Research paper
Simple criss-cross model of epidemic for heterogeneous populations

https://doi.org/10.1016/j.cnsns.2019.104920Get rights and content

Highlights

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    The model of tuberculosis is universal and based on the real data.

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    The epidemic dynamics in heterogeneous population is described by the model.

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    The criss-cross infection between two subpopulations is considered.

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    The model can be applied to describe other infectious diseases spread.

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    Active case detection has great impact on reducing incidence rate.

Abstract

In this paper we consider a class of criss-cross models describing the dynamics of epidemic of infectious illness. We follow the ideas presented in the paper by Romaszko et al., where the authors described actions of active detecting of tuberculosis (TB) among homeless subpopulation in Warmian-Masurian province of Poland. However, the class of epidemiological models analyzed in this paper, because of their universalism, can be applied in modeling dynamics of epidemics of various kinds of illnesses. In the original model, the whole population is divided into subpopulations of non-homeless and homeless people. Each of the subpopulations consists of two groups – susceptible and infected individuals. We consider the division of the whole population into two subpopulations described by different model parameters. We focus on the analysis of the basic criss-cross model depending on the form of a function describing transmission of illness. We consider two types of this function – first, called standard incidence function, was used in the original work of Romaszko et al., and second, bilinear function. The most important property of this model is related to its Malthusian origin, and this property is independent of the transmission function. This means that in many cases the size of one of the subpopulations or the whole population grows boundlessly or the population goes to extinction. However, for the bilinear transmission function coexistence of both subpopulations is also possible. We also analyze the influence of active detection onto the model dynamics.

The basic criss-cross model is fitted to the demographic data from Poland which then allows for making a short-term prediction on TB dynamics.

Introduction

Mathematical criss-cross modeling is commonly applied in investigating phenomena where it is needed to consider interactions between various subpopulations specified in the whole population. In most cases, the sizes of different groups of population change in time, that is why criss-cross models are widely used in investigating dynamics of heterogeneous populations. In particular, criss-cross models are applied in modeling infectious diseases between different groups of the same species and, arguably even more extensively, between different species, for example brucellosis [1], gonorrhea [2], toxoplasmosis disease in human and cat populations [3] and others

One of the illnesses which is considered to be re-established nowadays is tuberculosis (TB). It is widely known that the spread of TB in particular subpopulations like homeless people is faster than among the rest of population [4]. On the basis of owned data about the incidence of TB in subpopulations of homeless and non-homeless people in Warmian-Masurian Province of Poland in the years 2001–2013, the model of TB spread among these subpopulations will be presented and analyzed.

As in most deterministic illness transmission models (cf. [5]), we assume that:

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    the population size is constant,

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    the probability of contact is the same for every two individuals,

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    the habitat is closed.

These are the assumptions that allow to propose the simplest epidemic model, as was done by Kermack et al. [6]. If these assumptions are fulfilled, then each individual has the same probability of infection and the number of new illness occurrences is directly proportional to the size of the whole population [7]. In order to capture the essential complexities in analysis of infectious diseases (like TB) transmission dynamics, the individuals in the whole population are divided into several classes (cf. [8], [9]):

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    susceptible (S) – those who are not infected but susceptible to the disease;

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    exposed (E) – those who are infected, but do not infect other people (so-called latent disease);

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    infectious (I) – those who have active disease and infect other people;

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    recovered (R) – those who are treated (from active or latent infection);

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    vaccinated (V) – those who have feasibly lower susceptibility to the disease.

In heterogeneous populations, each of the subpopulations (e.g. children and adults, homeless and non-homeless people) could be divided into all the classes mentioned above [10]. However, in reality, there is not enough data that allow to fit the parameters of the model when we divide the subpopulation into all the classes. Moreover, such a division leads to very complex criss-cross models for which analytical description would be probably impossible. For these reasons, following Romaszko et al. [11], in this paper we consider only two groups: susceptible and infected people. Furthermore, even with this simplification, the analysis of the model, particularly stability analysis, is complicated.

In order to reduce the number of the new TB infections, Active Case Finding (ACF) programs can be conducted. ACF steps involve screening groups of high risk (for example poor or homeless people communities) and contact tracing to increase the TB case identification ratio. The aim of contact tracing is to reduce the time needed for detection and treating a case, and hence reducing the possibility of illness transmission by infectious individuals [12], [13].

The problem of control of TB spread has been extensively studied by many authors. In [14], on a basis of deterministic model of TB which takes into account two classes of infective individuals, it is indicated how detection should be introduced into the population to reduce the number of individuals with active TB. The influence of TB screening to reduce the spread of the disease is also highlighted. Similar model for the TB transmission dynamics in sub-Saharan Africa was analyzed in [15]. Optimal time-dependent prevention policies which consider also the execution cost, are proposed in [16]. Based on actual data on TB in Cameroon, it is shown that chemoprophylaxis and education may reduce by 80% the TB burden in 10 years (cf. [16]).

Mathematical model proposed in [11] describes a short-term dynamics of TB in order to improve our understanding of this disease transmission dynamics and to evaluate the effectiveness of control strategy among the subpopulation of homeless people. In this paper we consider a generalization of the model proposed in [11] and present the mathematical analysis of the model. It needs to be pointed out that in the present work we would like to address the problem of short-term dynamics of TB, although we will make an asymptotic analysis focusing on the existence and stability of stationary states. We consider the cases without and with ACF. We also indicate the Malthusian background of the model. This approach is typical in such models (c.f. [17] and references therein) and is strongly related with the possibility of fitting parameters to demographic data, from which net reproduction rate per year is easily available. We make such fitting for the considered model with respect to demographic data from Poland. Numerical simulations complement the analytic results.

The paper is organized as follows. In Section 2 we introduce the model and notation which we use throughout the paper. Section 3 presents basic properties of the model without ACF, particularly those who are related to Malthusian origin of the model. Section 4 deals with the stability analysis. Section 5 is focused on the model considering ACF with accompanying stability analysis. Numerical simulations are presented in Section 6. The paper is finished with discussion presented in Section 7.

Section snippets

Simple criss-cross model of infectious illness

In this paper, following [11], we present a simple criss-cross model of the spread of infectious illness. Such type of models is often used in the description of epidemics spreading not only between different subpopulations of one population, but also among different populations; cf. [5]. We consider two subpopulations (which are non-homeless and homeless people in the original article). The illness is transmitted not only among one subpopulation, but also between individuals of different

Basic properties of the model

Let us denoteki=ηi+αi−Ai,κi=αiAi−1,Ai≠0,i=1,2.Notice thatki=ηi+Aiκi,η1=1,η2=η,and if Ai < 0, then κi < 0 and ki > 0, while κi ≥ 0 implies αi ≥ Ai > 0 and ki ≥ 0.

Local existence and uniqueness of solutions of Eq. (3) is a simple consequence of the assumptions that are fulfilled by the transmission function f. Due to these assumptions, the right-hand side of Eq. (3) fulfills local Lipschitz condition which yields desired properties of the solutions.

Assume now that xi(0) and yi(0) are positive and

Analysis of stationary states

In this section we investigate the conditions of existence and stability of stationary states for Eq. (3). First we assume that Ai ≠ 0. The cases for which A1=0 or A2=0 will be discussed at the end of this section.

Notice that from Eq. (4) any stationary state satisfies the relation0=Aiwi−αiyi.Hencexi=(αi−Ai)yiAi=κiyiforAi≠0.We see that the trivial stationary state, indicated by E0 ≔ (0, 0, 0, 0), always exists, regardless of the model parameters and the form of the transmission function f.

The model with active detection

In this section we consider the case of TB active detecting (ACF), mentioned in [11]. We denote by B the number of homeless infected people who are cured in programs of TB active detection. As a first approximation, we assume that B is a constant number, which is justified by the fact that each preventive action is limited by a specific budget allowing to diagnose a certain number of individuals. Then Eq. (3) takes the following formx1′=−f(x1,y1)−β1f(x1,y2)+y1+A1x1,y1′=f(x1,y1)+β1f(x1,y2)−k1y1,x

Numerical simulations

In this section we illustrate the model dynamics for the parameters fitted to TB data from Warmian-Masurian Province of Poland. Let us consider Eq. (1) with the transmission functions f(S,I)=SI and f(S,I)=SIS+I. We compare the actual epidemic data with the model to get the best-fitted parameters, which are summarized in Table 1.

The values of the parameters α1, α2, γ1, γ2, A1 and A2 are taken from [11]. A best-fit technique was used to estimate the values of the transmission coefficients βij, i,

Discussion

In this paper we have presented the analysis of the criss-cross model of infectious illness spread in the population consisting of two subpopulations. The model has been used in the context of TB epidemic for non-homeless and homeless people. The most essential finding is that our system has properties referring to the Malthusian model – for definite set of parameters each of the subpopulations can grow boundlessly or become extinct. However, as we use the model for short-term predictions, this

Conflict of interest

The author declare that there is no conflicts of interest regarding the publication of this article.

Acknowledgment

The authors would like to thank Dr. Małgorzata Zdanowicz for her help in working on the model analysis.

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