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Research article

Fecal carriage of ESBL-producing Escherichia coli in Egyptian patients admitted to the Medical Research Institute hospital, Alexandria University

  • Received: 25 July 2020 Accepted: 29 October 2020 Published: 03 November 2020
  • Commensal ESBL-producing E. coli represent a reservoir for resistance genes therefore, their detection is crucial to restrain the spread of beta-lactam resistance. Hence, the aim of the present study was phenotypic and genotypic characterization of commensal ESBL-producing E. coli obtained from the stool of patients at the time of admission and at the time of discharge from the Medical Research Institute hospital. A total of 70 E. coli isolates were collected from 35 patients and were categorized into Group A (samples obtained on admission) and Group B (samples obtained at the time of discharge). Phenotypically, 30 isolates were ESBL producers (40% of E. coli isolates collected on admission and 45.7% of the strains obtained at the time of discharge were ESBL producers). Most of them harbored one to three plasmids with sizes ranging from one kbp to ten kbp. Upon genotypic investigation, blaCTX-M was the most detected gene in 80% of ESBL strains, followed by blaTEM in 53.3% and the least detected was blaSHV in only 13.3%. By comparing group A and group B, ten patients were found to carry commensal ESBL-producing E. coli, in two patients these isolates carried ESBL genes that were identical on admission and on discharge. However, in eight patients, these isolates carried different ESBL genes, which were newly harbored during hospital stay. The high abundance of MDR commensal E. coli 48.57% together with the presence of 42.86% ESBL-producing commensal E. coli among our isolates represents an alarming threat, as they are frequently associated with the increased risk of infection, higher costs and longer hospital stay.

    Citation: Amira ElBaradei, Dalia Ali Maharem, Ola Kader, Mustafa Kareem Ghareeb, Iman S. Naga. Fecal carriage of ESBL-producing Escherichia coli in Egyptian patients admitted to the Medical Research Institute hospital, Alexandria University[J]. AIMS Microbiology, 2020, 6(4): 422-433. doi: 10.3934/microbiol.2020025

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  • Commensal ESBL-producing E. coli represent a reservoir for resistance genes therefore, their detection is crucial to restrain the spread of beta-lactam resistance. Hence, the aim of the present study was phenotypic and genotypic characterization of commensal ESBL-producing E. coli obtained from the stool of patients at the time of admission and at the time of discharge from the Medical Research Institute hospital. A total of 70 E. coli isolates were collected from 35 patients and were categorized into Group A (samples obtained on admission) and Group B (samples obtained at the time of discharge). Phenotypically, 30 isolates were ESBL producers (40% of E. coli isolates collected on admission and 45.7% of the strains obtained at the time of discharge were ESBL producers). Most of them harbored one to three plasmids with sizes ranging from one kbp to ten kbp. Upon genotypic investigation, blaCTX-M was the most detected gene in 80% of ESBL strains, followed by blaTEM in 53.3% and the least detected was blaSHV in only 13.3%. By comparing group A and group B, ten patients were found to carry commensal ESBL-producing E. coli, in two patients these isolates carried ESBL genes that were identical on admission and on discharge. However, in eight patients, these isolates carried different ESBL genes, which were newly harbored during hospital stay. The high abundance of MDR commensal E. coli 48.57% together with the presence of 42.86% ESBL-producing commensal E. coli among our isolates represents an alarming threat, as they are frequently associated with the increased risk of infection, higher costs and longer hospital stay.


    Prevention can help curb the HIV epidemic. Many studies [1,2,3,4,5] have demonstrated the efficacy of pre-exposure prophylaxis (PrEP) in reducing HIV acquisition in at-risk populations. New biomedical tools, including long-acting injections [6,7,8], slow-release implants [9,10], and multi-dose HIV vaccines [11,12], are being tested and may provide additional preventive measures.

    In the future, scientists will need to design strategies to account for both the variety of preventive measures and the exact distribution of HIV risk as determined by age, gender, mixing preference, sexual behavior, and other individual characteristics. The goal of these new strategies will be to both increase the proportion of individuals practicing prevention and to decrease HIV transmission using newly approved tools. Mathematical modeling will likely become the tool of choice for evaluating these strategies due to its ability to investigate multiple intervention scenarios with differing coverage and targeting criteria.

    Despite an increase in the use of individual-based network models in the last decade, deterministic compartmental models built around systems of ordinary differential equations (ODEs) still dominate HIV modeling. Many investigators find deterministic systems easier to both implement and calibrate. In an extensive survey of models studying the effects of HIV/AIDS intervention [13], 86.9% of the models were deterministic. In a recent analysis of antiretroviral therapies [14], 9 of 12 models were deterministic compartmental models, while 3 were individual-based microsimulation models.

    Compartmental models for isolated populations deal with HIV risk in a variety of ways. Many models simply ignore individual differences and assume that risk is spread equally across the entire population.

    Another group of models [15,16,17,18] divide the population into several (usually 2 to 6) distinct risk groups and assign specific behavioral attributes (number of partners, sexual frequency, condom use) to each group. These models are typically built around either transmission or contact matrices [19] or contact fraction or mixing matrices [20] with many elements. Because of this complexity, these models are often hard to analyze.

    More rarely [15,21,22], risk is treated as a continuous variable, leading to integrodifferential equations with bivariate functions (kernels) rather than matrices. In general, continuous-risk models are more complicated than models with a finite number of risk classes, and most analyses of continuous-risk models rely on numerical methods. For separable kernels, however, analyses may simplify, as stressed by [23] and [24].

    There is also a growing literature on heterogeneity and risk in metapopulation models. Metapopulation models are compartmental models with a large number of distinct populations. Quite commonly, each population has its own homogeneous dynamics, and the populations are coupled together linearly, due to migration or movement. Many published models [25,26,27,28] contain susceptible-infectious-susceptible (SIS) dynamics, and these papers often focus on the spread of flu-like diseases between cities due to air travel. Other authors [29,30], however, have used metapopulation models to analyze the effects of migrant workers on HIV transmission.

    In this paper, we consider a model in which the internal transmission dynamics are not homogeneous. Rather, we focus on a model for a single, structured population with an arbitrary number of discrete risk classes and direct, nonlinear transmission between all of the risk classes. To use the terminology of [31], we focus on a cross-coupled model rather than on a mobility metapopulation model. Interestingly, methods developed for continuous-risk models can sometimes simplify the analysis of models with discrete risk classes. In this study, we borrow methods from the study of integral and integrodifferential equations to analyze our model, and we investigate how the number of risk classes affects the model's dynamics. We consider distinct scenarios in which added groups have lower risk due to increased awareness or in which average risk across classes remains constant.

    In the next section, we describe our model and spell out our simplifying assumptions. Then, in section 3, we find the equilibria of our model. In section 4, we outline methods for determining the stability of these equilibria. We complement these general analyses with two examples. In section 5, we add low-risk groups, while in section 6, we add groups of intermediate risk. Finally, in section 7, we summarize our work, discuss its limitations, and suggest future avenues for research.

    Let r and s be continuous and dimensionless risk variables on a scale from zero to one. Zero is the lowest risk. One is the highest risk.

    Suppose that there are n risk classes, each of width Δr=1/n or Δs=1/n, and that i and j are indices for these risk classes. Let Si(t) be the number of susceptible individuals in risk class i at time t, Ii(t) be the number of sexually-active infectious individuals in risk class i at time t, and Ai(t) be the number of infected individuals in risk class i at time t who have developed AIDS and are no longer sexually active. We now consider the model

    dSidt=αiSi(nj=1βijIj)μSi, (2.1a)
    dIidt=Si(nj=1βijIj)γIiμIi, (2.1b)
    dAidt=γIiμAiδAi, (2.1c)

    where i=1,...,n.

    System (2.1) is a risk-structured susceptible-infectious-removed (SIR) model with demography. We assume that each risk class has its own (constant) birth rate, αi>0, and that the natural per-capita death rate (μ), AIDS-induced per-capita death rate (δ), and per-capita AIDS development rate (γ) are all constant and independent of risk class. The βij are the transmission coefficients between individuals of class j and class i. That is, βij is the per-capita rate of infection of class-i susceptibles per class-j infective. Model (2.1) should, arguably, contain standard incidence, but we have assumed mass-action incidence to greatly simplify subsequent calculations.

    To ease comparison with an integrodifferential model, we now shift from numbers to densities. In particular, let

    xi(t)=nSi(t),yi(t)=nIi(t),zi(t)=nAi(t) (2.2)

    be the densities of susceptible, infectious, and inactive individuals in class i at time t. (We have, in other words, divided Si, Ii, and Ai by the width Δr=1/n.) Since equations (2.1a) and (2.1b) do not depend on equation (2.1c), we now consider the reduced system

    dxidt=νixi(1nnj=1βijyj)μxi, (2.3a)
    dyidt=xi(1nnj=1βijyj)γyiμyi, (2.3b)

    i=1,...,n, where

    νi=nαi. (2.4)

    As the number of risk classes goes to infinity, we expect system (2.3) to approach the integrodifferential system

    xt=ν(r)x(r,t)10β(r,s)y(s,t)dsμx(r,t), (2.5a)
    yt=x(r,t)10β(r,s)y(s,t)dsγy(r,t)μy(r,t), (2.5b)

    for suitably defined birth-rate density ν(r) and transmission kernel β(r,s).

    Students of integral equations [32,33] often focus on separable kernels. These are kernels that can be written as a finite sum of products of pairs of univariate functions, with

    β(r,s)=ni=1ai(r)bi(s) (2.6)

    for some n. In epidemiology [23,24], the term separable is most commonly used for a single product,

    β(r,s)=a(r)b(s), (2.7)

    in the transmission kernel.

    For models with discrete risk classes, the term separable is again used [34,24], when the elements of the transmission matrix can be written as the simple product

    βij=aibj. (2.8)

    The transmission matrix [βij] is now just the dot product of a column vector and a row vector. This matrix is of rank one, and it has a one-dimensional range.

    In our analysis of system (2.3), we will focus on symmetric and separable transmission matrices with nonnegative elements

    βij=bibj. (2.9)

    In section 5, we will let

    βij=c2injn (2.10)

    so that

    bi=cin. (2.11)

    In section 6, we will let

    βij=c2(i1n1)(j1n1) (2.12)

    for n2 so that

    bi=c(i1n1). (2.13)

    We thus assume well-mixed populations, with encounters proportional to densities, but with transmission coefficients that vary multiplicatively, as products of simple functions of susceptible and of infective risk. Risk levels, in turn, differ due to differences in behavior or prevention (e.g., vaginal versus anal sex or safe versus unprotected sex). Assumption (2.9) greatly simplifies our analyses, but still allows us to capture a broad range of behavioral interactions, ranging from very low- to very high-risk encounters.

    We begin our analysis of system (2.3) by finding the system's equilibria. To do so, we set the time derivatives of system (2.3) equal to zero and look for roots of the system

    νixi(1nnj=1βijyj)μxi=0, (3.1a)
    xi(1nnj=1βijyj)γyiμyi=0, (3.1b)

    for i=1,...,n.

    For separable transmission matrices, the above system has two equilibria. There is, first and foremost, a disease-free equilibrium at

    (xi,yi)=(νiμ,0), (3.2)

    for i=1,...,n.

    To find our second, endemic equilibrium, we begin by noting that equation (3.1a) implies that

    xi=νiμ+1nnj=1βijyj. (3.3)

    If

    βij=bibj, (3.4)

    as in equation (2.9), then

    xi=νiμ+abi, (3.5)

    where the sum

    a=1nnj=1bjyj (3.6)

    still needs to be determined. We call a the infectivity potential.

    To find the densities of infectious individuals at equilibrium, we sum equations (3.1a) and (3.1b),

    νiμxi(γ+μ)yi=0, (3.7)

    and obtain

    yi=1(γ+μ)(νiμxi). (3.8)

    In light of equation (3.5), it now follows that

    yi=νi(γ+μ)(abiμ+abi). (3.9)

    Inserting equation (3.9) into the right-hand side of equation (3.6) gives us

    1(γ+μ)1nnj=1νjb2jμ+abj=1. (3.10)

    This equation can be solved, either analytically or numerically, for the infectivity potential a.

    The product abi is the force of infection, at equilibrium, for class i. The infectivity potential a, in turn, acts like the amplitude for the endemic equilibrium. Indeed, when a=0, the endemic equilibrium with coordinates (3.5) and (3.9) reduces to disease-free equilibrium (3.2). Equation (3.10), in turn, reduces to the equation

    R01μ(γ+μ)1nnj=1νjb2j=1. (3.11)

    We thus expect the endemic equilibrium to pass through the disease-free equilibrium in a transcritical bifurcation as the basic reproduction number, R0, passes through one. For R0=1, a=0, while for R0>1, a>0, and we have a unique endemic equilibrium.

    We have seen that we have a disease-free equilibrium with coordinates

    (xi,yi)=(νiμ,0) (4.1)

    and an endemic equilibrium with coordinates

    (xi,yi)=[νiμ+abi,νi(γ+μ)(abiμ+abi)] (4.2)

    for infectivity potentials a>0 satisfying equation (3.10). As usual, here and throughout, i=1,...,n.

    Let us now analyze the stability of these equilibria by introducing small perturbations, ξi(t) and ηi(t), about the coordinates of the equilibria,

    xi(t)=xi+ξi(t),yi(t)=yi+ηi(t), (4.3)

    into system (2.3). After simplifying, using system (3.1), and linearizing, we find that

    dξidt=(μ+1nnj=1βijyj)ξixi1nnj=1βijηj, (4.4a)
    dηidt=(1nnj=1βijyj)ξi(γ+μ)ηi+xi1nnj=1βijηj. (4.4b)

    For separable kernel (2.9), system (4.4) simplifies to

    dξidt=(μ+abi)ξibixi1nnj=1bjηj, (4.5a)
    dηidt=abiξi(γ+μ)ηi+bixi1nnj=1bjηj, (4.5b)

    where the infectivity potential a is given by equations (3.6) and (3.10).

    We now look for solutions of the form

    ξi(t)=uieλt,ηi(t)=vieλt. (4.6)

    After substituting these solutions into system (4.5) and canceling exponentials, we get the linear algebraic system

    λui=(μ+abi)uibixi1nnj=1bjvj, (4.7a)
    λvi=abiui(γ+μ)vi+bixi1nnj=1bjvj. (4.7b)

    Let us now define

    w=1nnj=1bjvj. (4.8)

    Our eigenvalue equations now simplify to

    (λ+μ+abi)ui=bixiw, (4.9a)
    abiui+(λ+γ+μ)vi=bixiw. (4.9b)

    Adding these two equations, we also note that

    (λ+μ)ui+(λ+γ+μ)vi=0. (4.10)

    We must infer the 2n (allowing for multiplicity) eigenvalues λ for each equilibrium from these equations.

    For disease-free equilibrium (4.1), with a=0, eigenvalue equations (4.9a) and (4.9b) reduce to

    (λ+μ)ui=biνiμw, (4.11a)
    (λ+γ+μ)vi=biνiμw. (4.11b)

    We now have several possibilities for our eigenvalues and eigenvectors.

    If bi is zero, the above equations reduce to

    (λ+μ)ui=0, (4.12a)
    (λ+γ+μ)vi=0, (4.12b)

    and we get two eigenvalues, λ=μ and λ=(γ+μ). The first eigenvalue has an eigenvector with ui nonzero and all other components zero. The second eigenvalue has an eigenvector with vi nonzero and all other components zero. These eigenvalues recur for each zero bi.

    More commonly, bi is positive. In that case, if the vi are all zero, the right-hand sides of equations (4.11a) and (4.11b) still vanish, as does the left-hand side of equation (4.11b). This leaves us with

    (λ+μ)ui=0 (4.13)

    and with the eigenvalue λ=μ. The corresponding eigenvector has ui nonzero and all other components zero. This eigenvalue recurs for each positive (or zero) bi. It thus occurs n times.

    For positive bi, we also have eigenvalues λ=(γ+μ). These eigenvalues are consistent with system (4.11) if all of the ui are zero and if, in addition,

    nj=1bjvj=0 (4.14)

    so that w is zero. In general, there are n1 vectors orthogonal to a non-zero vector with n components. Thus, if any of our bi are positive, λ=(γ+μ) occurs as an eigenvalue n1 times.

    For (at least one) positive bi, we also have one more eigenvalue, corresponding to w0. Now,

    ui=1λ+μbiνiμw,vi=1λ+γ+μbiνiμw. (4.15)

    Inserting the above expression for vi into our equation (4.8) for w, we get

    1(λ+γ+μ)1μ1nnj=1νjb2j=1. (4.16)

    It follows that our last eigenvalue is

    λ=1μ(1nnj=1νjb2j)γμ. (4.17)

    This last eigenvalue is positive if

    R01μ(γ+μ)1nnj=1νjb2j>1, (4.18)

    in which case the disease-free equilibrium is unstable. If, instead, R0<1, the disease-free equilibrium is asymptotically stable. The above formula for R0, the basic reproduction number is consistent with our earlier definition of R0 in equation (3.11).

    For endemic equilibrium (4.2), with a>0, eigenvalue equations (4.9a) and (4.9b) now take the form

    (λ+μ+abi)ui=biνiμ+abiw, (4.19a)
    abiui+(λ+γ+μ)vi=biνiμ+abiw. (4.19b)

    As with our disease-free equilibrium, we now have several possibilities for our eigenvalues and eigenvectors.

    If bi is zero, the above equations again reduce to

    (λ+μ)ui=0, (4.20a)
    (λ+γ+μ)vi=0, (4.20b)

    and we again get the eigenvalues λ=μ and λ=(γ+μ). These eigenvalues recur for each zero bi.

    For positive bi, we again have eigenvalues λ=(γ+μ). These eigenvalues are consistent with system (4.19) if all of the ui are zero and if w=0. Since there are n1 vectors orthogonal to a nonzero, n-component vector, λ=(γ+μ) now occurs as an eigenvalue n1 times.

    Finally, for some bi positive and w0, we have eigenvectors with components

    ui=biνi(μ+abi)(λ+μ+abi)w (4.21)

    and, in light of equation (4.10),

    vi=(λ+μ)(λ+γ+μ)biνi(μ+abi)(λ+μ+abi)w. (4.22)

    Inserting the above expression for vi into equation (4.8) for w, we find that our remaining eigenvalues satisfy the characteristic equation

    λ+μλ+γ+μ[1nnj=1νjb2j(μ+abj)(λ+μ+abj)]=1. (4.23)

    If all of our bi are positive, we get n+1 eigenvalues from this equation.

    The left-hand side of equation (4.23) is a nasty expression with many singularities in λ. Rather than hurting us, however, these singularities instead help us locate eigenvalues of characteristic equation (4.23). We will explore and illustrate this phenomenon more fully in the next section, in the context of an example.

    Let us now consider a simple example in which the birth-rate densities are all constant, νi=ν, and the transmission coefficients take the form

    βij=bibj, (5.1)

    with

    bi=cin (5.2)

    for i=1,...,n. All of these bi are positive.

    For n=1, there is only one risk class, and b1=c. As we increase n (see Table 1), we, in effect, add low-risk classes. The mean of the bi,

    1nni=1bi=c2(1+1n), (5.3)
    Table 1.  Transmission components for the low-risk scenario.
    n bi
    1 c
    2 c/2,c
    3 c/3,2c/3,c
    4 c/4,c/2,3c/4,c
    5 c/5,2c/5,3c/5,4c/5,c

     | Show Table
    DownLoad: CSV

    decreases towards c/2 as n goes to infinity.

    The basic reproduction number, equation (4.18), is now

    R0=νμ(γ+μ)1nnj=1c2(jn)2=νc2μ(γ+μ)(n+1)(2n+1)6n2, (5.4)

    and the right-most fraction decreases from 1, at n=1, to 1/3, as n approaches infinity. For R0>1, the disease-free equilibrium is unstable and we have a unique endemic equilibrium.

    Endemic equilibrium (4.2) now has coordinates

    (xi,yi)=[νnμn+iac,ν(γ+μ)(iacμn+iac)] (5.5)

    for infectivity potentials a satisfying

    c2ν(γ+μ)1n2nj=1j2μn+jac=1. (5.6)

    In principle, we could specify the number of classes n and the parameters ν, γ, μ, and c for any particular problem, use equation (5.6) to solve for a, and use equation (5.5) to find the coordinates of the endemic equilibrium. Since, however, a and c often occur as a product in equations (5.5) and (5.6), we instead take a simpler and more efficient approach.

    Let

    τ=ac. (5.7)

    We now treat τ>0 as a parameter and use equations (5.6) and (5.7) to write c and a as the parametric equations

    c=n(γ+μνω)12,a=τn(νωγ+μ)12, (5.8)

    where

    ω=nj=1ωj=nj=1j2μn+jτ. (5.9)

    These equations may be used to plot the infectivity potential a as a function of c, as in Figure 1.

    Figure 1.  A plot of the infectivity potential a as a function of the contact or transmission parameter c for the low-strata example of section 5. This curve was plotted using parametric equations (5.8) for μ=0.01 per year, γ=0.03 per year, ν=1.0 susceptibles per year, and n=10 risk classes. The infectivity potential a acts like the amplitude for the endemic equilibrium and increases with increasing transmission parameter c.

    As an alternative, c and a may each be plotted as a functions of the parameter τ, as in Figure 2. This means that you can pick a value of τ>0 and simply read off the corresponding values of c and a.

    Figure 2.  Plots of the infectivity potential a (left) and contact or transmission parameter c (right) as functions of the parameter τ for the low-strata example of section 5. These curves were plotted using parametric equations (5.8) for μ=0.01 per year, γ=0.03 per year, ν=1.0 susceptibles per year, and n=10 risk classes. Using these curves, we can pick a value of the parameter τ and read off the corresponding values of both c and a.

    For a given τ, the coordinates of the endemic equilibrium in equation (5.5) are now just

    (xi,yi)=[νnμn+iτ,ν(γ+μ)(iτμn+iτ)]. (5.10)

    We may thus plot the equilibrium densities of susceptibles and infectives, by class, for different τ values, as in Figure 3. Increasing τ decreases the equilibrium densities of susceptibles and increases the equilibrium densities of infectives.

    Figure 3.  Plots of the equilibrium densities of susceptibles (top) and infectives (bottom) as a function of risk class for the low-strata example of section 5. The curves were plotted using endemic equilibrium equation (5.10) for μ=0.01 per year, γ=0.03 per year, ν=1.0 susceptibles per year, n=10 risk classes, and various τ values. Equilibrium densities of susceptibles decrease and equilibrium densities of infectives increase as τ increases.

    We now wish to determine the stability of the endemic equilibrium. This equilibrium has 2n eigenvalues and we know, from section 4.2, that n1 of the eigenvalues have the value λ=(γ+μ). Using νi=ν and equations (4.23) and (5.2), the remaining n+1 eigenvalues satisfy the characteristic equation

    νc2n2λ+μλ+γ+μnj=1ωjλ+ρj=1, (5.11)

    where the ωj are defined in equation (5.9) and

    ρj=μ+jnτ. (5.12)

    Dividing both sides of characteristic equation (5.11) by the first fraction on its left-hand side and substituting c from parametric equations (5.8), we see that the remaining n+1 eigenvalues satisfy

    λ+μλ+γ+μnj=1ωjλ+ρj=ωγ+μ. (5.13)

    If all of the eigenvalues have negative real part, the endemic equilibrium is asymptotically stable.

    The right-hand side of equation (5.13) does not depend on λ and is a positive constant. The function on the left-hand side approaches zero through positive values as λ goes to plus infinity, and it approaches zero through negative values as λ approaches minus infinity. (Feel free to look ahead to the next subsection and to Figures 47 for graphical illustrations.) The left-hand function has an obvious zero at λ=μ, to the right of all singularities, and it has singularities (vertical asymptotes) at λ=(γ+μ) and at λ=ρi\, , for i=1,...,n. That is, it has n+1 singularities. (For ease of exposition, we now assume that all of these singularities are distinct.) These n+1 singularities help us locate the eigenvalues of equation (5.13).

    Figure 4.  Plots of both sides of characteristic equation (5.13) as a function of lambda for n=4, μ=0.01 per year, γ=0.03 per year, and τ=0.3 per year. The dashed lines are vertical asymptotes. The horizontal line is at the value of the right-hand side. The characteristic equation has eigenvalues at λ0.2961, 0.2205, 0.1469, 0.0719, 0.0547 per year that correspond to intersections of the left-hand and right-hand curves.
    Figure 5.  Plots of both sides of equation (5.13) for n=4, μ=0.01 per year, γ=0.03 per year, and τ=0.2 per year. We now have real eigenvalues at λ0.1972, 0.1447, 0.0917, and a complex-conjugate pair of eigenvalues at λ0.0532±0.0179i.
    Figure 6.  Plots of both sides of characteristic equation (5.13) for n=4, μ=0.01 per year, γ=0.03 per year, and τ=0.08 per year. Equation (5.13) now has real eigenvalues at λ0.0814, 0.0575, 0.0323 and a complex conjugate pair of eigenvalues at λ0.0344±0.0305i.
    Figure 7.  Plots of both sides of equation (5.13) as a function of lambda for n=4, μ=0.01 per year, γ=0.03 per year, and τ=0.02 per year. Equation (5.13) now has real eigenvalues at λ0.0271, 0.0211, 0.0153 and a complex conjugate pair of eigenvalues at λ0.0133±0.021i.

    As we increase λ, from left to right, between neighboring singularities that are both to the right of the singularity at λ=(γ+μ), the function on the left-hand side of equation (5.13) increases (monotonically) from minus infinity to plus infinity. If we instead increase λ, from left to right, between neighboring singularities that are both to the left of the singularity at λ=(γ+μ), the function on the left-hand side of equation (5.13) decreases (monotonically) from plus infinity to minus infinity. If we look for eigenvalues by plotting each side of equation (5.13) as a function of lambda, checking for intersections, these intervals each give us one real eigenvalue.

    The situation is more complicated as we move between neighboring singularities, one of which is λ=(γ+μ). We have either one or two such intervals depending on whether λ=(γ+μ) is an endpoint or an interior singularity.

    On one interval, the function on the left-hand side of equation (5.13) increases to plus infinity at both ends of the interval. We call this interval the parabolic interval.

    On the second interval, if it occurs, our left-hand function either increases from minus infinity to plus infinity or decreases from plus infinity to minus infinity, depending on whether the second interval is to the right or to the left of the parabolic interval. This second interval thus gives us another real eigenvalue. We thus have n1 real roots between the n singularities λ=ρi, not counting real roots from the parabolic interval. For two intervals with λ=(γ+μ) as an endpoint, which interval has one real eigenvalue and which is the parabolic interval depends on the exact location of the singularities.

    Let us now return to the parabolic interval. If the function on the left-hand side of equation (5.13) dips low enough to intersect the right-hand-side constant, we have two real eigenvalues in the parabolic interval. Otherwise, we have two complex conjugate eigenvalues or, possibly, two real eigenvalues that live elsewhere.

    It is tempting to think that the real parts of the aforementioned complex eigenvalues always lie within the parabolic interval, but this is incorrect. We can, however, show that the complex eigenvalues always have negative real part.

    To do so, we first rewrite characteristic equation (5.13) in polynomial form,

    ωγ+μ(λ+γ+μ)ni=1(λ+ρi)(λ+μ)nj=1[ωjij(λ+ρi)]=0, (5.14)

    and note that the leading coefficient, the coefficient in front of λn+1, is

    a0=ωγ+μ. (5.15)

    The next coefficient, the one in front of λn, is, in turn,

    a1=ω+ωγ+μni=1ρinj=1ωj=ωγ+μni=1ρi. (5.16)

    It is a well-known result, from the theory of equations [35], that the negative of the ratio of these two coefficients is just the sum of the roots of the polynomial. Thus,

    n+1i=1λi=a1a0=ni=1ρi. (5.17)

    Since we know that n1 of these roots are real and nested between the n singularities λ=ρi,

    ρi+1<λi<ρi,i=1,2,...,n1, (5.18)

    it quickly follows that

    λn+λn+1<ρ1. (5.19)

    Thus, if these last two eigenvalues are a complex-conjugate pair, they must have negative real part with

     Re λn<ρ12, Re λn+1<ρ12. (5.20)

    Alternatively, consider the possibility that the last two eigenvalues, λn and λn+1, are real and lie to the right of all of the singularities. The rightmost singularity is, depending on our parameters, either λ=ρ1 or λ=(γ+μ). In either case, the current assumptions imply that λn>ρ1 and λn+1>ρ1. In light of inequality (5.19), it now follows that the last two eigenvalues must lie in the region bounded by the inequalities

    λn>ρ1,λn+1>ρ1,λn+λn+1<ρ1. (5.21)

    This is a triangular region that lies entirely within the interior of the third quadrant of the (λn,λn+1) plane. It follows that both λn and λn+1 must be negative.

    In summary, our endemic equilibrium has 2n eigenvalues. At most, two of these eigenvalues are complex. All of the eigenvalues are negative or have negative real part. Our endemic equilibrium is thus asymptotically stable.

    To illustrate some possibilities and to show how the loci of the eigenvalues of the endemic equilibrium can change, we start with μ=0.01 per year, γ=0.03 per year, and n=4. In Figure 4, for τ=0.3 per year, we plot both sides of characteristic equation (5.13) as a function of λ. We see three real eigenvalues in the intervals bounded by the four singularities at λ=ρi, i=1,...,4, and two real eigenvalues in the parabolic interval bounded by the singularities at λ=ρ1 and λ=(μ+γ)=0.04 (per year). In addition, we have n1=3 eigenvalues (not shown) at λ=(μ+γ). Since all of our eigenvalues are negative real numbers, our endemic equilibrium is asymptotically stable.

    As we decrease τ to 0.2 per year (see Figure 5), we lose the two real eigenvalues in the parabolic interval. A standard numerical eigenvalue program [36] applied to system (4.7) reveals complex eigenvalues at λ0.0532±0.0179i. In this example, the real part of the complex eigenvalues lies within the parabolic interval. Our endemic equilibrium is again asymptotically stable.

    In Figure 6, we kept all other parameters as in Figures 4 and 5, but set τ=0.08 per year. As we decreased τ, n of our n+1 singularities shifted to the right, and our parabolic interval, now (0.05,0.04), shifted to the left relative to the other intervals. Our complex eigenvalues, however, lagged behind. Our numerical eigenvalue program indicates that the complex eigenvalues are now at λ0.0344±0.0305i.

    After we decrease τ even further, to τ=0.02 per year (see Figure 7), the parabolic interval, (0.04,0.03), lies to the left, but the complex eigenvalues, λ0.0133±0.021i, lie to the right. Even so, inequalities (5.20) guarantees that the real parts of the two complex eigenvalues lie to the left of λ=ρ1/2=0.0075.

    Finally, we may plot solutions of system (2.3), for birth-rate densities νi=ν and transmission coefficients (5.1) and (5.2), by computing these solutions numerically. Figure 8 shows, as an example, the densities of susceptible and infectious individuals for system (2.3) for μ=0.01 per year, γ=0.03 per year, τ=0.2 per year, and n=100 risk classes. The solutions were computed using a standard, variable-step, Runge-Kutta algorithm [37]. The solutions do eventually approach the asymptotically stable endemic equilibrium. At the same time, the eigenvalues for the endemic equilibrium may not tell the whole story, since our solutions quickly overshoot the endemic equilibrium before slowly returning to the equilibrium. We will return to this topic in the discussion.

    Figure 8.  Plots of solutions of system (2.3), as a function of time (in years), for μ=0.01 per year, γ=0.03 per year, τ=0.2 per year, ν=1.0 susceptibles per year, and n=100 risk classes. Solutions were computed using a fourth-order Runge-Kutta algorithm with adaptive step-size control starting with initial conditions corresponding to susceptibles at their disease-free equilibrium, xi(0)=100, and a small, uniform density of infectives, yi(0)=0.1 for i=1,...,100. Numerical solutions were recorded for time steps of Δt=0.01 years but were only plotted at 2.5 year intervals for aesthetic reasons. Infectives (bottom) rapidly increase before slowly decreasing to endemic equilibrium levels.

    Let us now briefly consider a second example. We again assume that the birth-rate densities are constant, νi=ν, but we now let the transmission coefficients take the form

    βij=bibj, (6.1)

    with

    bi=c(i1n1) (6.2)

    for n2 and i=1,...,n. The first of these bi is always zero. All of the other bi are positive.

    For n=2, there are two risk classes, with b1=0 and b2=c. As we increase n (see Table 2), we add intermediate risk classes. The mean of the bi,

    1nni=1bi=c2, (6.3)
    Table 2.  Transmission components for the intermediate scenario.
    n bi
    2 0,c
    3 0,c/2,c
    4 0,c/3,2c/3,c
    5 0,c/4,c/2,3c/4,c
    6 0,c/5,2c/5,3c/5,4c/5,c

     | Show Table
    DownLoad: CSV

    remains constant while the variance of the bi,

    (1nni=1b2i)(c2)2=c22nn1c24, (6.4)

    decreases from 3c2/4, at n=2, to c2/4, as n approaches infinity.

    If we again use equation (4.18), we see that the basic reproduction number is now

    R0=νμ(γ+μ)1nnj=1c2(j1n1)2=νc2μ(γ+μ)(2n1)6(n1). (6.5)

    In this instance, the right-most fraction decreases from 1/2, at n=2, to 1/3, as n approaches infinity. This reduction occurs despite the fact that the mean of the bi remains constant.

    Endemic equilibrium (4.2) now has coordinates

    xi=ν(n1)μ(n1)+(i1)ac, (6.6a)
    yi=ν(γ+μ)[(i1)acμ(n1)+(i1)ac]. (6.6b)

    for infectivity potentials a satisfying

    c2ν(γ+μ)1n(n1)nj=1(j1)2μ(n1)+(j1)ac=1. (6.7)

    We now proceed much as we did in section 5. We let

    τ=ac, (6.8)

    treat τ>0 as a parameter, and use equations (6.7) and (6.8) to write c and a as the parametric equations

    c=[n(n1)(γ+μ)νω]12, (6.9a)
    a=τ[νωn(n1)(γ+μ)]12, (6.9b)

    with

    ω=nj=1ωj=nj=1(j1)2μ(n1)+(j1)τ. (6.10)

    For a given τ, the coordinates of the endemic equilibrium are now just

    xi=ν(n1)μ(n1)+(i1)τ, (6.11a)
    yi=ν(γ+μ)[(i1)τμ(n1)+(i1)τ]. (6.11b)

    We now wish to determine the stability of the endemic equilibrium for τ>0. Since b1=0, we know, from section 4.2, that λ=μ is an eigenvalue. Since the remaining bi are positive, we also know that n1 of the eigenvalues have the value λ=(γ+μ).

    For the remaining n eigenvalues, we proceed much as we did in section 5. Using νi=ν and equations (4.23), (6.2), and (6.9a), we quickly determine that the remaining n eigenvalues satisfy the characteristic equation

    λ+μλ+γ+μnj=1ωjλ+ρj=ωγ+μ (6.12)

    with ω and ωj as in (6.10) and

    ρj=μ+(j1)(n1)τ. (6.13)

    Because b1=0, the first term of each series drops out. So, counting really starts with j=2.

    The function on the left-hand side of the above equation has singularities (vertical asymptotes) at λ=(γ+μ) and at λ=ρi for i=2,...,n. These n vertical asymptotes constrain and help determine the location of the n eigenvalues of characteristic equation (6.11), much as in our previous example. All of the eigenvalues are negative or have negative real part, and the endemic equilibrium is asymptotically stable.

    In this paper, we used a structured infectious-disease model with demography, mass-action incidence, and an arbitrary number of risk classes to investigate the effects of risk on the spread of HIV. Using a simple, separable transmission matrix, we obtained equations for the disease-free equilibrium, the basic reproduction number, the endemic equilibrium, and the eigenvalues that determine the stability of the endemic equilibrium.

    Separable transmission matrices can capture a broad range of behavioral interactions. For real populations, the actual distribution of interactions will also depend on the initial distributions of risk amongst susceptible and infective individuals. (Interactions may also, of course, depend on other details, such as the exact mixing pattern [38] and sexual network [39], that we did not consider.) Scientists [40] have recently made tremendous progress in classifying the risk-structure of real populations. Our model thus holds the promise that it may help us assess the effects of risk for real populations in an analytically tractable way.

    We complemented our general analysis with two specific examples. In our first example, we stratified risk so that the mean of the separable components of the transmission coefficients decreased as we added more risk classes. We found a simple, closed-form expression for the basic reproduction number, and this number decreased as we added more risk classes. Thus, initiating HIV-prevention strategies that add low-risk categories lowers average risk and reduces the basic reproduction number, as expected.

    We also determined the nontrivial endemic equilibrium and characterized the spectrum of eigenvalues at this equilibrium. For n risk classes, the endemic equilibrium had 2n eigenvalues. No more than two of the eigenvalues are complex and all eigenvalues are negative or have negative real part. Thus, when the endemic equilibrium exists, it is asymptotically stable.

    Despite the asymptotic stability of our endemic equilibrium, the number of infectives can greatly overshoot endemic levels, as seen in Figure 8. This is not surprising since many mass-action disease models have density thresholds, above which infectives increase and below which infectives decrease [41,42,43]. In other words, our model burns through many susceptibles before equilibrating.

    It is also important to remember, moreover, that the short-term behavior of a perturbation from an equilibrium may differ from its long-term behavior. Transient amplification can, for example, occur near asymptotically stable equilibria in reactive systems governed by nonnormal matrices [44,45,46]. In this paper, we focused solely on the stability of our endemic equilibria. Preliminary analyses (not shown) suggest that our endemic equilibria may be reactive, despite their asymptotic stability. We hope to look at the reactivity of our endemic equilibria in future work.

    In our second example, we briefly looked at a model in which the mean of the separable components of the transmission coefficients remained constant but the variance decreased as we added intermediate risk classes. Here, the basic reproduction number again decreased as we added more risk classes.

    This second example highlights the value of getting the risk structure right, since different risk structures with comparable mean properties can lead to different predictions. The seemingly unimportant decision of the number of risk strata can influence the dynamic behavior of our system, even if every effort has been made to keep the overall risk identical. Choosing the wrong number of strata can even lead to qualitatively different behavior if the basic reproductive number falls below one. There has recently been great progress in analyzing data-driven structured-population models in other fields [47], and we hope to apply some of these new methods to our own model. In a future study, we plan to look at real risk data to explore how feasible the above scenarios are for realistic HIV epidemics.

    In formulating our models, we used mass-action rather than standard incidence and a separable, well-mixed transmission matrix rather than assortative mixing. We made these assumptions to increase analytic tractability. We hope to loosen these assumptions in future work. We note, however, that our use of mass-action incidence may, in fact, make our model useful for the study of other diseases, such as coronavirus (COVID-19).

    We also introduced but did not analyze integrodifferential system (2.5). Our analysis of discrete system (2.3), in the limit as the number of risk classes goes to infinity, suggests that the characteristic equation at the endemic equilibrium of system (2.5) will have a continuum of singularities and that it would be hard to understand system (2.5) without first understanding discrete system (2.3). At the same time, many investigators find it easier to estimate parameters and to assess goodness of fit for models with continuous traits [47]. We hope to analyze integrodifferential system (2.5) more carefully in the future.

    We thank Shane D. Wilson for useful discussions that helped motivate this study. We also wish to thank the anonymous reviewers for their hard work and helpful comments.

    All authors declare that there are no conflicts of interest regarding the publication of this paper.



    Funding



    This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

    Conflict of interest



    The authors declare no conflict of interest.

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