A Discrete SIS Model of Epidemic for a Heterogeneous Population without Discretization of its Continuous Counterpart

In this paper we propose a model of an infectious disease transmission in a heterogeneous population consisting of two different subpopulations: individuals with accordingly low and high susceptibility to an infection. This is a discrete model which was built without discretization of its continuous counterpart. It is not a typical approach. We assume that parameters describing particular processes in each subpopulation have different values. This assump - tion makes model analysis more complicated comparing to models without this assumption. We investigate conditions for existence and local stability of stationary states. The novelty of this paper lies in presenting the explicit condition concerning stationary states, including stability. We compute the basic reproduction number R 0 of the given system, which determines the local stability of the disease-free stationary state. Additionally, we consider a situation when there is no illness transmission in the subpopulation with the low susceptibility. Theoretical result are complemented with numerical simulations in which we fit the model to epidemic data from the Warmian-Ma - surian province of Poland. These data reflect the case of the tuberculosis epidemic for which the homeless people were treated as a group with the high susceptibility.


INTRODUCTION
In order to investigate the epidemic dynamics from the mathematical point, continuous models are commonly used.However, discrete models are also gaining attention.In most cases, these discrete systems arise from discretization of their continuous counterparts.We introduced and analyzed models of this type in [1,2] and [3].There are various methods of discretization, which are described and analyzed in the literature.In our previous papers we used the explicit Euler method and nonstandard discretization methods, including the strictly positive scheme.In this paper we show an unusual concept of building a discrete model of epidemic dynamics without discretization of its continuous counterpart.This concept is presented in [4] and [5].In the constructed model we will include demographic and epidemiological processes which were considered in models from our previous papers.
Here as a homogeneous population we understand a population in which we cannot distinguish individuals concerning the risk of being infected.A heterogeneous population is a population in which this distinction can be done.Here we assume that the heterogeneous population consists of two homogeneous subpopulations.The first subpopulation is formed by individuals with low susceptibility to an infection and the second subpopulation is composed of the people with high one.We will call these subpopulations the low subpopulation (LS) and high subpopulation (HS), respectively.The infection can be transmitted among each subpopulation separately and from HS to LS.We will reasonably assume that there is no transmission from LS to HS.
The proposed model is an example of SIS (susceptible-infected-susceptible) models -an infected individual after recovery does not gain immunity and can be infected again.In this paper we assume that every process for each subpopulation is independent.This assumption results in different values of coefficients describing specific processes in every subpopulation.According to our knowledge, there are no papers in which this case is analyzed in such a type of SIS models which is presented here, particularly in the context of presenting explicit results for stability analysis.The reason of neglecting it is obtaining complicated computations.In this paper transmission of the infection is described with a general form of functions which values can be interpreted as probability of remaining susceptible by an individual.These functions will be defined later.
Although discrete SIS models built with the discretization are not widely used, one find some examples of them in literature.Authors in [4] analyze two-dimension SIS model for a homogeneous population.This model was extended in [6], where environment seasonality is considered.To do so, the authors introduced a periodic function reflecting the birth and recruitment process.They emphasized formerly omitted an impact of demographic processes on population density.Martcheva in her book [5] applied the approach from [4] to a model for a heterogeneous population.However, she did not conduct stability analysis of stationary states.An interesting approach to discrete modelling is depicted in [7].The authors indicated in the subpopulation four compartments: susceptible, exposed (infected but not infectious), infective, and recovered one.Each compartment is divided into n patches.The author introduced two time scales: a slow and fast one, reflecting accordingly disease dynamics and movement between patches.The authors focused on analysis of the disease-free stationary state stability.The concept from [7] is continued in the very recent paper [8] -here the author analyzed the additional case when disease dynamics is faster than movement between patches.
In epidemiological modeling the concept of the basic reproduction number, customarily denoted with R 0 , is an important issue.Considering an epidemic in a heterogeneous population, we define R 0 as a number of new infections produced by an infective individual in a population at the disease-free stationary state.In this paper we compute R 0 for the given system and formulate condition for local stability of the disease-free stationary state, defined later, in the context of R 0 .
Motivation of presented research arose from the case of tuberculosis (TB) spread in the Warmian-Masurian province of Poland in years 2001-2018.In the population of this province two subpopulations were indicated -the non-homeless people, being LS, and the homeless people, which comprise HS.In the community of homeless people programs of Active Case Finding (ACF) were conducted.As a result, the TB incidence dropped not only among homeless individuals, but in the whole population in the province.This observation emphasized usefulness of indicating LS and HS in the population so TB preventive actions can be conducted only for specified subpopulation.It is therefore reasonable to investigate TB dynamics in a population where its heterogeneity is considered.A description of the ACF programmes held in the Warmian-Masurian province and their impacts are presented in [10].In the cited paper authors introduced a continuous model of TB dynamics for the given heterogeneous population.The heuristic assumptions of this continuous model are considered in our proposed system.We should stress that both models (mentioned in [10] and proposed by us) can be used for other diseases and other subpopulations of a heterogeneous population.
In this paper we assume, if it is not written otherwise, that n ∈ N ∪ {0}.

DESCRIPTION OF THE MODEL
Let us now describe elements of the analyzed model.Each variable and parameter corresponding to LS and HS has a lower subscript equal to 1 and 2. respectively.If the lower subscript is denoted by i, then i ∈ {1,2}.Following this notation, we will denote by S 1 and S 2 a size of the group of healthy people in HS and LS, analogically.The sizes of the infected individuals in each subpopulation are expressed with I1 and I 2 .With N 1 and N 2 we indicate the sizes of the whole subpopulations.Naturally we have N i = S i + I i .We consider that there are constant inflows C i into both subpopulations.The probabilities of individuals' survival are equal to r i .The ratios of recovery and mortality related to the illness are denoted by γ i and α i .

As an example of G and H one can
Later we will relate to the values of G'(0) and H'(0) for the general form of functions G and H.If G(x) is a surjection, we formulate the lemma:  (5) Eq. ( 4) can be written also as Si+Ii = Ci +ri Si+(ri -αi )Ii, giving Let us define   : = 1 −   +   > 0 (7) and express Eq. ( 6) as For Ii = 0 we immediately get the disease-free stationary state ,0), which always exists.From Eq. ( 8) for Ii ,Si ≠0 we obtain that Si>0 if and Ii>0 if In a further analysis we will use a notation Naturally we have κi ∈ (0,1).See that dynamics of Eqs.(1c)-(1d) is independent on Eqs.(1a)-(1b), hence Eqs.(1c)-(1d) can be analyzed solely.
Before investigating the existence of the stationary states, let us introduce auxiliary functions and describe their properties.

Auxiliary functions
Let us define a function and Let us introduce an another function ].The functions F and F a, because of their forms, have similar properties.Hence, F a is The definition of σ 2 in Eq. ( 7) yields . Hence, we restrict the ].The supremum of the values' set of F a for the narrowed domain equals Monotonicity of F a gives (16) Now let us investigate a function   ( 2 ): = 1 − (  ( 2 )).
(17) Looking at the properties of H and Fa, we state that the composition H(Fa) is decreasing.Remind that the domain of H is [0, 1].The first and the third conditions from Eqs. (15) and the dependence (16) yield that we do not have additional conditions because of the composition.

Case I2>0
Let us analyze a case when I2>0, meaning that in the population there is at least one infected individual from HS.We formulate the theorem: Theorem 1.If I2>0 for any stationary state of System (1), then there is a unique pair of the coordinates: ( 2 ,  2 ) = ( 2 ,  2 ), where  2 is a solution of an equation The pair ( 2 ,  2 ) Proof.Considering Eqs. ( 5), ( 8) and (11) in Eq. (1d) for any stationary state gives Eq. ( 18).This equation can be written, using the definitions ( 12) and (17), as F(I2) = Fb (I2).Let us investigate the intersection point of F and Fb for the domain  2 ∈ (0, ) determined in Subsection 3.1.Observe that H is decreasing and concave up, so Fb is increasing and concave down.Both Fa and Fb intersect at I2=0.
Let us compare the values of  1 and  1 .Remind that they are the positive unique solutions of Eqs.accordingly (24) and ( 27).Since G'<0, we have Hence, we state that  1 <  1 .
We will investigate the local stability of the states, computing eigenvalues of J. Hence, it is sufficient to consider eigenvalues of J1 and J2.
Recall Ineq.( 25), what is the condition for the E1 existence.This inequality stays on the contrary to Ineq. ( 28), what is one of the conditions for the Edf local stability.The similar reasoning can be done for Ineq.(19) being the condition for the Ee existence.We conclude that Corollary 5.If E1 or Ee exists, then Edf loses stability.
(39) Character of eigenvalues of J1(E1) depends on the sign of the discriminant of P(λ).We will denote this discriminant with Δ .The non-generic case when Δ=0 will be omitted.
1. Firstly we assume that Δ>0.The eigenvalues are real and equal to It is sufficient to check λ1>-1 and λ2<1.From the first inequality we have √ 2 − 4 < 2 − .
2. Now let Δ<0.The eigenvalues are complex with non-zero imaginary part and equal to where i is an imaginary unit.The dependence λ1 λ2=| λ|<1 guaranties the local stability of E1.See that Hence, only the condition c<1 has to hold.This inequality can be written as Ineq.(38).

Local stability of Ee
Here we determine local stability of Ee.We will use notations  =  ( ).
We conduct similar approach as for the determinant of J1(E1)-λI.Hence, we state that Corollary 6. Existing state Ee is locally stable if one of sets of conditions:

The basic reproduction number
Now we compute R 0 of System (1) with the use of the next-generation approach.This approach was introduced and described in [9].Firstly we rearrange System (1) so that two first equations correspond to variables describing the groups of infected individuals.We obtain The disease-free state in System (46) has a form   : = (0,0, ).
In a further analysis we will use a definition : The Jacobian matrix at   for System (46) can be written as a block matrix (   0     ) , where We express Ja as F+T, where F reflects new infections and T corresponds to the other process for infective states.The matrices F and T read We write I-T and (I-T) -1 and F(I-T) -1 as

NUMERICAL SIMULATIONS
Finally we illustrate the dynamics of System (1) for values of parameters fitted to real data.These data reflect the case of the TB epidemic mentioned in Section 1. Simulations were performed with Matlab software.Numerical results of the simulations and actual epidemic data were compared so that the bestfitted values of the parameters were obtained.These values are shown in Table 1.
Numbers of the homeless people were obtained from the Regional Center for Social Policy, Office of the Marshall of the Warmian-Masurian province in Poland.Demographic figures were taken from statistical yearbooks [12].Epidemic data were anonymized by the Independent Public Tuberculosis and Lung Diseases Unit in Olsztyn, Poland.Only numerical details were provided.All data are fully available without any restrictions.The numbers of healthy and infected individuals for each subpopulation for year 2001 were chosen as the initial condition for System (1).The values of β1, β2 and β were estimated with the use of the built-in lsqcurvefit function in Matlab.This function is based on the Gauss-Newton algorithm [13].The values of the remaining parameters were obtained from the Central Statistical Office of Poland.In Figure 1 comparison between the simulated values and the actual data is presented.

CONCLUSIONS
In this paper we introduced and analyzed the discrete model of the illness transmission in the heterogeneous population.The model was constructed without using discretization of the continuous model, what is not a usual approach.We defined stationary states of the proposed system -we determined the conditions for their existence.Later we investigated the conditions for their local stability and computed the basic reproduction number R 0 of the system.Finally we discussed the case of the lack of the illness transmission in the subpopulation with low susceptibility to infection.
We provided the mathematical analysis of the proposed system.We indicated appearing stationary states and expressed the explicit conditions of their existence and local stability.What is important, we assumed that the parameters describing each subpopulation have different values.Furthermore, we did not define specific functions describing the illness transmission.These assumptions make our model general.
The results obtained in this paper are in line with those concerning similar continuous For E e we obtained explicit conditions for its local stability, what is not obvious for a four-dimensional system.Besides E df and E e , there exists also the stationary state E 1 for which the infection appears only in LS, what is desirable from the epidemiological point.System (1), because of the mentioned properties appearing also in analogical continuous systems, can be exploited in cases when it is reasonable to investigate the discrete nature of epidemic spread.What is important, System (1) does not include a step size of the discretization method, which appears in discrete systems built with discretization of their continuous counterpart.Thanks to that, there are no cases where some conditions, for example for stability of stationary states, depend on the step size.
To conclude, the model proposed and analyzed in this paper can be treated as an exemplary one for researchers investigating epidemic dynamics in heterogeneous populations.From the theoretical point, the obtained outcomes can be used for analyzing the systems of four discrete equations which are built from coupling two-dimensional discrete systems.

1 𝐼𝐼 1 𝑁𝑁 1 Theorem 4 .
variable's value in the above definitions relate to E1.We will indicate the derivative of G at by G' and the identity 2x2 matrix by I. Existing E1 is locally stable if (

Figure 1 .
Figure 1.Tuberculosis in the Warmian-Mazurian province over the years 2001-2018 (number of the infected non-homeless individuals).Comparison between the model and the actual data If G(x) is a surjection, then Ineq.(25) from Corollary 3 holds if  1 >  1  1 .Existence of an endemic state Now let us investigate existence of a postulated positive (endemic) stationary state.For this state we have I2>0, what is the assumption of Theorem 1.Hence, for the positive state we have ( 2 ,  2 ) = ( 2 ,  2 ).We formulate the theorem:

Table 1 .
(1) values of the parameters for the model described by System(1)