key: cord-0487575-yqf8z4o0
authors: Medaglia, Andrea; Zanella, Mattia
title: Kinetic and macroscopic epidemic models in presence of multiple heterogeneous populations
date: 2021-11-10
journal: nan
DOI: nan
sha: 031094c47a6dd397a40b453cd65d55701a6b37ce
doc_id: 487575
cord_uid: yqf8z4o0

We study the impact of contact heterogeneity on epidemic dynamics. A system characterized by multiple susceptible populations is considered. The description of the spread of an infectious disease is obtained through the study of a system of Boltzmann-type equations for the number densities of social contacts of the introduced compartments. A macroscopic system of equations characterizing observable effects of the epidemic is then derived to assess the impact of contact heterogeneity.

the present setting models with saturated incidence rates can be easily derived with minimal assumptions [7, 17, 24] . Other recent contributions stressed were centered on the effects of the structure of contacts of agents, we mention in this direction the works [12, 16, 19] .

In the present contribution, we concentrate on the influence of contact heterogeneity on the dynamics of the disease in presence of multiple susceptible populations. Each susceptible compartment can be characterized by its mean number of connections. This situation is very common when non-pharmaceutical interventions have different impacts on the population [6] or in presence of sanitary cordon measures, where a portion of the territory results highly affected by the disease. We mention recent contributions in this direction using mobility data [4, 9, 13, 18, 22] .

The mathematical tools that we consider are based on kinetic theory for large interacting systems [8, 14, 21] for which we are able to derive the evolution of observable quantities from microscopic, often unobservable, dynamics. In details, we will show how heterogeneity in the contact structure plays a central role in the evolution of an epidemic. In particular, preliminary results will highlight that, in several regimes of parameters, the asymptotic number of recovered can be unexpectedly high in societies with small contact heterogeneity, compared to the ones with high contact heterogeneity. These results are coherent with the recent findings presented in [5] .

In more details, the contribution is organized as follows: in Section 2 we introduce a kinetic compartmental model of interest and we briefly discuss the contact formation dynamics. A macroscopic system of equations is then derived for the coupled evolution of mass fractions and the mean number of connections. In Section 3 we present some results highlighting the impact of contact heterogeneity in the evolution of the disease. Some conclusions and research are then reported in the last Section.

In this section, we introduce a kinetic model to describe the spreading of an infectious disease depending on an additional variable describing the number of social contacts. Coherently with the modeling approach introduced in the recent works [10] , we subdivide the total population into three main compartments: susceptible, who can contract the disease, infected infectious, who can transmit the disease and recovered, corresponding to formerly infected patients that are not infectious. Furthermore, to mimic the early effects of the epidemic, where the collective compliance to reduce the number of daily contacts is often not accepted, we subdivide the susceptible population into two main categories + , − in relation to their average number of contacts, + and − respectively.

The contact distribution of the whole population is therefore recovered as

Hence, we obtain the mass fractions of population in each compartment and their momentum of order > 0 as

Unambiguously, in the following we will indicate the mean of contact in the compartment , corresponding to = 1, by .

Coherently with [11] , we can define a process of contact formation based on microscopic transitions for the variation of contacts of a single agent. At aggregate level, the evolution of the distributions , ∈ { ± , , }, can be obtained through a Boltzmann-type equation. As shown in the aforementioned work, to obtain an explicit formulation of the large time distribution, it is possible to derive the following Fokker-Planck-type equation

with > 0, 2 > 0 and > 0 the mean number of contacts. The emerging large time contact distributions ∞ ( ), ∈ { ± , , } of (1) can be explicitly computed and are of Gamma-type [23] coherently with experimental results in [2] . In particular, if = / 2 > 0, we have

whose momenta of order are

. We remark that (2) is explicitly dependent on the positive parameter = / 2 that measures the contact heterogeneity of a population in terms of the variance of the distribution of social contacts. More precisely, small values of correspond to a larger heterogeneity of the individuals in terms of social contacts.

The resulting system of kinetic equations is given by

where > 0 and > 0 is the recovery rate. The infection transmission is taken into account by the operator

with ( , * ) > 0 expressing the dependency of the disease transmission by the number of contacts and such that (0, ) = ( , 0) = 0. In [10] it has been proposed as possible example

The operators ( ), ∈ { ± , , } characterize the thermalization of the distributions ( , ) and as discussed in Section 2.1 are given by

that are mass and momentum preserving. From now on we will omit time dependency. Integrating both sides of (4) we get

that is not closed like classical compartmental modeling since it depends on the evolution of local mean values ( ). A possible way to obtain a closed system of equations is obtained by resorting to a limit procedure that is classical in statistical physics. Indeed, for ≪ 1 the distribution functions ( , ) collapse to Gamma-type densities with mass fractions ( ) and local mean values ( ). After multiplication by we get

Hence, in view of (3) we obtain the following closed system for the evolution of the mass fractions and mean connections in each compartment

and

We observe that the obtained social SIR model with generalized interaction forces reduces to the one obtained in [10] in the case of a unique susceptible population with the choice 1 = 2 = 1.

Fixing ( ) =˜ > 0 from the first two equations in (6) we get

2 , ˜ 2 ( ) complemented with the initial condition ± (0). The exact solution of the above equation reads

Hence, with the introduced assumption we get the following set of first order macroscopic equations with saturated incidence rate

where ± ( , ( )) =˜ 2 1

0 ( ) is a generalization of the classical saturated incidence rate.

To understand the influence of contact heterogeneity we divide the equations for ± ( ) in (7) by / . Hence, in the limit → +∞ we have

since ∞ ± ( ∞ = 0) = ± (0). For large times we have ∞ + + ∞ − + ∞ = 1. Taking advantage of the explicit solution (9) we can rewrite the last relation as

whose solution defines the dependence of ∞ by the introduced contact heterogeneity.

In this section, we present several numerical experiments for the system (5)-(6) and the system (7) with saturated incidence rate. In particular, we focus on the relation between the fraction of the recovered at the equilibrium ∞ and the coefficients ± measuring the heterogeneity of the population of the compartments ± in terms of the variance of the contact distribution. More specifically, small values of ± correspond to a larger heterogeneity of the individuals with respect to the social contact, since ± = ± / 2 ± . Since ∞ defined in (10) depends on several parameters defining the initial set-up of the contact distribution, to understand the influence of the contact heterogeneity, we fix the following values Therefore, with these choices ∞ is function of + and − with a parametric dependence on , that is linked to the reproduction number

The relation between the contact structure of the agents and the spreading of the epidemics has been recently studied in literature [5, 12] . In particular, regarding the COVID-19 pandemic, it has been pointed out that a smaller heterogeneity could be associated to a larger value of the recovered at the equilibrium. Similarly to [5] , we consider first the case R 0 = 2.5. This choice is then compared with the case of an infectious disease characterized by R 0 = 1.25. In details, we solve numerically equation (10) for ∞ , varying ± and taking different values of R 0 . As we can observe from Figure 1 , ∞ is an increasing function of both the coefficients ± regardless of the considered values of the reproduction number R 0 . A rather different behavior can be observed for the non-saturated system (5)-(6). In particular, ∞ exhibits in this case a maximum for small ± . This means that a higher heterogeneity is linked to a larger value of the recovered at the equilibrium. As a consequence, we note also that different conditions of the heterogeneity of the social contacts could be associated to the same ∞ , despite they have a distinct time evolution.

As we can easily observe from the left panel of Figure 2 , ∞ in the R 0 = 1.25 scenario has a maximum for high conditions of heterogeneity and then it decreases as the parameters ± increase. On the contrary, the R 0 = 2.5 case in the right panel of Figure 2 shows the same trend of the system with saturated incidence rate. In more details, in Figure 3 we show the time evolution of the system (5)-(6) for R 0 = 2.5. We clearly see that a decreasing contact heterogeneity is associated to bigger fraction of recovered for large times. In the end, we observe that for high values of heterogeneity parameters the model with saturated incidence rate, mimicking non-pharmaceutical protection measures such as a lockdown strategy, exhibits a lower fraction of recovered at the equilibrium than the system (5)- (6) . In particular, for small R 0 , a fixed mean of contacts in the infected compartment is able to avoid the maximum for small ± shown in the left panel of Figure 2 .

In this short note, we focused our attention on a kinetic compartmental model describing the spread of an infectious disease. The process of contact formation is coupled with the epidemic dynamics. We show that the presence of contact heterogeneity is central for the assessment of the evolution of a disease. The interplay between the process leading to the formation of social contacts and Maxwellian models with multiple interactions studied in [3] is currently under deeper investigation.

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