Digital Object Identifier (DOI):10.1007/s00285-004-0288-0

J. Math. Biol. 51, 123–143 (2005) Mathematical Biology

Vladimir M. Veliov

On the effect of population heterogeneity on dynamicsof epidemic diseases

Received: 4 December 2003 / Revised version: 14 April 2004 /Published online: 13 July 2005 – c© Springer-Verlag 2005

Abstract. The paper investigates a class of SIS models of the evolution of an infectiousdisease in a heterogeneous population. The heterogeneity reflects individual differences inthe susceptibility or in the contact rates and leads to a distributed parameter system, requiringtherefore, distributed initial data, which are often not available. It is shown that there exists acorresponding homogeneous (ODE) population model that gives the same aggregated resultsas the distributed one, at least in the expansion phase of the disease. However, this ODEmodel involves a nonlinear “prevalence-to-incidence” function which is not constructivelydefined. Based on several established properties of this function, a simple class of approx-imating function is proposed, depending on three free parameters that could be estimatedfrom scarce data.

How the behaviour of a population depends on the level of heterogeneity (all otherparameters kept equal) – this is the second issue studied in the paper. It turns out that bothfor the short run and for the long run behaviour there exist threshold values, such that moreheterogeneity is advantageous for the population if and only if the initial (weighted) preva-lence is above the threshold.

1. Introduction

Epidemic models that explicitly take into account the heterogeneity of a populationinvolve distributed parameter systems, see e.g. Diekmann, Heesterbeek, and Metz[3], Coutinho et al. [1], Diekmann and Heesterbeek [2]. Such models are not onlymore complex for numerical processing, but require distributed data that are usuallynot available. For this reason it is desirable to deal with non-distributed (aggregated)models, usually obtained by an appropriate averaging. In the present paper, startingfrom a rather general distributed SIS system, modeling the dynamics of an epi-demic disease in a heterogeneous population, we pass to a non-distributed systemwhose solution coincides with the aggregated solution of the distributed system,at least in the expansion phase of the disease. This ordinary differential system isnot explicitly defined, but we establish that it represents an ordinary SIS model in

V.M. Veliov: Institute of Mathematical Methods on Economics, Vienna University of Tech-nology, Argentinierstrasse 8/119, 1040 Vienna, Austria, and Institute of Mathematics andInformatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.e-mail: vveliov@eos.tuwien.ac.at

This research was partly supported by the Austrian Science Foundation under contract N0.15618-OEK.

Key words or phrases: Epidemic disease – Population heterogeneity – Incidence function

124 V.M. Veliov

a homogeneous population which involves a nonlinear “prevalence-to-incidence”function as a multiplicative component of the infection rate. We establish someproperties of this nonlinear incidence function and propose a class of approximat-ing functions depending on three parameters only, which could be identified bya modest amount of measurements of the prevalence. The analysis is motivatedmainly by HIV/AIDS and Hepatitis, but could be useful also for other diseases.

Moreover, we compare two populations with the same parameters (relevantto the considered model), and the same mean value of risk, the two populationsdiffering only in the distribution of the individuals along the risk scale: one ofthe populations is more heterogeneous with respect to the risky behaviour than theother. We establish that there is a threshold value such that less heterogeneity is moreadvantageous for the population in the short run if the initial weighted prevalenceis below this threshold, and less advantageous in the opposite case. In addition, thisthreshold value turns out to be independent of data and equals 0.5. In the long runthe asymptotic state of the system could be history dependent. Moreover, a similarpattern of the dependence of the long run behaviour on the level of heterogeneityholds, as in the short run. However, the threshold is data dependent and can evendegenerate to zero for some data specifications. The result concerning the long runbehaviour is obtained for a substantially simplified version of the general model,but the conclusions are supported by numerous numerical tests carried out withmore general specifications.

The exposition is organized as follows. In Section 2 we describe a general SISmodel in its homogeneous and heterogeneous (distributed) versions, and discussthe assumptions. In Section 3 we investigate the dynamics of the integral momentsof the susceptible and of the infected heterogeneous populations. Then in Section4 we pass from the heterogeneous model to a homogeneous one with a nonlin-ear “prevalence-to-incidence” function and support by analytical and numericalarguments the relevance of the proposed class of approximate incidence functions.Section 5 is devoted to the study of the effect of the heterogeneity on the evolutionof the disease in the short and in the long run.

2. The homogeneous and the heterogeneous models

The model below is of SIS-type, that is, it involves only susceptible and infectedindividuals. This is the core component of many more detailed epidemiologicalmodels being at the same time structurally simplest. On the other hand, within thisstructure, the model is a rather general one, since nonlinear dependence of fertility,mortality and recovery rates on the population size is allowed.

2.1. The homogeneous model

The main variables in the model are the size S(t) of the susceptible population, andthe size I (t) of the infected population at time t .

The dynamics of the disease is described by the equations

S(t) = −σpyS + λ(S, I )S + γ (S, I )I, S(0) = S0, (1)

I (t) = σpyS − δ(S, I )I, I (0) = I0, (2)

On the effects of population heterogeneity on dynamics 125

where

y(t) = I (t)

S(t)+ I (t).

Here

λ = η − µ, whereη is the birth rate of the susceptible individuals 1,µ is the mortality rate of the susceptible individuals;γ = ν+εη is the inflow rate of susceptible individuals resulting from the infectedpopulation:ν is the recovery rate from the infection,η is the fertility rate of the infected individuals,ε is the fraction of susceptible “babies” of infected mothers;δ = µ+ ν − (1 − ε)η is the net out-flow rate of infected individuals, whereµ is the mortality rate of the infected individuals.

The rate of infection σpy(t) consists of three multipliers:σ represents the infectiousness (strength of infection),p is the (average) level of risk of the population, depending on the average intensityof participation in risky interactions, on the average immunity, etc.,y(t) is the prevalence of the disease at t .

The models that include the above one as a base component, have a drawback thatcould be significant in the expansion phase of the disease if the population is con-siderably heterogeneous with respect to risky behaviour of the individuals, in whichcase the individual values of p may be rather different from the average. In reality,individuals who are more vulnerable become infected with higher probability thanaverage. For many diseases individuals who are more vulnerable as susceptibleare more infective if they become infected. As a result, the rate of infection in theearly stage of the disease tends to be higher than σpy(t). On the other hand, ifthe mortality of the infected individuals is higher than that of the susceptible onesand the recovered individuals do not increase their level of risk after recovery, thenthe average vulnerability to risk in the susceptible population decreases with thetime. As a result, the value σpy(t) overestimates the rate of infection in the latestage of the expansion phase. In our subsequent analysis we formally support theseobservations and try to avoid the resulting distortion in predicting the evolution ofthe disease.

2.2. Modeling the heterogeneity

In this subsection we explicitly take into account the heterogeneity of the popula-tion, supposing that the value of p is specific for each individual. Similarly as inDiekmann et al. [3] we introduce a variable ω that characterizes individual featuresthat are relevant to the disease. As in [3] the variable ω will be called h-state (hstands for heterogeneity). We assume that ω ∈ , where is a measurable subsetof a finite dimensional space.

1 It is assumed that the “babies” of the susceptible individuals are all susceptible.

126 V.M. Veliov

Correspondingly we introduce the following variables on :

S(t, ·) – the density of the susceptible population at time t ;I (t, ·) – the density of the infected population at t .

Moreover, p(ω) ≥ 0 will denote the level of risk at h-state ω. In principle p(ω)could combine the individual susceptibility and the individual contact rate. How-ever, as explained later, more consistent with the other suppositions below is theinterpretation of p as the individual rate of potentially risky contacts.

To avoid confusion we note that S(t, ·), I (t, ·) : �→ R are not probabilitydensities, since their integrals over , denoted further by S(t) and I (t), give thetotal size of the susceptible and of the infected populations at t .

The dynamics of the heterogeneous population is described by the followingmodel where “dot” means derivation with respect to t :

˙S(t, ω) = −σp(ω)z(t)S(t, ω)− µS(t, ω)

+η∫

ψ0(S(t, ω), ω, ω′)S(t, ω′) dω′

+ γ∫

ψ(S(t, ω), ω, ω′)I (t, ω′) dω′, (3)

˙I (t, ω) = σp(ω)z(t)S(t, ω)− δI (t, ω).

The meaning of the rates µ, η, γ and δ is as in the previous subsection. It issupposed that these rates are independent of ω. The density ψ0(S(t, ω), ω, ω

′)represents the “probability” that an offspring of an individual of h-state ω′ is ofh-state ω. Similarly,ψ(S(t, ω), ω, ω′) represents the “probability” that an infectedindividual of h-state ω′ passes to an h-state ω after recovery. These probabilitiesare allowed to depend on the current size of the susceptible population of h-stateω. As before the rates µ, η, γ and δ, but also the densitiesψ0 andψ may depend onthe total susceptible and infected populations S(t) and I (t), which is not explicitlyindicated in the above formulae.

In this paper ω has more behavioral than purely biological meaning. That is,it represents habits or vulnerability to risk, rather than natural immunity or frailty.Examples are the HIV disease in Central and South Africa (see e.g. Thieme andCastillo-Chavez [8], Sanderson [7], Feichtinger et al. [4]) and the Hepatitis incommunities of heroin users (cf. Gavrila et al. [5]). Therefore we assume that thenewborn individuals have the same h-distribution as the current susceptible popu-lation 2. Similar assumption we make also for the recovered individuals. The latteris certainly fulfilled if there is no recovery from the disease, as in our main concerns– AIDS and Hepatitis C.

2 This assumption is an idealization. However, it certainly holds in the case of a geneticallydetermined factors of the risk level (in this case – of the susceptibility), provided that onlythe susceptible individuals give birth to non-infected children. Alternatively, if p is relatedto behaviour, then one can argue that the susceptible individuals represent a more attractivegroup to follow than the infected ones, therefore the newborn individual accept the behaviourof the former.

On the effects of population heterogeneity on dynamics 127

In other words we assume further that (suppressing the time variable in thenotations)

ψ0(S, I, S(ω), ω, ω′) = ψ(S, I, S(ω), ω, ω′) = S(ω)

S. (4)

The term z(t) in (3) represents the infectivity of the environment in which thesusceptible individuals live. It is called weighted prevalence and is defined as

z(t) = J (t)

R(t)+ J (t),

where

R(t) =∫

p(ω)S(t, ω) dω, J (t) =∫

q(ω)I (t, ω) dω.

That is, z(t) is the probability to randomly pick up an infected individual out ofthe pool of all individuals, if the individuals are counted according to their weightsp(ω), for the susceptible, and q(ω), for the infected individuals. Thus the structureof the first term in (3) assumes the so-called separable mixing: z(t) applies to allindividuals, rather than allowing mixing preferences depending on p.

In the sequel we assume that

q(ω) = κp(ω), (5)

which seems reasonable in the present context 3.The overall heterogeneous model, under the simplifying assumptions (4),(5),

and with the notations λ and δ from the previous section, becomes

˙S(t, ω) = −σp(ω) J (t)

R(t)+ J (t)S(t, ω)+ λ(S, I )S(t, ω)+ γ (S, I )

I

SS(t, ω), (6)

˙I (t, ω) = σp(ω)J (t)

R(t)+ J (t)S(t, ω)− δ(S, I )I (t, ω), (7)

S(t) =∫

S(t, ω) dω, (8)

I (t) =∫

I (t, ω) dω, (9)

R(t) =∫

p(ω)S(t, ω) dω, (10)

J (t) = κ

∫

p(ω)I (t, ω) dω, (11)

3 Assumption (5) of proportional mixing is easily justifiable, for example, if the level ofrisk is determined by the frequency with which the individual is involved in a risky interac-tion, and if this frequency does not change (or changes proportionally) when the individualbecomes infected.

128 V.M. Veliov

with initial conditions

S(0, ω) = ϕS0 (ω)S0,

I (0, ω) = ϕI0 (ω)I0.

The initial conditions are given in terms of the initial size of the susceptible and ofthe infected sub-populations, S0 and I0, respectively, and the probabilistic densitiesϕS0 (·) and ϕI0 (·) of their h-distributions.

The measurable mapping (S(·, ·), I (·, ·), S(·), I (·), R(·), J (·)) is a solution of(6)–(11) if for almost every ω the functions S(·, ω) and I (·, ω) are absolutely con-tinuous and (6)–(11) hold almost everywhere. Below we suppose that the functionsλ, γ and δ are at least continuous, or several times continuously differentiable,wherever appropriate. Moreover, we assume that is a closed subset of Rr withpositive Lebesgue measure, that p is a nonnegative measurable function on, andthat

meas{(ω1, ω2) ∈ × : p(ω1) = p(ω2)} = 0. (12)

This condition is obviously fulfilled if = [0, 1] and p is strictly monotone, or if = [0, 1]r and p is continuous and strictly monotone along every line throughthe origin.

Remark. A homogeneous population could be considered as a particular case whereϕS0 (·) and ϕI0 (·) are concentrated at a single point ω, that is ϕS0 (·) = ϕI0 (·) = δω(·),where δω(·) is the Dirac delta function.An alternative way is to take arbitraryϕS0 andϕI0 and a constant function p(·). In both cases we come up with a model equivalentto (1),(2), but in the second way involvement of Radon measures is not necessary.

The above heterogeneous model is designed to avoid the shortcomings of thehomogeneous models mentioned in the end of the previous subsection. Howeverit has its own disadvantages: (i) it involves distributed parameter integro-differ-ential equations, therefore is more complicated for calculation (especially in itsage-structured version, which we do not discuss in this paper); (ii) it requires theinitial densities ϕS0 (·) and ϕI0 (·) which are usually not available.

The first disadvantage is not critical in simulation tasks4 but could be an obsta-cle for the design of optimal treatment or prevention strategies, especially in theage-structured case 5. The second disadvantage makes the direct use of the modelimpossible even for simulation tasks, due to the lack of data. For the HIV diseasein Botswana, for example, even the initial data S0 and I0 are vague, while for thedensities ϕS0 (·) and ϕI0 (·) there is no data available at all (cf. Sanderson [7]).

3. Approximating the heterogeneous system by an infinite system of ODEs

For the two reasons formulated in the end of the previous section we study the issueof approximating the heterogeneous model by a homogeneous one, in such a way

4 The author has developed an efficient solver for rather general (descriptive or control)heterogeneous models, see Veliov [9].

5 Which should be the case in modeling HIV in the African countries.

On the effects of population heterogeneity on dynamics 129

that the results obtained by the homogeneous model are similar to those that couldbe obtained by the heterogeneous model if the relevant data were available.

Integrating (6) and (7) with respect to ω we obtain

S = −σρ∗(t)S + λ(S, I )S + γ (S, I )I, S(0) = S0, (13)

I = σρ∗(t)S − δ(S, I )I, I (0) = I0, (14)

where

ρ∗(t) = z(t)R(t)

S(t)= J (t)

R(t)+ J (t)

R(t)

S(t). (15)

Note that the only information that one needs to recover the aggregated solutionsS(·) and I (·) of the heterogeneous system by solving the ODE system (13),(14) isthe function ρ∗(t).

Define the normalized moments

mSk (t) =∫

(p(ω))kS(t, ω)

S(t)dω,

mIk(t) =∫

(p(ω))kI (t, ω)

I (t)dω, k = 0, 1, . . . .

Obviously

R(t) = mS1 (t)S(t), J (t) = κmI1(t)I (t), (16)

From here we obtain

z(t) = κmI1(t)I (t)

mS1 (t)S(t)+ κmI1(t)I (t), (17)

ρ∗(t) = z(t)mS1 (t) = κmS1 (t)mI1(t)y(t)

(1 − y(t))mS1 (t)+ κy(t)mI1(t), (18)

where, as in Section 2.1, we use the notation y(t) = I (t)/(S(t) + I (t)) for theprevalence.

Using equations (6) and (13) in the expression for mSk one can obtain the fol-lowing equations:

mS1 = −σz(t)(mS2 −mS1mS1 ),

mS2 = −σz(t)(mS3 −mS2mS1 ),

. . . . . . . . . . . . . . . . . . . . .

mSk = −σz(t)(mSk+1 −mSkmS1 ),

. . . . . . . . . . . . . . . . . . . . .

130 V.M. Veliov

and similarly, using (7) and (14),

mI1 = σz(t)1 − y(t)

y(t)(mS2 −mI1m

S1 ),

mI2 = σz(t)1 − y(t)

y(t)(mS3 −mI2m

S1 ),

. . . . . . . . . . . . . . . . . . . . .

mIk = σz(t)1 − y(t)

y(t)(mSk+1 −mIkm

S1 ),

. . . . . . . . . . . . . . . . . . . . .

The initial conditions

mSk (0) =∫

(p(ω))kϕS0 (ω) dω, mIk(0) =∫

(p(ω))kϕI0 (ω) dω

are known, provided that the initial distributions ϕS0 and ϕI0 are given.Having in mind (17) and (18) we establish that the above infinite system of

differential equations, together with (13),(14) determines the solution (S(·), I (·)).It can be used for numerical approximation of (S(·), I (·)) in a version of the methodof Poincare by a truncation of the infinite system. Such an approximating proce-dure, however, still makes use of the initial data, ϕS0 (·) and ϕI0 (·), which shouldbe avoided, as we argued in the end of Section 2, therefore we do not discuss thedetails.

Another consequence of the above reformulation of the heterogeneous systemis the following.

Proposition 1. Assume that J (0) > 0. Then for each k ≥ 1 the moment mSk (·) isstrictly monotone decreasing. For k > 1 also the normalized momentmSk (·)/mS1 (·)is strictly monotone decreasing. Moreover, if mSk (t) ≤ mIk(t) for t = 0, then thisinequality holds for all t ≥ 0.

Proof. The proof makes use of the following Lyapunov type inequalities: if φ : �→ R is nonnegative, measurable probability distribution, and

mkdef=

∫

(p(ω))kφ(ω) dω,

then for every nonnegative integers p < q < r < s with p + s = q + r

mpms > mqmr. (19)

Here the inequality is strict due to assumption (12). Applying this inequality withp = 0, s = k+ 1, q = k, r = 1 we obtainmSk+1 −mSkmS1 > 0. From J (0) > 0 and(7) it follows that J (t) > 0 for every t . Thus z(t) > 0, which implies mSk (t) < 0.

To prove the second claim we calculate(mSk

mSi

)′= −σz(m

Sk+1 −mSkm

S1 )m

Si − (mSi+1 −mSi m

S1 )m

Sk

(mSi )2

= −σzmSk+1m

Si −mSi+1m

Sk

(mSi )2

.

On the effects of population heterogeneity on dynamics 131

The last quantity is strictly negative according to (19) applied withp = i, q = i+1,r = k, s = k + 1.

To prove the invariance of the area mSk ≤ mIk with respect to the differen-tial equations for the moments it is enough (this is obvious, however, for a strictreasoning one can apply the Nagumo invariance theorem) to verify that mSk (t) ≤mIk(t)whenevermSk (t) = mIk(t). At such a point t we have mSk (t) ≤ 0 and mIk(t) ≥0 by the same inequality used in the proof of the first claim. �Corollary 1. The heterogeneous system (6)–(11) does not have a periodic solutionwith J (0) > 0, whatever are the functions λ, γ and δ.

We remind that the strict monotonicity in Proposition 1 and the above corollaryare obtained under the standing assumptions (12), which obviously excludes thecase of a homogeneous system. We mention that the homogeneous system (1),(2)may have a periodic solution for appropriate λ, γ and δ depending on S and I .

The value

mS2 (t)

mS1 (t)−mS1 (t)

(which is “variance/mean” of p(·)) can be viewed as a measure of heterogeneity ofthe current susceptible population. If the two populations have the same mean riskmS1 (0) at time 0, then the one with the higher value ofmS2 (0) is more heterogeneous.

4. Encapsulating heterogeneity in a homogeneous model

In this section we continue the analysis of the heterogeneous model (6)–(11) aim-ing at obtaining a homogeneous system of the form of (1), (2) which simulates theheterogeneous one. The key point is that we replace the multiplier y in (1), (2) witha nonlinear function of the prevalence, ρ(y). It turns out that there exists such afunction ρ(y) for which the solution of (1), (2) (with y replaced by ρ(y)) coincideswith the (S, I )-part of the solution of the heterogeneous system (6)–(11), in theexpansion phase of the disease (where y increases). The advantage of knowingthe appropriate function ρ(y) is that one would not need to know the distributionsϕS0 and ϕI0 in order to simulate the evolution of the disease in a heterogeneouspopulation. However, the exact function ρ will not be constructively defined. Ourapproach suggests to approximate the function ρ(y) by measuring the prevalencey(t) at several moments t and applying standard identification technique. For thispurpose one has to restrict the search of the function ρ(y) to a class of functions �depending on a few parameters. In order to justify a choice of the class � we firstestablish some qualitative properties of the function ρ(y).

We stress that the approach below is appropriate for the expansion phase ofthe disease, where the prevalence y(t) is increasing. Let (S(·, ·), I (·, ·), S(·), I (·),R(·), J (·)) be a solution of the heterogeneous system of (6)–(11), let ρ∗(t) be thefunction defined in (15), and y(t) = I (t)/(S(t) + I (t)) be the prevalence. Wesuppose that y(0) > 0, thus there is an interval [0, t∗) (t∗ could be +∞) such thaty(t) > 0 on [0, t∗) and y(t∗) = 0 (in the case t∗ = +∞ the last equation shouldbe disregarded). Let [y0, y

∗) be the set of values of y(t) when t runs in [0, t∗).

132 V.M. Veliov

Thanks to the strict monotonicity of y(·), the equation

ρ(y(t)) = ρ∗(t) (20)

defines in a unique way the function ρ : [y0, y∗) �→ R. Then the solution of the

system

S = −σρ(

I

S + I

)S + λ(S, I )S + γ (S, I )I, S(0) = S0, (21)

I = σρ

(I

S + I

)S − δ(S, I )I, I (0) = I0, (22)

coincides with the (S, I )-part of the solution of the heterogeneous model (6)–(11).The definition of the nonlinear “prevalence-to-incidence” function ρ, however, isnot constructive, since it requires knowledge of the solution of (6)–(11). Thereforethe next step to constructive approximation will be to establish some properties ofthe function ρ(y).

If the heterogeneity condition (12) holds, then according to (19) we have forthe normalized dispersion

d = mS2 (0)

mS1 (0)−mS1 (0) > 0.

Below in this section we assume that 6

mI1(0) ≤ mS1 (0)+ ed, where e ∈ (0, 1). (23)

For a homogeneous population d = 0 and assumption (23) reduces to an equality,which is automatically satisfied in this case. If the population is heterogeneous,then (23) allows mI1(0) to be somewhat larger than mS1 (0), which is usually thecase, since individuals of higher level of risk get infected with a higher probability.

Solving the differential equation formI1 and denoting q(t) = σz(t)(1 − y(t))/

y(t) we have

mI1(t) = mI1(0)e− ∫ t0 q(θ)mS1 (θ)dθ +

∫ t

0e− ∫ tξ q(θ)mS1 (θ)dθ q(ξ)mS2 (ξ) dξ

and integrating by parts and rearranging the terms we obtain

mI1(t)− mS2 (t)

mS1 (t)=(mI1(0)− mS2 (0)

mS1 (0)

)e−

∫ t0 q(θ)m

S1 (θ)dθ

−∫ t

0

d

dξ

(mS2

mS1

)(ξ)e

− ∫ tξ q(θ)mS1 (θ)dθ dξ

6 This supposition is used below only to prove that the function ρ is increasing if theprevalence is sufficiently small.

On the effects of population heterogeneity on dynamics 133

Substituting this in the equation for mI1 we have

mI1(t) = q(t)mS1 (t)e− ∫ t0 q(θ)mS1 (θ)dθ

[(mS2 (0)

mS1 (0)−mI1(0)

)

+∫ t

0

d

dξ

(mS2

mS1

)(ξ)e

∫ ξ0 q(θ)m

S1 (θ)dθ dξ

].

The first term in the last brackets is constant and positive, according to (23). Thederivative of mS2/m

S1 is strictly negative according to Proposition 1. Therefore, the

second term is zero at t = 0 and monotone decreasing. We come to the following.

Claim 1. If y(0) ∈ (0, 1) and t∗ > 0, then there exists tI ∈ [0, t∗] such that mI1(t)> 0 on [0, tI ) and mI1(t) < 0 on (tI , t∗).

It may happen that tI = t∗, that is, mI1 is increasing in the whole expansionphase.

From the definition of the function ρ we have

ρ′(y(t)) = ρ∗(t)y(t)

.

Having that (see (18))

ρ∗(t) = κmS1mI1y

(1 − y)mS1 + κymI1

(t),

one obtains

ρ′(y(t)) = κmI1(m

S1 )

2 + mI1(mS1 )

2y(1 − y)/y + κmS1 (mI1)

2y2/y[(1 − y)mS1 + κymI1

]2 (t). (24)

Claim 2. Denote c=mS2 (0)/(mS1 (0))2. If y(0)=y0 ∈(

0,(1 + c

√κ/(1 − e)

)−1)

,

then the function ρ(·) : [y0, y∗) �→ R is strictly increasing close to y = y0. If

t∗ < +∞ and either y∗ = 1 or tI < t∗, then ρ(·) is strictly decreasing closeto y∗.

To prove the first part of this claim we substitute in (24) mS1 and mI1 by theright-hand sides of the corresponding differential equations in Section 3. Thisgives

ρ′(y(t))=κmI1(m

S1 )

2+ σzy

[(1−y)2(mS2 −mI1mS1 )(mS1 )2−κy2(mS2 −(mS1 )2)(mI1)2

][(1−y)mS1 +κymI1

]2Using the inequality (23), and the definitions of d and c we obtain for t = 0

ρ′(y0) ≥ κmI1(m

S1 )

2 + σzy(mS1 )

3d[(1 − e)(1 − y)2 − κc2y2

][(1 − y)mS1 + κymI1

]2 ,

which implies the first statement in Claim 2.

134 V.M. Veliov

To prove the second part we notice that ˙y(t∗) = 0. Then the sign of ρ′ close toy∗ is determined by that of mI1(m

S1 )

2y(1 − y)+ mS1 (mI1)

2y2, evaluated at t = t∗.The second term is negative, according to Proposition 1. The first term is eitherzero (if y∗ = 1) or negative (if tI < t∗), according to Claim 1.

The last consideration and (24) imply also the following.

Claim 3. ρ(·) has a bounded derivative in every compact subinterval of [y0, y∗),

but ρ′(y) may converge to −∞ at y∗.We mention that it is a known fact that a nonlinear “prevalence-to-incidence”

function ρ may be more relevant (due to the heterogeneity) than a linear one,like in (21)–(22). Sanderson [7], for example, uses the function ρ(y) = ayα inthe context of the HIV disease in Botswana, with α identified from real data asα ≈ 0.3. Notice, however, that this function does not fulfill neither Claim 2 norClaim 3. Indeed, it is everywhere monotone increasing and has infinite derivativeat y = 0.

One can look for a simple class of functions that have the above propertiesand could be used as approximations of the “prevalence-to-incidence” functiony −→ ρ(y). First of all, we chose the form ayψ(y) – a “deformation” of thelinear functions ay which corresponds to a homogeneous population. Second,the function ρ can be defined only on a subinterval (0, y∗) ⊂ [0, 1], in whichcase ρ′(y∗) = −∞. Therefore, we chose the functional form ayψ(1 − y/y∗),where now ψ is a nonnegative differentiable function defined on (0, 1), withψ ′(0) = ∞. If, in addition, ψ(1) > 0 then the property in Claim 2 is auto-matically satisfied. A simple class of such functions is {ψ(x) = xβ : β ∈[0, 1]}.

Based on the above phenomenological argument we propose the following classof functions depending on four parameters a, α, b, β to be used for approximationsof the function y −→ ρ(y):

ρ(y)def= ayα(1 − by)β, (25)

where

a ≥ 0, α ∈ [0, 1], b ≥ 1, β ∈ [0, 1].

Here, in fact, b = 1/y∗, but the last value is not known in advance. Notice thataccording to Claim 3 we may fix α = 1. We deliberately keep a redundancy byallowing values of α ≤ 1, but we shall see later that identifying the four param-eters from experimental data leads to α = 1 (or to a value close to one, in someexceptional cases). Thus, in fact, only the three parameters a, b, β need to be iden-tified from data in a real application. Notice that no data about the heterogene-ity of the population is required. Provided that the other parameters in (21), (22)are known, one needs only measurements of y(t) for at least three moments oftime.

We illustrate the proposed approximation by several examples, where simula-tion of the heterogeneous system (6)–(11) is used for determination of the “true”“prevalence-to-incidence” function ρ(y) by (15), (20).

On the effects of population heterogeneity on dynamics 135

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

h=0.2

h=0.4

h=0.6

h=0.8

h=1

ρ(y)

y = I/(S+I)

Fig. 1. The “prevalence-to-incidence” function ρ(y) for different levels of heterogeneity h:the bigger h, the more heterogeneous the population.

Figure 1 shows the “true” function ρ(y) arising from a heterogeneous modelwith artificial fertility/mortality data (roughly calibrated for a human population).In all plots the mean risk at time 0, (mS1 (0),m

I1(0)), is the same. The only differ-

ence is the level of heterogeneitymS2 (0) = mI2(0). Bigger values of h correspond tohigher heterogeneity. The value h = 0 corresponds to a homogeneous population,and the function ρ(y) is linear in this case. The higher is the heterogeneity, themore significant is the deviation from the straight line. Notice that for h = 0.8 stilly∗ = 1, but for h = 1 we see y∗ < 0.9.

Figure 2 represents the functions ρ(y) for two different levels of heterogene-ity, together with their best uniform approximations (the dotted lines) with func-tions from the class (25). Here measurements of ρ(y) are taken over the wholeinterval [y0, y

∗). In figures 3 and 4, only five (resp. ten) measurement in the inter-vals indicated by vertical lines are used for finding the parameters in (25). The dottedline represents the best uniform approximation in the class of functions (25). Thedash-dotted lines represent the best approximation based on the same data, in theclass of functions used in Sanderson (2002). The forecast based on the dash-dottedextrapolation of ρ(·) would be rather pessimistic. Notice also that in Figure 4the function ρ(·) is rather flat in the interval [0.1, 0.2] where the measurements aretaken. Nevertheless, the tendency of future decrease is captured by the approximatingclass (25).

136 V.M. Veliov

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

a=0.9524 α=0.9136 b=1.0000

h=0.4:

β=0.0425

a=1.4777 α=1.0000 β = 0.5944 b=1.1182 h=1.0:

Fig. 2. The functions ρ(·) for h = 1 and h = 0.4 and their best uniform approximations byfunctions from the class (25) (the dotted lines).

5. The Effect of the heterogeneity on the evolution of an epidemic disease

Now we shall investigate how the level of heterogeneity of the population influ-ences, ceteris paribus, the evolution of the disease. It turns out that the effect of theheterogeneity on a short and on a long time horizons could be opposite, thereforewe study these two cases separately.

5.1. Short run behaviour

Proposition 2. Let (S, I , S, I, R, J ) and ( ˜S, ˜I , S, I , R, J ) be two solutions of(6)–(11) corresponding to initial data (S0ϕ

S0 (·), I0ϕ

I0 (·)) and (S0ϕ

S0 (·), I0ϕ

I0 (·)),

such that mS1 (0) = mS1 (0), mI1(0) = mI1(0) and mS2 (0) < mS2 (0). If

J (0)

R(0)+ J (0)<

1

2(26)

then there exists t > 0 such that

S(t) > S(t), I (t) < I (t), ∀ t ∈ (0, t]. (27)

If the opposite strict inequality holds in (26), then there exists t > 0 such that theopposite strict inequalities hold in (27).

On the effects of population heterogeneity on dynamics 137

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ayα

ayα(1−by)β

y

ρ(y)

a=1.968α=1b=1.192β=0.461

a=0.873α=0.429

Fig. 3. Extrapolations of ρ(·) by functions from the class (25) (dotted line) and by ayα

(dash-dotted) from measurements in [0.4, 0.5].

Proof. Let us denote �S(t) = S(t) − S(t) and �I (t) = I (t) − I (t). Similarnotation �g(t) will be used also for other variables g.

Obviously ρ∗(0) = ρ∗(0), therefore from the conditions of the proposition and(13),(14) we obtain that �′

S(0) = �′I (0) = 0. We shall investigate the sign of

�′′S(0). Differentiating (13) and using the above equalities and the assumptions we

obtain that

�′′S(0) = −( ˙ρ∗

(0)− ρ∗(0))S0.

We have

ρ∗(0) =(z(t)

R(t)

I (t)

)′

t=0= z′(0)mS1 (0)+ z(0)mS1 (0)

= z′(0)mS1 (0)− z2(0)(mS2 (0)−mS1 (0)m

S1 (0)

).

The derivative of z can be found by differentiating J (t)/(R(t)+ J (t)), where thederivatives ofR and J are calculated from (10),(11) using also (8),(9). After certaincalculations one can represent

z(0) = mS2 (0)S0z(0)

R(0)+ J (0)− ζ,

˙z(0) = mS2 (0)S0z(0)

R(0)+ J (0)− ζ,

138 V.M. Veliov

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

1.2

1.4

ayα

ayα(1−by)β

a=1.835α= 1

b=1.192

β=0.303

a=1.522α=0.937

y

ρ(y)

Fig. 4. Extrapolations of ρ(·) by functions from the class (25) (dotted line) and by ayα

(dash-doted) from measurements in [0.1, 0.2].

where it is important that the term ζ is the same in the two formulas. Notice thatR(0) = mS1 (0)S0 = mS1 (0)S0 = R(0) and similarly for J . Also z(0) = z(0). Thenwe obtain

�′′S(0) =

[mS2 (0)

S(0)z(0)

R(0)+ J (0)− ζ

]R(0)

S(0)− z2(0)

(mS2 (0)−mS1 (0)m

S1 (0)

)

−[[mS2 (0)

S(0)z(0)

R(0)+ J (0)− ζ

]R(0)

S(0)− z2(0)

(mS2 (0)− mS1 (0

)mS1 (0))

]

= z(0)

[R(0)

R(0)+ J (0)− z(0)

] (mS2 (0)− mS2 (0)

).

Since the last multiplier is negative, �′′S(0) will be negative (therefore �S(t) will

be negative for small t) if the multiplier in the brackets is positive. The latter isequivalent to R(0) > J (0), which coincides with (26). �

The meaning of the above proposition can be explained in the following informalway. Let us compare two populations with the same mortality, birth and recoveryrates, with the same initial size and same initial number of infected individuals,and also with the same mean level of risk at time t = 0. Let only the initial levelof heterogeneity of the susceptible sub-populations be different: mS2 (0) < mS2 (0).Then in the short run the less heterogeneous population has

On the effects of population heterogeneity on dynamics 139

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

Noninfected population

m2(0)=0.76

m2(0)=0.53

m2(0)=0.45

m2(0)=0.39

t

Fig. 5. Evolution of the susceptible part of populations differing only with their level ofheterogeneity. The initial prevalence is y(0) = 0.02.

(i) more susceptible individuals and less infected individuals, if the weightedprevalence at t = 0 is less than 0.5;

(ii) less susceptible individuals and more infected individuals, if the weightedprevalence at t = 0 is bigger than 0.5.

The fact that for a very high weighted prevalence, z(0), the higher heterogeneityis advantageous for the population is obvious. Not that obvious is that for verylow weighted prevalence the lower heterogeneity is advantageous in the short run.What is, to our opinion, somewhat unexpected, is that the threshold value is alwaysz = 0.5, which is independent of the particular functions λ, γ and δ, as well as onthe other data involved.

The result is illustrated in figures 5, 6 for initial weighted prevalence z(0) < 0.5,and in Fig. 7 for z(0) > 0.5.

5.2. Long run behaviour

From figures 5 and 7 one may expect that the more heterogeneous population willmaintain more susceptible individuals in the long run. This conclusion, however,is false. The analysis of the asymptotic behaviour of the solution of (6)–(11) is acomplicated task at this level of generality. Therefore we shall consider a simplified

140 V.M. Veliov

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30 Infected population

m2(0)=0.76

m2(0)=0.53

m2(0)=0.45

m2(0)=0.39

t

Fig. 6. Evolution of the infected part of populations differing only with their level of heter-ogeneity. The initial prevalence is y(0) = 0.02.

situation where we shall obtain a complete analytic description of the asymptoticbehaviour, depending on the initial prevalence.

Namely, we compare two populations with the same initial number ofinfected individuals and the same mean risk at time zero. One of these populationsis assumed homogeneous, with p(ω) = p for all individuals. The second popula-tion is assumed heterogeneous with the set = {ω1, ω2} consisting of only twovalues: at time zero a fraction α of the individuals have no risk at all (p(ω1) = 0),the rest have risk level p2 > 0. In order to have the same mean value of risk attime t = 0 we have to take p2 = p/(1 − α). Moreover, we assume that there is norecovery, that is, γ = 0, that the rates λ and δ are constants, and that κ = 1.

The equations of the homogeneous population become

S(t) = −σp I

S + IS + λS, S(0) = S0,

I (t) = σpI

S + IS − δI, I (0) = I0.

Denoting Si(t) = S(t, ωi), Ii(t) = I (t, ωi), i = 1, 2, we rewrite the equations forthe heterogeneous population as

On the effects of population heterogeneity on dynamics 141

0 5 10 15 20 25 30 35 40 45 5014

16

18

20

22

24

26

28

30

32

Noninfected population

t

m2(0)=0.76

m2(0)=0.53

m2(0)=0.45

m2(0)=0.39

Fig. 7. Evolution of the susceptible part of populations differing only with their level ofheterogeneity. The initial prevalence is y(0) = 0.75.

S1 = λS1, S1(0) = αS0,

I1 = −δI1, I1(0) = αI0,

S2 = −σ p

1 − α

I2

S2 + I2S2 + λS2, S2(0) = (1 − α)S0,

I2 = σp

1 − α

I2

S2 + I2S2 − δI2, I2(0) = (1 − α)I0.

Each of the above two systems can be transformed to a Ricati-type system and canbe solved analytically. The solution depends qualitatively on the quantities

r = λ+ δ − σp, and rα = λ+ δ − σp

1 − α.

Denoting by Shom(t) = S(t) and Sheter(t) = S1(t)+ S2(t) the size of the suscepti-ble part of the homogeneous (respectively heterogeneous) population, and solvingthe corresponding equations one can obtain

Shom(t) = eλtS0 ×{(y(0)e−rt + 1 − y(0)

) σpr , if r �= 0,

e−σpy(0)t , if r = 0,

142 V.M. Veliov

Sheter(t) = eλtS0 ×{α + (1 − α)

(y(0)e−rαt + 1 − y(0)

) σp(1−α)rα , if rα �= 0,

α + (1 − α)e−σp

1−α y(0)t , if rα = 0,

As above, y(0) is the initial prevalence in the two populations.We compare the limits at infinity of the “detrended” populations:

Shomdet (∞) := lim

t→+∞ e−λtShom(t)=S0×

{(1−y(0)) σpr , if r >00, if r≤0,

Sheterdet (∞) := lim

t→+∞ e−λtSheter(t)=S0 ×

{α+(1−α)(1−y(0)) σp

(1−α)rα , if rα >0α, if rα ≤ 0.

Three cases could be distinguished:

Case 1. 0 < rα < r . By an elementary argument one can prove that the equation

(1 − y)σpr = α + (1 − α)(1 − y)

σp(1−α)rα

has a unique solution y ∈ [0, 1]. Then we have

Shomdet (∞) > Sheter

det (∞) if y(0) < y, (28)

Shomdet (∞) < Sheter

det (∞) if y(0) > y, (29)

Case 2. rα ≤ 0 < r . In this case the same conclusions (28), (29) hold, with theonly difference that y is determined from the equation

(1 − y)σpr = α,

which has a unique solution y ∈ [0, 1].

Case 3. rα < r ≤ 0. In this case 0 = Shomdet (∞)<Sheter

det (∞) holds for every positiveinitial prevalence y(0).

We summarize the above conclusions as follows:

(i) the asymptotic steady state depends on the initial data S0 and y(0);(ii) there is a threshold value y ∈ [0, 1] (which for some values of the parameters

is strictly inside this interval) such that the more homogeneous populationis asymptotically larger than the more heterogeneous one if y(0) < y. Theconverse is true for y(0) > y.

The above conclusions are drawn on the basis of a rather simplified model. How-ever, the history dependence property (i) is common for the models with endogenousreproduction (both heterogeneous and homogeneous), which exhibits a certain sin-gularity (indeterminacy) at the moment of initiation of the disease 7. What concernsthe finding (ii), it might have a limited validity, but is supported also by some ofour numerical experiments with various models of the form (6)–(11).

7 The system (1), (2), for example, has a variety of equilibria (detrended equilibria, ifλ > 0) with I = 0. The initial prevalence determines which one will be approached atinfinity).

On the effects of population heterogeneity on dynamics 143

6. Concluding remarks

Heterogeneous models with a single parameter of heterogeneity are numericallytractable, but nevertheless are often inapplicable due to lack of h-distributed data.Sections 2–4 propose a bridge from heterogeneous to homogeneous models of thedynamics of infectious diseases. The analysis shows that while a linear “prevalence-to-incidence” function is reasonable for a homogeneous population, the presenceof heterogeneity tends to deform this function in a way qualitatively describedin Section 4. We show that the evolution of the heterogeneous population in theexpansion phase of the disease can be, in principle, exactly described by a non-dis-tributed (homogeneous-type) model which however, is not constructively defined.Based on the analysis in Section 4 we propose an appropriate class of “prevalence-to-incidence” functions which could be used to obtain reasonable approximationsto the evolution of the disease within a non-distributed model. The functions fromthis class depend on three free parameters that could be estimated from a modestamount of (non-distributed) data for the history of the prevalence. The approach isespecially appropriate for diseases such as HIV in Africa, Hepatitis C in societiesof drug users, etc., where distributed data are scarce, while on the other hand, theheterogeneity plays an essential role.

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