggest that more workis demanded toward better understanding of this issue. In addition, Bertuzzo andco-workers [4] developed a simple partial differential equation (PDE) model, building

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on the framework of Codeco’s model [6], to investigate the spatial movement of thepathogen in cholera epidemic setting.

In the present paper, we aim to address the aforementioned challenges in choleramodeling by proposing new models that incorporate both intrinsic bacterial dynamicsand spatial variation. We will start with an ordinary differential equation (ODE)model that allows a general representation of the direct and indirect transmissionpathways and a general description of the bacterial growth. We then derive the basicreproduction number of this model and analyze the local and global stabilities forthe disease-free equilibrium and endemic equilibrium. Particularly, we will apply thegeometric approach originally introduced by Li and Muldowney [18] to the endemicglobal stability analysis. Next, we extend this model to a PDE system by addingdiffusion terms for both the human population and the pathogen. We investigatethe threshold dynamics of this PDE model by analyzing and estimating its basicreproduction number. To our knowledge, no prior work has been published on thethreshold dynamics of PDE cholera models. In addition, we verify our analysis usingnumerical simulation results.

The remainder of this paper is organized as follows. In Section 2, the ODE model ispresented and necessary assumptions are stated, followed by a careful analysis of thethreshold dynamics based on the derived basic reproduction number. In Section 3,the PDE model is introduced, the possibility of Turing instability is investigated, andthe basic reproduction number is analyzed and compared to that of the ODE model.In Section 4, numerical results are presented to validate the analytical predictions.Finally, conclusions are drawn and some discussion is presented in Section 5.

2. A generalized SIRS-B cholera epidemic model

2.1. Model Description. Cholera infection consists of two populations: humanhosts and bacteria. Since cholera infection does not invoke a long-lasting immune re-sponse [19, 23, 44], we apply the standard susceptible-infected-recovered-susceptible(SIRS) epidemic framework for the infection with the host population. Moreover,we assume that (a) a susceptible host becomes infected either by direct contact withinfectious hosts or via indirect contact with bacteria in contaminated water; (b)infectious hosts contaminate the environment through shedding of the bacteria. Acompartmental diagram of a generalized SIRS-B epidemic model (where B stands forthe bacterial concentration in the environment) describing the dynamics of cholerainfection is displayed in Figure 1. This generalized model can be written as

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Figure 1. A compartmental diagram of the SIRS-B model (2.1).

(2.1)

dS

dt= b− Sf1(I)− Sf2(B)− dS + σR,

dI

dt= Sf1(I) + Sf2(B)− (d+ γ)I,

dR

dt= γI − (d+ σ)R,

dB

dt= ξI + h(B)− δB.

Here S, I andR represent the number of susceptible, infectious and recovered hosts,respectively, and B is the concentration of the bacteria (vibrios) in the contaminatedwater. The parameter b describes the influx (or, recruitment) of susceptible hosts.The functions f1(I) and f2(B) depict the direct and indirect transmission rate, re-spectively. For example, f1(I) = 0 and f2(B) = aB/(B + κ) in the model of Codeco[6] (where a is the contact rate with contaminated water, and κ is the half saturationrate that describes the infectious dose in water sufficient to produce disease in 50%of those exposed), f1(I) = βI and f2(B) = λB in the model proposed by Gosh et al.[10] (where β and λ represent the direct and indirect transmission parameters due tothe human-to-human and the environment-to-human interaction, respectively), andf1(I) = βhI and f2(B) = βeB/(B+κ) in the model of Mukandavire et al. [21] (whereβh and βe represent the direct and indirect transmission parameters). In addition,d is the natural death rate of each host class, γ is the recovery rate of infectiousindividuals, σ denotes the rate at which recovered individuals lose immunity, ξ is the

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shedding rate of bacteria by infectious hosts, and δ is the natural death rate of thebacteria. All these aforementioned model parameters are assumed to be positive.

Furthermore, as most existing cholera models (e.g., [6, 12, 16, 17, 21]), disease-induced mortality is assumed to be negligible. A reason for making this assumptionis that, with a few exceptions, the death rate for cholera is generally quite low (about1%). For example, WHO states that “In 2012, the overall case fatality rate for cholerawas 1.2%”[43].

Finally, the function h(B) is introduced to describe the intrinsic growth of thebacteria. The functions f1, f2 and h are assumed to be all differentiable in thisstudy.

We assume that

(H1) (1) f1(I) ≥ 0; (2) f1(0) = 0; (3) f ′1(I) > 0; (4) f ′′1 (I) ≤ 0.(H2) (1) f2(B) ≥ 0; (2) f2(0) = 0; (3) f ′2(B) > 0; (4) f ′′2 (B) ≤ 0.(H3) (1) h(0) = 0; (2) h′′(B) ≤ 0.

Biologically, the assumptions (H1) and (H2) state that the disease transmission ratesare monotonically increasing, but also subject to saturation effects. The incidencefunctions employed in most of the existing cholera models (e.g., [6, 10, 21, 34]) satisfythese conditions. Meanwhile, the assumption (H3) states that the bacterial growthrate is also subject to saturation effects. We mention, however, that (H3) excludessome more complicated growth models such as the logistic growth with a thresholdTB (i.e., h(B) = rB(1 − B/K)(B/TB − 1))) and extended logistic growth models[24].

2.2. Basic Reproduction Number. In epidemic models, one of the main concernsis to quantify the infection risk so as to effectively control the disease. An importantdisease control threshold is the basic reproduction number, R0, which measures theexpected number of secondary infections caused by one infectious individual duringits infectious period in an otherwise susceptible population [9, 37]. It can be com-puted using the next generation matrix theory [37]. We will focus our attention onthe basic reproduction number in this paper, but see Appendix A for two other typesof threshold quantities which are closely related to R0 .

Let N∗ = b/d. One can easily see from direct calculation that the disease freeequilibrium (DFE) is given by (N∗, 0, 0, 0). The Jacobian matrix J at the DFEis decomposed as follows: J = F − V, where F is the matrix characterizing thegeneration of secondary infectious cases/agents, and V is the matrix of transitionrates between compartments. We assume that bacteria population in the environ-ment is regarded as a reservoir of the infection. The free-living bacteria adapt tothe environment through physiological and genetic changes that can promote their

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survival and growth [7]. Thus, we assume that the bacteria can possibly be self-maintained through growth in the environment, and secondary bacteria can be intro-duced through pathogen shedding by infectious humans. For simplicity of notations,we write

f(I, B) = f1(I) + f2(B), fI = f1′(0), fB = f ′2(0), and g = h′(0).

Based on the standard next-generation matrix technique [37] and our assumptions,matrices F and V have the following forms:

F =

[N∗fI N∗fBξ g

]and V =

[(d+ γ) 0

0 δ

].(2.2)

Hence the next generation matrix is

K = FV−1 =

N∗fId+ γ

N∗fBδ

ξ

d+ γ

g

δ

.(2.3)

The basic reproduction number R0 is given by

(2.4) R0 = ρ(K) =1

2

[ N∗fId+ γ

+g

δ+

√(N∗fId+ γ

− g

δ

)2

+ 4ξN∗fBδ(d+ γ)

].

Here ρ(M) denotes the spectral radius for any n× n matrix M.In particular, the quantity Rh

0 := N∗fI/(d + γ) (resp. Re0 := g/δ) represents

the average number of secondary infections through human-to-human (resp. theenvironment-to-human) transmission produced by one infectious host during its in-fectious period. g/δ depicts the average number of secondary free-living bacteriacaused by one bacterium during its lifetime in the environment.

Recall that ξ > 0 by the model assumption. It is straightforward to see fromassumptions (H1)-(H3) that

R0 > max{Rh

0 , Re0

}.

This result shows that (a) the cholera infection with bacterial presence is more severethan that in the absence of bacteria, which indicates that if an outbreak occurs inhuman population without bacteria (Rh

0 > 1), then an outbreak will certainly takeplace in the presence of bacteria (R0 > 1); (b) if the free-living bacteria can sustainthemselves in the environment without hosts (Re

0 > 1), then disease will persist inthe presence of hosts (R0 > 1).

Suppose that all the parameters except g are constants. Consider R0 as a functionof g, i.e., R0 = R0(g). Based on our model assumptions, one can verify by directcalculation that

R0(g) > R0(0), if g > 0.

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This indicates that the basic reproduction number of (2.1) in the presence of bacterialgrowth is higher than that without inclusion of bacterial growth. In other words,the persistence and intrinsic growth of vibrios would increase the risk of choleraepidemics. Moreover, it is easy to see that R0 > 1 if g ≥ δ. This implies that withstrong growth and prevalence of vibrios, utilizing only vaccination and/or antibioticswould not be able to reduce R0 below 1. Instead, improving infrastructure andproviding clear water would be of fundamental importance in controlling cholera inthe long run.

2.3. Equilibrium Analysis. It follows by direct calculation that an equilibrium forthe system (2.1) is a solution of the following equations:

S + I +R = N∗,(2.5)

R =γ

d+ σI,(2.6)

I = φ(B) :=−h(B) + δB

ξ,(2.7)

Sf1(I) + Sf2(B) = (d+ γ)I.(2.8)

By (2.5) and (2.6), S := S(I) = N∗ − (1 + γ/(d+ σ))I. By assumption (H2),f ′2(B) > 0, and hence there exists a unique function f−1

2 , such that f−12 (f2(B)) = B.

Let α = 1 + γ/(d+ σ). In view of (2.8),

(2.9) B = ψ(I) := f−12

( (d+ γ)I

N∗ − αI− f1(I)

).

The intersection points of the curves I = φ(B) and B = ψ(I) in R2+ determine the

equilibria. First, let’s consider the concavity of the curves (2.7) and (2.9) on the B-Iplane. By (H3),

φ′′(B) = −h′′(B)/ξ ≥ 0,

and φ(B) is concave up. On the other hand, by a direct calculation, we have

(2.10) ψ′′(I) =[2α(d+ γ)N∗

S3− f ′′1 (I)− f ′′2 (B)(ψ′(I))2

]/f ′2(B).

By assumptions (H1) and (H2), (2.10) yields ψ′′(I) > 0, and hence ψ(I) is concave up.So, there exist at most two points of intersection in R2

+. Notice that φ(0) = ψ(0) = 0.There always exists a unique disease-free equilibrium (DFE), (I, B) = (0, 0), whereS = N∗ and R = 0. There may be a second, endemic equilibrium (EE), where thetwo curves intersect in the interior of R2

+. This depends on the slopes of the curvesat zero: ψ′(0) =

[(d+ γ)/N∗ − f ′1(0)

]/f ′2(0) and φ′(0) = (−g + δ)/ξ. There are two

cases where the two curves intersect in the interior of R2+, leading to an EE: (a)

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g ≥ δ or (b) g < δ and ψ′(0) < [φ′(0)]−1. The stability of the equilibrium will beestablished in Theorem 2.1.

First, let’s consider ρ(M) for a square matrix M of size 2. Denote “if and onlyif” as ⇐⇒ .

Lemma 2.1. Let M =

[a bc d

]with a+ d ≥ 0 and bc ≥ 0. Then

(a) ρ(M) < 1 ⇐⇒ a < 1, d < 1 and trace(M) < det(M) + 1.

(b) ρ(M) = 1 ⇐⇒ a ≤ 1, d ≤ 1 and trace(M) = det(M) + 1.

(c) ρ(M) > 1 ⇐⇒ trace(M) ≥ 2, or trace(M) < 2 and trace(M) > det(M) + 1.

Proof. (a) By the assumption on M, it is easy to verify that

(2.11) ρ(M) =1

2

[a+ d+

√(a− d)2 + 4bc

].

Thus,

ρ(M) < 1 ⇐⇒√

(a− d)2 + 4bc < 2− (a+ d),

⇐⇒ (a− d)2 + 4bc < (2− (a+ d))2, and 2− (a+ d) > 0,

⇐⇒ a+ d < ad− bc+ 1, and (a+ d) < 2,

⇐⇒ trace(M) < det(M) + 1, and trace(M) < 2.

(2.12)

By (2.11), one can directly verify that ρ(M) ≥ max{a, d} ≥ 1 if d ≥ 1 or a ≥ 1.Thus, ρ(M) < 1 implies a < 1 and d < 1. Together with (2.12), we show that

(2.13) ρ(M) < 1 =⇒ a < 1, d < 1 and trace(M) < det(M) + 1.

The other direction of (2.13) is trivial by virtue of (2.12). Likewise, one can prove(b) and (c). �

Remark of Lemma 2.1 (c): If bc > 0, one can verify that

ρ(M) = 1 ⇐⇒ a < 1, d < 1, and trace(M) = det(M) + 1.

ρ(M) > 1 ⇐⇒ a ≥ 1, or d ≥ 1, or a < 1, d < 1 and trace(M) > det(M) + 1.

(2.14)

Based on the expression of R0 in (2.4), we can easily obtain that when g ≥ δ , R0 > 1.Meanwhile, note that

φ′(B) =δ − h′(B)

ξand ψ′(I) =

N∗(d+γ)(N∗−αI)2 − f

′1(I)

f ′2(B),

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which yields

[φ′(0)]−1 =ξ

δ − gand ψ′(0) =

d+ γ −N∗fIN∗fB

.

Lemma 2.2. The following statements hold:(1) R0 ≤ 1 ⇐⇒ g < δ, and ψ′(0) ≥ [φ′(0)]−1.(2) R0 > 1 ⇐⇒ g ≥ δ , or g < δ, and ψ′(0) < [φ′(0)]−1.

Proof. (1). Applying Lemma 2.1 (a), and (2.14) to K, we obtain

R0 ≤ 1 ⇐⇒ N∗fI < d+ γ, g < δ, and N∗fBξ ≤ (d+ γ −N∗fI)(δ − g),

⇐⇒ g < δ andd+ γ −N∗fI

N∗fB≥ ξ

δ − g,

⇐⇒ g < δ and ψ′(0) ≥ [φ′(0)]−1.

(2.15)

This proves (1). Similarly, we can show (2). �

We now prove the following theorem regarding the local stabilities of the DFEand EE of the system (2.1). The local stability of the DFE; that is, the DFE islocally asymptotically stable for R0 < 1 ans unstable for R0 > 1, can be obtaineddirectly from Theorem 2 of [8]. To demonstrate the local stability of the DFE whenR0 = 1, however, we provide a proof below which automatically integrates the casesfor R0 < 1 and R0 = 1.

Theorem 2.1.

1. If R0 ≤ 1, then (2.1) has only the DFE and it is locally asymptotically stable.2. If R0 > 1, then (2.1) has two equilibria: the DFE and the EE. Furthermore,

the DFE is unstable and the EE is locally asymptotically stable.

Proof. Linearizing (2.1) at the equilibrium (S, I, R,B), we obtain the Jacobian ma-trix

(2.16) J =

−(f1(I) + f2(B))− d −Sf ′1(I) σ −Sf ′2(B)

f1(I) + f2(B) Sf ′1(I)− (d+ γ) 0 Sf ′2(B)0 γ −(d+ σ) 00 ξ 0 h′(B)− δ

Case 1: R0 ≤ 1. By Lemma 2.2, R0 ≤ 1 implies g < δ and ψ′(0) ≥ [φ′(0)]−1. Hencewe see that (2.1) has only one equilibrium and it is the DFE. Evaluating J at theDFE, we find that λ1 = −d, λ2 = −(d + σ), λ3 + λ4 =

(N∗fI − (d + γ)

)+ (g − δ)

and λ3λ4 =(N∗fI − (d+ γ)

)(g − δ)−N∗ξfB. Clearly, λ1 and λ2 are both negative.

Hence the local stability of the DFE relies on the sign of λ3 and λ4. In view ofψ′(0) ≥ [φ′(0)]−1 = ξ/(δ − g) > 0, we have ψ′(0) =

[(d+ γ)/N∗ − fI

]/fB > 0.

fB > 0 implies N∗fI − (d + γ) < 0. Together with g < δ, it gives λ3 + λ4 < 0. By

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[φ′(0)]−1 < ψ′(0) and g < δ, one can directly verify that λ3λ4 > 0. This impliesλ3 < 0 and λ4 < 0. Therefore, the DFE is locally asymptotically stable.

Case 2: R0 > 1. In view of Lemma 2.2, R0 > 1 if and only if (a) g ≥ δ; (b)g < δ and ψ′(0) < [φ′(0)]−1. In either case, (2.1) has the DFE and EE. By a similarargument as that of Case 1, one can verify that the DFE is unstable. It remains toshow that the EE is locally asymptotically stable.

Let N(t) = (S + I +R)(t). We consider an equivalent system of (2.1)

dN

dt= b− dN,

dI

dt= (N − I −R)(f1(I) + f2(B))− (d+ γ)I,

dR

dt= γI − (d+ σ)R,

dB

dt= ξI + h(B)− δB.

(2.17)

Then the Jacobian matrix of (2.17) evaluated at the EE (N∗, I, R,B) is given by

(2.18) J =

−d 0 0 0

f(I, B) −f(I, B) + Sf ′1(I)− (d+ γ) −f(I, B) Sf ′2(B)0 γ −(d+ σ) 00 ξ 0 h′(B)− δ

where f(I, B) = f1(I) + f2(B), and S = N∗ − I − R. It is easy to verify that the

characteristic polynomial of J is

det(λI − J) = (λ+ d)(λ3 + a1λ2 + a2λ+ a3),

where

a1 = −(c1 + c3 + c4),

a2 = c1c4 + c3(c1 + c4) + c0 + ξc2,

a3 = −c3

(c1c4 + c0

)− ξc2c4,

(2.19)

with

c0 = f(I, B)γ,

c1 = −f(I, B) + Sf ′1(I)− (d+ γ),

c2 = −Sf ′2(B),

c3 = h′(B)− δ,c4 = −(d+ σ).

(2.20)

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Clearly, λ1 = −d < 0. Thus, the stability of the EE is determined by the zeros of

λ3 + a1λ2 + a2λ+ a3 = 0.

It follows from the Routh-Hurwitz criterion that to verify that the EE is stable, itsuffices to show that

(2.21) a1 > 0, a2 > 0, a3 > 0, a1a2 > a3,

where

a2 = c1c4 + c0 + c3c4 + (c1c3 + ξc2),

a3 = (−c4)(c1c3 + ξc2) + (−c3)c0,

a1a2 − a3 = −(c1 + c3)(

(c1c3 + ξc2) + c4(c3 + c4) + c1c4

)− (c1 + c4)c0.

(2.22)

We now claim that

(1) c0 ≥ 0, ci < 0 (i = 2, 2, 3, 4),

(2) c1c3 + ξc2 > 0.(2.23)

Clearly, (2.21) is satisfied if (2.23) holds. It remains to verify (2.23). In the followingstatement, (S, I, R,B) ∈ R+

4 is assumed to the EE. It is clear that c0 ≥ 0, c2 < 0 andc4 < 0 hold, since f(I, B) ≥ 0, γ ≥ 0, σ ≥ 0, S > 0, f ′2(B) > 0 and d > 0. First, wewant to show that Sf ′1(I)−(d+γ) ≤ 0. In view of I ′ = 0, S(f1(I)+f2(B)) = (d+γ)I.It follows from f1(I) ≥ f ′1(I)I that

Sf ′1(I)− (d+ γ) =S

I

(f ′1(I)I − f1(I)− f2(B)

)≤ 0.

Together with f(I, B) > 0 implies c1 < 0. Second, we will show c3 < 0. Noticeδ = (ξI + h(B))/B. We find that

h′(B)− δ = h′(B)− (ξI + h(B))/B = (h′(B)B − h(B)− ξI)/B < 0,

by virtue of h(B) ≥ h′(B)B and ξ > 0. This gives c3 < 0. Last, we will verify (??)(2) holds. Notice that

c1c3 + ξc2 = −f(I, B)(h′(B)− δ) +(

(h′(B)− δ)(Sf ′1(I)− (d+ γ)

)− ξSf ′2(B))

).

To show c1c3+ξc2 > 0, it suffices to show (h′(B)−δ)(Sf ′1(I)−(d+γ)−ξSf ′2(B)) ≥ 0.By h(B) ≥ h′(B)B, ξI + h(B)− δB = 0 implies ξI + (h′(B)− δ)B ≤ 0. Thus,

B0 := B − ξI/(δ − h′(B)) ≥ 0.

On the other hand, a Taylor expansion of f2 at B yields 0 ≤ f2(B0) = f2(B) +f ′2(B)(B0 − B) + f ′′2 (θB)(B0 − B)2/2, for some θB depending on B0 and B. Byassumption (H2), we see that

f2(B) + f ′2(B)(B0 −B) ≥ 0,

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and hence

f2(B) ≥ f ′2(B)ξI

δ − h′(B).

Together with f1(I) ≥ f ′1(I)I gives

0 = Sf1(I) + Sf2(B)− (d+ γ)I ≥ Sf ′1(I)I + Sf ′2(B)ξI

δ − h′(B)− (d+ γ)I.

Thus,

(h′(B)− δ)(Sf ′1(I)− (d+ γ)

)− ξSf ′2(B) ≥ 0.

The proof is complete. �

Building on the result in Theorem 2.1, we now proceed to analyze the globalstabilities of the equilibria for our cholera model (2.1).

2.4. Global Dynamics. Adding the first three equations of (2.1), we getdN

dt=

b − dN . Hence 0 ≤ N(t) ≤ N∗ if 0 ≤ N(0) ≤ N∗. Now consider the growthof bacteria, which is described by h(B). We introduce another regulation on theintrinsic bacterial growth:

(H4) lim supB→∞ h(B) ≤ (−r + δ)B for some positive constant r .

Note that the rate of change of the bacterial concentration due to the intrinsicgrowth and death is h(B) − δB. Thus, biologically, the assumption (H4) impliesthat in the absence of external contribution (e.g., shedding from infected people),the bacteria would eventually decay (at least in a linear manner) when the bacterialpopulation size is large. Based on this assumption, there exists M > 0 such that0 ≤ B(t) ≤M as long as 0 ≤ B(0) ≤M .

Define the domain

∆ = {(S, I, R,B) : S, I, R,B ≥ 0, S + I +R ≤ N∗, B ≤M} .

Clearly, if the solution of system (2.1) is initially in ∆, it will remain in ∆. Hence, ∆is positively invariant for model (2.1). The following theorem summarizes the globaldynamics of model (2.1).

Theorem 2.2. Suppose that assumptions (H1)-(H4) hold.

(1) If R0 < 1, then the deterministic model (2.1) has a unique DFE that isglobally asymptotically stable in ∆.

(2) If R0 > 1, then the EE of (2.1) is globally asymptotically stable in the interiorof ∆, provided b = dN and sup(S,I,B)∈∆{2(Sf ′1(I))}+ σ ≤ γ.

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Proof. Case (1): R0 < 1.Then Theorem 2.1 implies that the system (2.1) has a unique DFE, and it is locallystable. By assumption (H1)-(H3), f1(I) ≤ fII, f2(B) ≤ fBB and h(B) ≤ gB for allI and B. Hence (2.1) implies

dI

dt≤ N∗(fII + fBB)− (d+ γ)I,

dB

dt≤ ξN∗ + (g − δ)B.

(2.24)

Let X = (I, B)T . (2.24) gives

(2.25)dX

dt≤ (F−V)X.

By the Perron-Frobenius theorem, there exists a nonnegative left eigenvector u of thenonnegative matrix V−1F with respect to the eigenvalue R0 = ρ(FV−1) = ρ(V−1F).Motivated by [32], we define the Lyapunov function

(2.26) L = uTV−1x.

Differentiating L along solutions of (2.1) gives

(2.27) L′ = uTV−1dX

dt≤ uTV−1(F−V)X = (R0 − 1)uTX.

If R0 < 1, then L′ ≤ 0, and L′ = 0 implies uTX = 0. Hence either (1) I = 0 or(2) B = 0. It follows from the second the fourth equations of (2.1) and assumption(H1)-(H3) that either (1) B = 0 or (2) I = 0. Hence I = B = 0. In view of thethird and the first equations of (2.1), we see that R = 0 and S = N∗. Thus, thelargest invariant set where L′ = 0 is the singleton (N∗, 0, 0, 0). Therefore, by LaSalle’sInvariant Principle [15], the DFE is globally asymptotically stable in ∆ if R0 < 1.

Case (2): R0 > 1.By Theorem 2.1, the system (2.1) has two equilibria: the DFE and the EE, andthe DFE is unstable. We now employ the geometric approach based on the secondcompound matrix [18] to analyze the endemic global stability of (2.1). Essentialassumptions and results of the geometric approach are provided in Appendix B.Since the DFE is unstable and located on the boundary of ∆, (2.1) is uniformlypersistent; i.e., there is a constant c > 0, such that

lim inft→∞

{S(t), I(t), R(t), B(t)} > c.

The compactness of ∆ and the uniform persistence of the system (2.1) imply thatthis system has a compact absorbing set. On the other hand, (2.1) has a uniqueequilibrium in the interior of ∆ in Case (2) by Theorem 2.1. We proceed to verify

14

the Bendixson criterion q2 < 0, where q2 is as defined in equation (B.4) in AppendixB.

The Jacobian matrix of the system (2.1), after dropping the equation for R , atthe endemic equilibrium is

(2.28) J =

−(d+ σ)− (f1(I) + f2(B)) −Sf ′1(I)− σ −Sf ′2(B)f1(I) + f2(B) Sf ′1(I)− (d+ γ) Sf ′2(B)

0 ξ h′(B)− δ

,and the associated second compound matrix is(2.29)

J [2] =

−(d+ γ) + θ(I, B) + δ + Sf ′1(I) Sf ′2(B) Sf ′2(B)ξ θ(I, B) + h′(B) −Sf ′1(I)− σ0 f1(I) + f2(B) Sf ′1(I) + h′(B)− (d+ γ + δ)

where θ(I, B) = −(d+ σ + δ)− (f1(I) + f2(B)).

Define

P = diag[1,I

B,I

B

].

Then

PFP−1 = diag

[0,I ′

I− B′

B,I ′

I− B′

B

],

and(2.30)

PJ [2]P−1 =

[−(d+ γ) + θ(I,B) + δ + Sf ′1(I) SBf ′2(B)/I SBf ′2(B)/IIξ/B θ(I,B) + h′(B) −Sf ′1(I)− σ0 f1(I) + f2(B) Sf ′1(I) + h′(B)− (d+ γ + δ)

].

The matrix Q = PFP−1 + PJ [2]P−1 can be written in the block form as follows:

(2.31) Q =

[Q11 Q12

Q21 Q22

],

in which

Q11 = −(d+ γ) + θ(I, B) + δ + Sf ′1(I),

Q12 =[SBf ′2(B)/I SBf ′2(B)/I

],

Q21 =

[Iξ/B

0

],

Q22 =

[θ(I, B) + h′(B) + I ′/I −B′/B −Sf ′1(I)− σ

f1(I) + f2(B) Sf ′1(I) + h′(B)− (d+ γ + δ) + I ′/I −B′/B

].

We now define the vector norm for R3 as

|(x1, x2, x3)| = max{|x1|, |x2|+ |x3|}

15

for any (x1, x2, x3) ∈ R3. Let µ denote the Lozinskii measure with respect to thisnorm. By direct calculation, we find

µ(Q) = sup{g1, g2},

with

g1 = µ1(Q11) + |Q12|,g2 = |Q21|+ µ1(Q22),

where |Q12| and |Q21| are the matrix norms induced by L1 norm, µ1 denotes theLozinskii measure with respect to L1 norm. Specifically,

g1 = Q11 + |Q12| = (−(d+ γ) + θ(I, B) + δ + Sf ′1(I)) + SBf2(B)/I,

g2 = |Q21| − (d+ σ + δ) + h′(B) + I ′/I −B′/B + sup{0, 2[Sf ′1(I) + σ]− γ}.

By assumptions (H1)-(H2),

f ′1(I)I ≤ f1(I), f ′2(B)B ≤ f2(B)

for all (S, I, B). Note that I ′/I = Sf1(I)/I + Sf2(B)/I − (d+ γ). It gives

−(d+ γ) = I ′/I −[Sf1(I)/I + Sf2(B)/I

].

Thus, we obtain

g1 = −(d+ σ) +I ′

I− S

If1(I)− S

If2(B)− (f1(I) + f2(B)) + Sf ′1(I) +

SB

If ′2(B)),

=I ′

I− (d+ σ)− (f1(I) + f2(B))− S

I

((f2(B)− f ′2(B)B) + (f1(I)− f ′1(I)I)

),

≤ I ′

I− (d+ σ).

(2.32)

By the similar argument together with assumption (H3), we find

g2 =I

Bξ − (d+ σ + δ) + h′(B) + I ′/I −B′/B + sup{0, 2

(Sf ′1(I) + σ

)− γ},

≤ I ′

I− (d+ σ) + sup{0, 2

(Sf ′1(I) + σ

)− γ}

If sup{2(Sf ′1(I))}+ σ ≤ γ, then

sup{

2(Sf ′1(I) + σ

)}− γ ≤ σ.

It gives

(2.33) g2 ≤I ′

I− d.

16

By (2.32) and (2.33),

µ(Q) ≤ I ′

I− d.

In view of 0 ≤ I(t) ≤ N , we have

ln(I(t))− ln(I(0))

t≤ d

2,

for t sufficiently large. Therefore,

1

t

∫ t

0

µ(s)ds ≤ 1

t

∫ t

0

(I ′(s)I(s)

− d)ds =

ln(I(t)))− ln(I(0))

t− d ≤ −d

2,

if t is large enough. This implies q2 ≤ −d/2 < 0. It completes the proof. �Finally, we mention that, realistically, σ � γ since for cholera, the typical immu-

nity wanning period is 3-5 years [23] whereas the normal recovery period is about5-7 days [12]. Hence, the inequality in Theorem 2.2 (2) basically sets an upper limitfor the rate of change of the direct transmission mode to ensure the global stabilityof the endemic equilibrium.

3. An SIRS-B cholera epidemic model with spatial movement

As mentioned before, spatial heterogeneity plays an important role in disease trans-mission and the movement of human hosts and dispersal of pathogens may be criticalin shaping a cholera epidemic. Thus, in this section we will extend our SIRS-B modelto a PDE system to investigate the spatial dynamics of cholera.

For a simple start, we consider a 1D spatial domain, 0 ≤ x ≤ 1 , and we assumethat both the human population and the bacteria undergo a diffusion process. LetDi > 0 (1 ≤ i ≤ 4) be the diffusion coefficients of S, I, R and B, respectively. Thenthe cholera model (2.1) with inclusion of diffusion takes the form:

∂S

∂t= b− Sf1(I)− Sf2(B)− dS + σR +D1

∂2S

∂x2,

∂I

∂t= Sf1(I) + Sf2(B)− (d+ γ)I +D2

∂2I

∂x2,

∂R

∂t= γI − (d+ σ)R +D3

∂2R

∂x2,

∂B

∂t= ξI + h(B)− δB +D4

∂2B

∂x2.

(3.1)

Meanwhile, we assume the entire spatial domain represents a closed community ofour interest; i.e., no individuals would cross the boundary. Hence, we impose the

17

no-flux Neumann boundary conditions:

(3.2)∂S

∂x=∂I

∂x=∂R

∂x=∂B

∂x= 0 , at x = 0, 1.

Although our spatial domain is oversimplified in some sense, such a PDE modelingframework could be useful in both analytical and applied aspects. For example, thecurrent model, with some further improvement, could be suitable to study the spreadof the infection during the first period of the 2010-2012 Haiti cholera outbreak. Thesuspected source of this outbreak was Artibonite River, the longest as well as themost important river in Haiti, and the initial spread of the disease was along theriver [41, 42].

3.1. Turing Instability. One common phenomenon in many reaction-diffusion sys-tems with multiple components is the occurrence of Turing instability [36]; i.e., lossof stability due to inclusion of diffusion. In what follows we first study the possibilityof a Turing instability regarding our PDE cholera model (3.1).

We linearize (3.1) at the DFE, p := (y∗1, y∗2, y∗3, y∗4)T = (N∗, 0, 0, 0)T , of the system

(2.1) in the absence of diffusion. Set

yi = y∗i + δyi, (i = 1, 2, 3, 4).(3.3)

Let Y = (δy1, δy2, δy3, δy4)T . The associated linearized system in vector form is givenby

∂Y

∂τ= JY + D

∂2

∂θ2Y,(3.4)

where J is the Jacobian matrix of the associated ODE system evaluated at the DFE,and

D = diag[D1, D2, D3, D4].

Consider the eigenvalue problem{−υxx(x) = ηυ(x), x ∈ (0, 1),

υx(x) = 0, x = 0, 1.(3.5)

One can easily verify that the eigenvalues ηk = (kπ)2 ≥ 0 and the correspondingeigenfunctions υk(x) = cos(kπx). We consider the ansatz

δyi(t, x) = eρtυ(x)ωi, (1 ≤ i ≤ 4),(3.6)

where υ is the solution of the eigenvalue problem (3.5), and ρ and ωi are constant.Substituting (3.6) into (3.4) yields

(ρI4 + ηD)ω = Jω.(3.7)

We are interested in whether there exists ρ such that Re(ρ) > 0 at the DFE.

18

Solve (3.7). We find that the associated eigenvalues are given by

ρ1 = −(k1 + k4)/2 +√

4k2k3 + (k1 − k4)/2)2,

ρ2 = −(k1 + k4)/2−√

4k2k3 + (k1 − k4)/2)2,

ρ3 = −(ηD1 + d),

ρ4 = −(ηD3 + d+ σ),

(3.8)

where k1 = (d+ γ)−N∗fI + ηD2, k2 = N∗fB, k3 = ξ and k4 = δ − g + ηD4. �

Proposition 3.1. If R0 < 1, inclusion of diffusive spatial spread into model (2.1)will not produce a Turing instability.

Proof. Suppose R0 < 1. The ODE model (2.1) has a unique DFE, and (d + γ) −N∗fI > 0 and δ > g. This gives ki > 0 for i = 1, 2, 3, 4. Thus, ρ2, ρ3 and ρ4 areall negative and have the same sign as the corresponding eigenvalue of J. The onlyeigenvalue that could have a sign change is ρ1. Thus, it suffices to study the sign ofρ1.

Notice that

ρ1ρ2 = k1k4 − k2k3

= ((d+ γ)−N∗fI + ηD2)(δ − g + ηD4)− ξN∗fB= ((d+ γ)−N∗fI)D4 + (δ − g)D2 +D2D4 + ((d+ γ)−N∗fI)(δ − g)− ξN∗fB.

(3.9)

It is clear that in the last equation of (3.9), the first three terms are positive. By theproof of Theorem 2.1, ((d+γ)−N∗fI)(δ−g)−ξN∗fB is a product of two eigenvaluesat the DFE, and hence it is positive. This yields ρ1ρ2 > 0. Hence, ρ2 < 0 impliesρ1 < 0. Therefore, all the eigenvalues associated with p0 will keep the same sign,and Turing instability will not occur. �

Proposition 3.1 states that inclusion of diffusion spatial spread in the original ODEsystem (2.1) tends to stabilize the system. An implication is that the PDE model(3.1) would exhibit threshold dynamics similar to that of the ODE system. Thequestion now is how to quantify the threshold in the spatial-temporal domain forour PDE cholera model. Below we will focus our attention on this point.

3.2. Disease Threshold. In contrast to the large body of work devoted to thethreshold dynamics of ODE epidemic models, there are very few such studies onPDE models. In a recent article by Wang and Zhao [39], the concept of the basicreproduction number is extended to reaction-diffusion epidemic systems with Neu-mann (no-flux) boundary conditions. Based on the theory of principle eigenvalues,

19

the authors defined the basic reproduction number R0 for a reaction-diffusion epi-demic system as the spectral radius of the operator

(3.10) L[φ(x)] =

∫ ∞0

F (x)T (t)φ dt = F (x)

∫ ∞0

T (t)φ dt .

Consequently, they showed that

(3.11)

∫ ∞0

T (t)φ dt = −B−1φ ,

where B := ∇ · (dI∇)− V . It then follows that

(3.12) L = −FB−1 .

Here F and V are analogues to the next-generation matrices associated with thecorresponding ODE system (i.e., without diffusion terms), T (t) is the solution semi-group for the linearized reaction-diffusion system, φ denotes the distribution of theinitial infection, and dI is the diffusion coefficient vector.

For our cholera model (3.1), we have

(3.13) B =

D2∂2

∂x2− (d+ γ) 0

0 D4∂2

∂x2− δ

.

In order to analyze the basic reproduction number of the PDE system, RPDE0 =

ρ(L), we proceed to calculate B−1 by solving B(φ1 , φ2

)T=(y1 , y2

)Tsubject to

homogeneous Neumann boundary conditions. Let us first consider the equation

B1[φ1] := D2∂2φ1

∂x2− (d+ γ)φ1 = y1 , 0 ≤ x ≤ 1 ;

φ′1(0) = 0, φ′1(1) = 0 .(3.14)

This problem can be conveniently solved by using the Laplace transform. Denotethe Laplace transforms of φ1(x) and y1(x) by Φ1(s) and Y1(s), respectively. We thenobtain

Φ1(s) =Y1(s)

D2s2 − (d+ γ)+

sD2φ1(0)

D2s2 − (d+ γ),

where we have applied the first boundary condition of φ1 . The inverse Laplacetransform then yields

φ1(x) =1√

D2(d+ γ)

∫ x

0

sinh[√d+ γ

D2

(x− τ)]y1(τ) dτ + φ1(0) cosh

(√d+ γ

D2

x).

20

Now differentiating φ1 and using the second boundary condition, we obtain

φ1(0) =−1√

D2(d+ γ) sinh(√

d+γD2

) ∫ 1

0

cosh[√d+ γ

D2

(1− τ)]y1(τ) dτ .

Hence,

φ1(x) = B−11 [y1] =

1√D2(d+ γ)

∫ x

0

sinh[√d+ γ

D2

(x− τ)]y1(τ) dτ

−cosh

(√d+γD2

x)

√D2(d+ γ) sinh

(√d+γD2

) ∫ 1

0

cosh[√d+ γ

D2

(1− τ)]y1(τ) dτ .(3.15)

In this expression, the second part represents the homogeneous solution, whereas thefirst part represents a particular solution to the original non-homogeneous equation.In a similar way, we can solve the boundary value problem

B2[φ2] := D4∂2φ2

∂x2− δφ2 = y2 , 0 ≤ x ≤ 1 ;

φ′2(0) = 0, φ′2(1) = 0(3.16)

to obtain

φ2(x) = B−12 [y2] =

1√D4δ

∫ x

0

sinh[√ δ

D4

(x− τ)]y2(τ) dτ

−cosh

(√δD4x)

√D4δ sinh

(√δD4

) ∫ 1

0

cosh[√ δ

D4

(1− τ)]y2(τ) dτ .(3.17)

For consistence in notations, below we will switch φ1 and y1 in equation (3.15),and φ2 and y2 in equation (3.17). Now we look at the eigenvalue problem L[φ] = λφ ;that is,

(3.18) − FB−1φ = λφ ,

where, in our cholera model,

(3.19) F =

[N∗fI N∗fBξ g

]=

[N∗f ′1(0) N∗f ′2(0)

ξ h′(0)

].

21

We thus obtain the following two equations

ki1

∫ x

0

sinh[√d+ γ

D2

(x− τ)]φ1(τ) dτ

+ ki2 cosh(√d+ γ

D2

x)∫ 1

0

cosh[√d+ γ

D2

(1− τ)]φ1(τ) dτ

+ ki3

∫ x

0

sinh[√ δ

D4

(x− τ)]φ2(τ) dτ

+ ki4 cosh(√ δ

D4

x)∫ 1

0

cosh[√ δ

D4

(1− τ)]φ2(τ) dτ = λφi(x), (i = 1, 2).

(3.20)

with the coefficients

k11 = − N∗fI√D2(d+ γ)

, k12 =N∗fI√

D2(d+ γ) sinh(√

d+γD2

) ,k13 = −N

∗fB√D4δ

, k14 =N∗fB

√D4δ sinh

(√δD4

) ,k21 = − ξ√

D2(d+ γ), k22 =

ξ√D2(d+ γ) sinh

(√d+γD2

) ,k23 = − g√

D4δ, k24 =

g

√D4δ sinh

(√ δ

D4

) .Analysis of the eigenvalue problem associated with the integral equations (3.20)

appears difficult. Instead, we may gain some insight into the eigenvalue problem(3.20) by looking at its discrete form. Let us partition the interval [0, 1] uniformlywith the nodes xi = i∆x (0 ≤ i ≤ M), where M∆x = 1 . Evaluating equations(3.20) at xi for i = 1, 2, · · · , M , and simply approximating each integral by usingthe right endpoint of each subinterval, we obtain a matrix equation

(3.21) AY = λY

with Y = [φ1(x1), φ1(x2), · · · , φ1(xM), φ2(x1), φ2(x2), · · · , φ2(xM) ]T . The prob-lem is now reduced to analyzing the spectral radius, ρ(A), of the coefficient matrixA; in particular, we have lim∆x→0 ρ(A) = ρ(L) = RPDE

0 .The coefficient matrix A in equation (3.21) can be written as

(3.22) A = A1 + A2 + A3 + A4 ,

where each matrix Ai (1 ≤ i ≤ 4), of dimension 2M × 2M , results from the dis-cretization of the ith integral in equations (3.20). Specifically, A1 can be represented

22

by a block form

(3.23) A1 = ∆x

[k11A1 0Mk21A1 0M

],

where 0M denotes the zero square matrix of dimension M ×M , and A1 = (a1,ij) isan M ×M lower-triangular matrix given by

(3.24) a1,ij =

{sinh

[√d+γD2

(xi − xj)], if i ≥ j

0, otherwise.

Similarly, we have

(3.25) A3 = ∆x

[0M k13A3

0M k23A3

],

where A3 = (a3,ij) is also an M ×M lower-triangular matrix:

(3.26) a3,ij =

{sinh

[√δD4

(xi − xj)], if i ≥ j

0, otherwise.

The matrix A2 takes the form

(3.27) A2 = ∆x

[k12A2 0Mk22A2 0M

]with the M ×M block A2 = (a2,ij) for which

a2,ij = cosh(√d+ γ

D2

xi)

cosh[√d+ γ

D2

(1− xj)], 1 ≤ i, j ≤M.

Finally, The matrix A4 takes the form

(3.28) A4 = ∆x

[0M k14A4

0M k24A4

]with the M ×M block A4 = (a4,ij) for which

a4,ij = cosh(√ δ

D4

xi)

cosh[√ δ

D4

(1− xj)], 1 ≤ i, j ≤M.

It is obvious that the spectral radius for each of A1 and A3 is 0; i.e.,

(3.29) ρ(A1) = ρ(A3) = 0 .

Meanwhile, using Lemma C.1 from Appendix C, we obtain the characteristic poly-

nomial of A2 as

(3.30) det(λIM − A2

)= λM−1(λ− λ2)

23

where

(3.31) λ2 :=M∑i=1

cosh(√d+ γ

D2

xi

)cosh

[√d+ γ

D2

(1− xi)].

Note that A2 is a strictly positive matrix and λ2 is the unique positive dominant

eigenvalue of A2. Similarly, the matrix A4 is also strictly positive and we have

(3.32) det(λIM − A4

)= λM−1(λ− λ4),

where

(3.33) λ4 :=M∑i=1

cosh(√ δ

D4

xi

)cosh

[√ δ

D4

(1− xi)]

is the unique positive dominant eigenvalue of A4. It is clear to see that

(3.34) ρ(A2) = ∆x k12λ2 , ρ(A4) = ∆x k24λ4 .

With some algebraic manipulation, we can observe that when ∆x→ 0,

(3.35) lim∆x→0

ρ(A2) =k12

2cosh

(√d+ γ

D2

)+k12

2

√D2

d+ γsinh

(√d+ γ

D2

)≈ NfId+ γ

provided that d+γD2� 1 . In a similar way,

(3.36) lim∆x→0

ρ(A4) =k24

2cosh

(√ δ

D4

)+k24

2

√D4

δsinh

(√ δ

D4

)≈ g

δ

provided that δD4� 1 . Note that NfI

d+γand g

δare two components in the expression

of R0 for our ODE model; indeed, in the ODE model, we have R0 ≥ max(NfId+γ

, gδ

).

There is no general relationship between ρ(A) and ρ(Ai) (1 ≤ i ≤ 4). Nevertheless,

if we assume d+γD2� 1 and δ

D4� 1 , then each entry of A1 and A3 is very small; it

is bounded between 0 and sinh(√

d+γD2

)if in A1 , and between 0 and sinh

(√δD4

)if

in A3 . Thus, A1 and A3 can be treated as small perturbation to A2 and A4 in thiscase, and

(3.37) ρ(A) ≈ ρ(A2 + A4) .

Next we explore lower and upper bounds for ρ(A2 + A4). We first have (see, e.g.,[14])

(3.38) max(ρ(A2), ρ(A4)

)≤ ρ(A2 + A4) ,

24

by noting that both A2 and A4 are non-negative matrices. In addition, some stan-dard results on the lower/upper bounds of non-negative matrices based on mini-mum/maximum row (or, column) sums [14] can be applied to estimate ρ(A2 + A4)and give a range for ρ(L). In order to compare the ODE and PDE thresholds, how-ever, we will make use of the following result [40] to determine an upper bound forρ(A2 + A4) .

Lemma 3.1. Consider a nonnegative irreducible square matrix Q of the block form

Q =

[Q1 Q2

Q3 Q4

], where Q1 and Q4 are both square matrices. If ρ(Q) > ||Q1||, where

|| · || denotes any consistent matrix norm, then

(3.39) ρ(Q) ≤ 1

2

[(||Q1||+ ||Q4||

)+

√(||Q1|| − ||Q4||

)2+ 4||Q2|| ||Q3||

].

If we treat the matrix A2 + A4 as Q in the lemma above, then we have Q1 =

∆xk12A2 , Q2 = ∆xk14A4 , Q3 = ∆xk22A2 , Q4 = ∆xk24A4 . Since A2 +A4 is strictlypositive, Q must be irreducible. For simplicity, let us use the induced ||·||1 norm; i.e.,the maximum column sum of each (nonnegative) matrix. Through direct calculation,it is easy to obtain the following:

lim∆x→0

||Q1||1 = k12

√D2

d+ γsinh

(√d+ γ

D2

)cosh

(√d+ γ

D2

)=

NfId+ γ

cosh(√d+ γ

D2

):= P1 ,

lim∆x→0

||Q3||1 = k22

√D2

d+ γsinh

(√d+ γ

D2

)cosh

(√d+ γ

D2

)=

ξ

d+ γcosh

(√d+ γ

D2

):= P3 ,

lim∆x→0

||Q2||1 = k14

√D4

δsinh

(√ δ

D4

)cosh

(√ δ

D4

)=NfBδ

cosh(√ δ

D4

):= P2 ,

lim∆x→0

||Q4||1 = k24

√D4

δsinh

(√ δ

D4

)cosh

(√ δ

D4

)=g

δcosh

(√ δ

D4

):= P4 .

Meanwhile, the condition ρ(Q) > ||Q1||1 is satisfied based on (3.35) and (3.38). Wethus obtain, as ∆x→ 0,

(3.40) ρ(A2 + A4) ≤ 1

2

[P1 + P4 +

√(P1 − P4)2 + 4P2P3

].

Furthermore, if we let d+γD2→ 0 and δ

D4→ 0 , we can easily observe that

(3.41) ρ(A2 + A4) ≤ 1

2

[ NfId+ γ

+g

δ+

√( NfId+ γ

− g

δ

)2

+ 4ξNfBδ(d+ γ)

]:= RODE

0 .

Note that the right-hand side of the inequality (3.41) gives R0 for our ODE model.The results in (3.40) and (3.41) imply that when the diffusion coefficients D2 andD4 are not so large, the value of ρ(L) can possibly exceed RODE

0 , since cosh(x) > 1

25

for x > 0 . However, when D2 and D4 are very large, we have RPDE0 = ρ(L) ≤

RODE0 ; that is, the basic reproduction number for our PDE model would be equal to

or lower than that for the ODE model. Indeed, we can establish a even strongerresult regarding the limit case.

Proposition 3.2. When d+γD2→ 0 and δ

D4→ 0 , we have

RPDE0 = RODE

0 .

Proof. Within such limits, the eigenvalue problem (3.20) is reduced to

NfId+ γ

∫ 1

0

φ1(τ)dτ +NfBδ

∫ 1

0

φ2(τ)dτ = λφ1(x) ,

ξ

d+ γ

∫ 1

0

φ1(τ)dτ +g

δ

∫ 1

0

φ2(τ)dτ = λφ2(x) .

From these two equations, it is straightforward to see that φ1(x) ≡ c1 and φ2(x) ≡ c2 ,where c1 and c2 are some nonzero constants. Let Z = c2/c1 . Then we have

λ =NfId+ γ

+NfBδ

Z =g

δ+

ξ

d+ γ

1

Z.

Solving the quadratic equation for Z , we obtain

Z± =−(NfId+γ− g

δ

)±√(

NfId+γ− g

δ

)2+ 4 ξNfB

δ(d+γ)

2NfBδ

.

Consequently, we have

λ± =1

2

[ NfId+ γ

+g

δ±

√( NfId+ γ

− g

δ

)2+ 4

ξNfBδ(d+ γ)

].

Clearly, |λ+| > |λ−| and ρ(L) = λ+ > 0. We thus obtain RPDE0 = λ+ = RODE

0 . �

As one can naturally expect, when the diffusion terms are incorporated into theoriginal ODE model (i.e., spatial variation and human/pathogen movement are takeninto account), the infection risk predicted by the model will be changed, leading toa different basic reproduction number. The result in Proposition 3.2, nevertheless,implies that when the diffusion is extremely strong, it tends to smooth out the spatialheterogeneity and reduce the system to a homogeneous setting, such that the ODEand PDE models would share the same disease threshold.

26

4. Numerical Results

We now verify our analytical results by numerical simulation. In particular, wewill numerically demonstrate (and compare) the threshold dynamics based on ourODE and PDE cholera models. To conduct the simulation, the direct and indirecttransmissions are assumed to take the form developed in Mukandavire et al. [21]:f1(B) = βhI andf2(B) = βeB/(B + κ) where βh and βe represent the direct andindirect transmission parameters, respectively; κ is the concentration of V. cholerain the environment depicting the half saturation rate. Cholera transmission throughingestion of bacteria from a contaminated environment can be greatly influencedby growth and survival of free-living bacteria in the environment [27]. Therefore,growth of the pathogen in the environment can not be ignored in a realistic model.We include bacteria growth through a logistic growth assumption [22], i.e., h(B) =gB(1 − B/K) where g = h′(0) is the intrinsic bacteria growth rate and K is themaximal capacity of free-living bacteria in the environment.

g

ξ

0 0.2 0.4 0.6 0.8 10

4

8

12

16

20

1

2

3

4

(a)

0 20 40 60 80 1000

200

400

600

800

Time (weeks)

Infe

cti

ou

s h

ost

R0<1

R0>1

(b)

Figure 2. (a) R0 associated with ODE model (2.1) as a function ofg and ξ. The straight line represents R0 = 1; (b) Number of infectedhosts associated with the ODE model (2.1) vs. time. The blue (resp.red) curve illustrates the case when R0 < 1 (resp. R0 > 1).

The model parameters of (2.1) are listed in Table (D.1). Our particular interestis the impact of the bacterial growth rate, g , on the value of R0 . Meanwhile,the shedding rate, ξ , describes how much an infected individual contributes to thebacterial concentration in the water reservoir, and is likely to vary widely [11]. Thus,we plot R0 associated with the ODE model (2.1) as a function of g and ξ in Figure 2

27

(a). Particularly, the black curve, denoted as Γ,

ξ = w(g) :=(δ − g)(d+ γ −N∗fI)

N∗fB,

indicates where R0 = 1. As one can see from Figure 2 (a) the bacterial growth rateg and shedding rate ξ play a vital role in shaping R0 . In particular, (i) When g < δ,the disease threshold changes from R0 < 1 to R0 > 1 provided that either of thefollowing happens:

(1) g is fixed at a constant value in the range and ξ increases to cross Γ;(2) ξ < w(0)

.= 10.85 cells ·ml−1·week−1 is fixed and g increases to cross Γ.

(ii) When g ≥ δ = 7/30 week−1, R0 > 1. Moreover, for each fixed value of g, thebacteria shedding rate ξ becomes less important and even irrelevant as g increases,and for each fixed value of xi, varying g can lead to a significant change in the valueof R0. An illustration of the number of infectious individuals as a function of timefor R0 < 1 and R0 > 1 is displayed in Figure 2 (b). It clearly shows that the infectiondies out when R0 < 1 , and persists (approaching the endemic equilibrium over time)when R0 > 1 .

Figure 3. The difference in the basic reproduction number betweenthe PDE model (3.1) and the ODE model (2.1): RPDE

0 −RODE0 .

Now let us consider the PDE model (3.1). Figure 3 shows the difference in thebasic reproduction numbers of the PDE model (3.1) and the ODE model (2.1).We observe, from Figure 3 (a), that RPDE

0 − RODE0 is nearly invisible when the

diffusion coefficients of the hosts and bacteria, D2 and D4, are getting large. Thisresult confirms that limD2,D4→∞R

PDE0 = RODE

0 . When D2 and D4 decrease to zero,

28

Figure 4. Number of infected hosts associated with the PDE model(3.1) vs. space and time. (a) R0 < 1; (b) R0 > 1.

Figure 3 (b) illustrates that RPDE0 − RODE

0 > 0, and for the fixed bacteria growthrate g and shedding rate ξ, RPDE

0 increases and becomes uniformly greater thanone for all the possible values of g > 0 and ξ > 0. In particular, (i) for a fixed g,RPDE

0 − RODE0 will be elevated if ξ increases; (ii) for a fixed ξ, RPDE

0 − RODE0 goes

up if g gets larger.Figure 4 illustrates the number of infectious individuals based on the PDE model

(3.1) with uniform initial distribution, as a function of space and time when theassociated R0 is lower or higher than the unity. We have also tested other initialconditions with various distribution types, and they all lead to similar patterns interms of the extinction (when R0 < 1) and endemic state (when R0 > 1) of cholerainfection.

5. Discussion

In this paper, we have proposed novel ODE and PDE cholera models that ad-dress the intrinsic bacterial dynamics and spatial movement of human and pathogen.Our ODE model employs general incidence functions for both the direct and indi-rect transmission routes, and incorporates a general representation of the bacterialgrowth. The resulting basic reproduction number, R0 , depends on all these factors.Particularly, we find that bacterial growth contributes to the increase of the valueof R0 , indicating that the risk of cholera outbreak would be higher than most ofprior model predictions in the absence of intrinsic bacterial dynamics. Our modelalso takes into account the waning cholera immunity in reality, by using an SIRS

29

framework for the human population dynamics. Our analysis shows that, despitethe incorporation of bacterial growth and partial immunity, the basic reproductionnumber remains a sharp threshold for cholera dynamics: when R0 < 1, the DFE isglobally asymptotically stable and the disease completely dies out; when R0 > 1, theunique endemic equilibrium is globally asymptotically stable and the disease persists.

Meanwhile, we have extended this ODE model to a PDE system so as to explic-itly track the spatial variation on cholera transmission. Our focus again is on thethreshold dynamics, and we have formulated the basic reproduction number R0 ofthis PDE model based on recent work in [39]. We have analyzed R0 and estimatedits bounds by reducing the operator eigenvalue problem to a matrix eigenvalue prob-lem, which also provides a practical means to compute R0 for the PDE system. Asa result of our analysis and simulation, we find that only if the diffusion coefficientsare very large, the basic reproduction number of the PDE model, RPDE

0 , is matchingthat of the corresponding ODE model, RODE

0 ; otherwise, RPDE0 is higher than RODE

0 .The implication is that, when we take into account the realistic spatial movementof human hosts and pathogen, the overall infection risk might be higher than whatwould be based on the (unrealistic) homogeneous setting, as the diffusion processmay help spread the disease more rapidly. In this regard, using the ODE modelmight underestimate the epidemic risk. Our findings could provide useful guidelinefor public health administration to properly scale their efforts in cholera control.

As mentioned before, Joh et al. [17] and Bani-Yaghoub et al. [3] have also consid-ered intrinsic bacterial dynamics in their cholera studies based on ODE systems. Themodel in [17] considered only the indirect transmission route and focused on logisticgrowth of the bacteria, whereas the model in [3] investigated the dual transmissionpathways with bilinear incidences and logistic bacterial growth. Compared to these,our SIRS-B ODE model is much more general that allows various formulation of theincidence factors and the bacterial growth dynamics. Our results also generalize thefindings in these previous studies.

In the present work, we have chosen to model the spatial effects of cholera trans-mission using a reaction-diffusion PDE system. Similar PDE modeling approach wasalso used by Bertuzzo and co-workers [4] in studying the spatial dynamics of cholera.Our PDE model formulation is more general than that of [4] in terms of varietyof incidence and pathogen functions, inclusion of multiple transmission pathways,and incorporation of bacterial intrinsic growth. The scope of our work is also differ-ent from that in [4] as our focus is on the analysis and simulation of the thresholddynamics for cholera transmission. Another common approach to investigate dis-ease spatial dynamics is based on metapopulation ODE models. Tuite et al. [35]performed a study on the Haiti cholera outbreak using a patch model. In a recentwork by Shuai and van den Driessche [31], a multigroup cholera model was presentedand analyzed. These metapopulation models can potentially reveal more details on

30

the disease dynamics of an individual group (e.g., a province, or a city). The cost,however, is that a large number of groups may be needed for modeling at differentspatial scales, which easily leads to a large dynamical system. Consequently, thiswould pose a challenge for both the analysis and computation of the model. Com-pared to the cholera models in [35] and [31], our PDE system appears simpler andis easy to construct, and straightforward for computation. Reducing the operatoreigenvalue problem to a matrix eigenvalue problem also allows us to easily calculatethe basic reproduction number for our PDE model.

Meanwhile, there are several limitations in our PDE model, and the current studybased on our PDE framework can be extended in a number of ways. First, spread offree-living bacteria may occur through the movement of human host, thus a cross-diffusion term would be appropriate in the equation for B in system (3.1). Practically,infected individuals might not be as active as the susceptible and recovered people,and the value of D2 might be much less than those of D1 and D3. (We note, though,that most cholera infections are minor and asymptomatic, and those people withminor infections could still pursue normal activities including travel and other move-ments.) Also, spread of free-living bacteria does not necessarily occur on the samespatial scale, which demands more careful modeling and analysis. Moreover, insteadof using a simplistic 1D space dimension, the system can be similarly constructed ona 2D spatial domain for more realistic cholera modeling. The diffusion coefficientsas well as several parameters of disease transmission rates can be taken as spacedependent, instead of constants, to better reflect the details of spacial heterogeneity.In addition, traveling wave solutions can be analyzed which will add useful insightinto the spatial spread of cholera.

Although we have considered several sources of variability, demographic and envi-ronmental, there are other variability factors that could be included in a more com-prehensive model. An important example among these is the individual heterogeneityin shedding density and the possibility of super-shedding [29] and hyper-infectivity[12]. The impact of such individual heterogeneity in shedding and infectivity oncholera epidemics will provide an interesting topic in future research.

Acknowledgment

This work was partially supported by a grant from the Simons Foundation (#317047to Xueying Wang). Jin Wang was partially supported by the National Science Foun-dation under Grant Nos. 1216936, 1245769 and 1412826. The authors thank theeditor and two anonymous referees for their valuable comments that improved thispaper.

31

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34

Appendix A. Other Kinds of Reproduction Numbers

Type reproduction number. If disease control is targeted at a particularhost type, a useful threshold is known as the type reproduction number, T . Thetype reproduction number defines the expected number of secondary infective casesof a particular population type caused by a typical primary case in a completelysusceptible population [13, 28]. It is an extension of the basic reproduction numberR0.

The type reproduction number T1 (resp. T2) for control of infection among humans(resp. bacteria in the contaminated environment) is defined in references [13, 28] as

(A.1) Ti = eTi K(I− (I−Pi)K)−1ei,

provided the spectral radius of matrix (I−Pi)K is less than one, i.e., ρ((I−Pi)K

)<

1, for i = 1, 2. Here I is the 2× 2 identity matrix, vector e1 = (1, 0)T , e2 = (0, 1)T ,K is the next generation matrix, and Pi is the 2 × 2 projection matrix with allzero entries except that the (i,i) entry is 1 (i = 1, 2). Write K = (kij). The typereproduction T can be easily defined in terms of the elements kij:

T1 = k11 +k12k21

1− k22

,

T2 = k22 +k12k21

1− k11

.

(A.2)

which exists provided k22 < 1. Based on (A.2), the type reproduction numberassociated with the infectious humans is

(A.3) T1 =N∗fId+ γ

+ξNfB

(δ − g)(d+ γ),

provided g < δ. The type reproduction number associated with bacteria in thecontaminated environment is

(A.4) T2 =g

δ+

ξN∗fBδ(d+ γ −N∗fI)

,

provided N∗fI < d+ γ .It is shown in [28] that R0 < 1(= 1, > 1) ⇐⇒ Ti < 1(= 1, > 1).Target reproduction number. Another threshold named the target reproduc-

tion number, TS, has been recently introduced by Shuai et al. [32]. It can be regardedas an extension of the type reproduction number. Let S = {(i1, j1), . . . , (im, jm)} de-note the set of the target indices. Define S1 = {i1, . . . , im} and S2 = {j1, . . . , jm}.The target reproduction number is defined as

TS = ρ(ES1PS1KPS2

(I − (K − PS1KPS2)

)−1ES1

),

35

provided ρ(K − PS1KPS2) < 1. Here ES1 = (eij) is a 2 × 2 matrix with eij = 1 ifi = j and i ∈ S1, and zero otherwise. PSk

= (pki,j) is a 2 × 2 matrix with pkij = 1 ifi = j and i ∈ Sk, and zero otherwise, for k = 1, 2.

There are a number of ways to control the cholera disease. If we target at thevaccination of humans, the target set will be S = {(1, 1), (1, 2)} and the associatedtarget reproduction number is the same as the type reproduction number of theinfectious hosts, i.e.,

TS = T1

provided g < δ. If we target at hygenic disposal of human feces, then S = {(2, 1)}and

TS = T21 =k12k21

(1− k11)(1− k22)=

ξN∗fB(δ − g)(d+ γ −N∗fI)

provided N∗fI < d+ γ and g < δ. If we target at the control of the contaminatedenvironment, such as food and water, then S = {(1, 2)} and

TS = T12 = T21

provided N∗fI < d+ γ and hE < δ. If the isolation of infectious hosts can be doneto reduce the direct transmission between humans, then S = {(1, 1)} and

TS = T11 =k11(1− k22)

(1− k22 − k12k21)=

N∗fI(δ − g)

(δ − g)(d+ γ)− ξN∗fB,

provided g(d+ γ) + ξN∗fB < δ(d+ γ).

Appendix B. The Geometric Approach

Here we present the main result of the geometric approach for global stability,originally developed by Li and and Muldowney [18].

We consider a dynamical system

(B.1)dX

dt= F (X)

where F : D 7→ Rn is a C1 function and D ⊂ Rn is a simply connected open set. LetP (X) be a

(n2

)×(n2

)matrix-valued C1 function in D, and set

(B.2) Q = PFP−1 + PJ [2]P−1 ,

where PF is the derivative of P (entry-wise) along the direction of F , and J [2] is thesecond additive compound matrix of the Jacobian J(X) = DF (X) . Let m(Q) bethe Lozinskii measure of Q with respect to a matrix norm; i.e.,

(B.3) m(Q) = limh→0+

|I + hQ| − 1

h,

36

where I represent the identity matrix. Define a quantity q2 as

(B.4) q2 = lim supt→∞

supX0∈K

1

t

∫ t

0

m(Q(X(s, X0))

)ds ,

where K is a compact absorbing subset of D. Then the condition q2 < 0 provides aBendixson criterion in D . As a result, the following theorem holds:

Theorem B.1. Assume that there exists a compact absorbing set K ⊂ D andthe system (B.1) has a unique equilibrium point X∗ in D . Then X∗ is globallyasymptotically stable in D if q2 < 0 .

Appendix C. A Lemma on a Special Matrix Determinant

Let us consider the following square matrix of dimension n× n (n ≥ 1):

(C.1) X1 =

N1P1 N1P2 · · · N1PnN2P1 N2P2 · · · N2Pn

......

. . ....

NnP1 NnP2 · · · NnPn

,

where Ni and Pi (1 ≤ i ≤ n) are arbitrary numbers.

Lemma C.1. The characteristic polynomial of the matrix in (C.1) is

(C.2) det(λIn −X1) = λn−1

(λ−

n∑i=1

NiPi

).

The result follows directly from the trace of X1, by noting that the rank of X1 is1.

Appendix D. Table of model parameters

The definition and numerical values of parameters in our ODE and PDE choleramodels are provided in the table here. Note that p (resp. y, w and d) represents aperson (resp. year, week and day).

37

Table D.1. Model parameters used in cholera models

.

Parameter Definition Value ReferencesN Total population size of host 12, 347 [21]d Natural death rate of human (43.5 y)−1 [42]βh Direct transmission rate 0.075 p−1w−1 [21]βe Indirect transmission rate 1.1−4 w−1 [21]κ Half saturation rate 106 cells ·ml−1 [12]γ Recovery rate (5 d)−1 [12]σ Rate of host immunity loss (3 y)−1 [23]K Maximal carrying capacity 2× 106 cells ·ml−1 Assumedδ Bacterial death rate (30 d)−1 [12]ξ > 0 Shedding rate Varied [cells ·ml−1w−1] Assumedg ≥ 0 Initial bacterial growth rate Varied [w−1] Assumed