In this paper, we study the output tracking control problem based on the event-triggered mechanism for cascade switched nonlinear systems. Firstly, an integral controller based on event-triggered conditions is designed, and the output tracking error of the closed-loop system can converge to a bounded region under the switching signal satisfying the average dwell time. Secondly, it is proved that the proposed minimum inter-event interval always has a positive lower bound and the Zeno behavior is successfully avoided during the sampling process. Finally, the numerical simulation is given to verify the feasibility of the proposed method.
Citation: Xiaoxiao Dong, Huan Qiao, Quanmin Zhu, Yufeng Yao. Event-triggered tracking control for switched nonlinear systems[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 14046-14060. doi: 10.3934/mbe.2023627
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In this paper, we study the output tracking control problem based on the event-triggered mechanism for cascade switched nonlinear systems. Firstly, an integral controller based on event-triggered conditions is designed, and the output tracking error of the closed-loop system can converge to a bounded region under the switching signal satisfying the average dwell time. Secondly, it is proved that the proposed minimum inter-event interval always has a positive lower bound and the Zeno behavior is successfully avoided during the sampling process. Finally, the numerical simulation is given to verify the feasibility of the proposed method.
Consider the mixing of two populations of hosts epidemiologically different with respect to the infection and transmission of a pathogen. What would be the outbreak outcome (e.g., in terms of attack rate) for each host population as a result of mixing in comparison to the situation with zero mixing? To address this question one would need to define what is meant by epidemiologically different and how mixing takes place.
To proceed, let's consider situations where mixing of epidemiologically different populations of hosts occurs. Such situations involve generalist (as opposed to specialist) pathogens capable of infecting multiple hosts and of being transmitted by multiple hosts [33]. Many of such pathogens cause zoonoses such as influenza, sleeping sickness, rabies, Lyme or West Nile, to cite a few [33]. In this paper, we focus on a specific example of a multi-host pathogen, the highly pathogenic avian influenza virus (HPAI) H5N1 -a virus considered as a potential pandemic threat by the scientific community.
The avian influenza virus can infect many hosts: wildfowl and domestic bird species, with occasional spill-over to mammals (including humans); the severity degree of the disease being species dependent: highly lethal (swans, chicken), few deaths (Common Pochards, humans), and asymptomatic (Mallards). Following the re-emergence of the highly pathogenic strain of H5N1 in China 2005 [6,7,28], a series of outbreaks spread throughout Western Europe, including France in 2006 [13,16,20]. The ensuing epizootics showed a need for adapted surveillance programs and a better understanding of the epidemiology of HPAI H5N1 [18]. In this context, this study is part of the French national project for assessing the risk of exposure of domestic birds and poultry farms to avian influenza viruses following introduction by wild birds; although human activities and commercial exchanges are also main sources for introduction of avian influenza [15,17,27,30].
The motivation for this study stems from the 2006 HPAI H5N1 outbreak that took place in France, in the Dombes wetlands. The area is one of the two main routes used by birds migrating across France, and an important stopover, breeding and wintering site for many wild waterfowl species. The outbreak was of minor size and affected mainly wild Anatidae bird species [13,16,20]: Common Pochards (Aythya ferina) and Mute Swans (Cygnus olor). Although the environmental conditions were conducive to the spread of the virus in the Dombes' ecosystem [31,34], it was suggested that the heterogeneity in the response to H5N1 viral infection of different bird species was a possible explanation for the reduced size of the outbreak [13]. Some studies have shown that averaging together different groups of a population, can only lead to a decrease (or no change) observed in the global reproduction number, compared to when no group structure of the population is considered [1]. Ref. [2] pointed out that the variance in the mixing rate between populations can have a substantial effect one the outbreak outcome. Other studies show that for multi-host pathogens, increasing host or species diversity may lead to either reduction or enhancement of the disease risk [12,24]. Therefore, addressing the question posed in the beginning of this section would provide insights and allow advances in the understanding of how avian influenza may spread in such ecosystems.
Our aim in this paper is to use a SIR compartmental model to investigate the effect of host heterogeneity on the disease outbreak in a multi-host population system. More precisely, we study how the outbreak outcome for each constituent population of hosts is affected in a multi-host population system with mixing in comparison with the single-host situation where individual populations are not mixed. The remainder of the paper is as follows. First, the key parameters and response functions characterizing the outbreak outcome are defined and determined for a single-host system in Section 2, and next the defined parameters are used to define the epidemiological heterogeneity in Section 3. Second, Section 4 is devoted to studying how the outbreak outcome in a multi-host population system is changed, due to mixing of epidemiologically heterogeneous hosts, compared to the outbreak outcomes in a single-host situation. Finally, the paper ends with the application of the results in the context of the Dombes area and concluding remarks in Section 5.
In this section we define the key characteristic parameters of the interacting population-pathogen system and the response function characterizing the outbreak outcome for such a system. To this end, consider a single species or single-host system in which the dynamics of an infection induced by a pathogen can be described within the framework of the compartmental susceptible-infected-recovered (SIR) model ([25]) in which susceptible individuals,
At any time
{dSdt=−λS[2ex]dIdt=λS−αI[2ex]dRdt=xαI[2ex] | (1) |
where
In writing Eq.(1) we have used the homogeneously mixing hypothesis and considered that the transmission of infection is frequency-dependent (i.e. the force infection is proportional to the inverse of the population size) like for the true mass-action kinetics [8]. For
The above SIR model is characterized by two (non independent) quantities: the generation time
1The derivation in Ref. [3] goes as follow. Consider a single infected individual applying a constant force of infection
R0=βN0β+αN0, | (2) |
where
To define a response function characterizing the outbreak outcome of the SIR model, we consider the following two indicators:
• the reduced persistence or extinction time,
• the attack rate,
To investigate
Bearing the distributions of
When
On the other hand, consider the probability
Thus, it follows from what precedes, that the mean attack rate
A=F(R0,g,x);R0=F−1[A(g,x)], | (3) |
where
Within the epidemiological framework as described in the Section 2, a host population interacting with a pathogen can be canonically characterized by two key parameters (or two dimensions): the basic reproduction number,
Hh=n∑i=1fih2i(n∑i=1fihi)2−1;hi=R0,g. | (4) |
It follows that a population of
For a single-host population,
Hh=y(z−1zy+1)2withy=f2f1andz=h2h1 | (5) |
where
Note that different demographic fractions
Now, we consider a heterogeneous system (in the sense of Section 3) constituted of
To proceed, consider
{dSidt=−λiSidIidt=λiSi−αiIidRidt=xiαiIi | (6) |
where
Assuming a hypothesis of homogeneous mixing of individuals for both within populations of hosts of the same kind (intra) and between host populations of different kind (inter), the elements of the matrix of contact probabilities can be written as,
{pii(t)=1Ni(t)[1−n∑j=1;j≠iϕijNj(t)Mi(t)]pij(t)=ϕijMi(t);Mi(t)=n∑j=1[1−δϕij,0]Nj(t) | (7) |
where
For the transmission of avian influenza viruses of interest here, we assume that infectious individuals of any kind are efficient sources of virus excretion such that the transmission of the infection to uninfected individuals only depends on the infection susceptibility of the receiver. That is to say that the infection transmission rate
λi(t)=[fiN0R0,ifiN0−R0,i]αi∑j=1pij(t)Ij(t)withR0,i=βifiN0βi+αifiN0, | (8) |
where
To go further and for the sake of simplicity, we specialize to the case of
For the mixing between
General considerations on the outbreak outcome can be drawn from the
{K1,1=(R0,1f1N0f1N0−R0,1)[1−ϕf2];K1,2=(R0,1f1N0f1N0−R0,1)(α1α2)ϕf1K2,1=(R0,2f2N0f2N0−R0,2)(α2α1)ϕf2;K2,2=(R0,2f2N0f2N0−R0,2)(1−ϕf1) | (9) |
In this approach,
R0=12[K2,2+K1,1+√(K2,2−K1,1)2+4(K2,1K1,2)]. | (10) |
Because of the term
• For a fixed nonzero heterogeneity
Rm=(f1N0f1N0−R0,1)R0,1f1+(f2N0f2N0−R0,2)R0,2f2. | (11) |
The decreasing of
• For a fixed nonzero mixing
- for any fixed ratio of reproductive numbers
- for fixed demography
The
To investigate the effects of mixing on individual outbreak outcomes at the level of each subsystem, we have run SIR stochastic simulations in a two-host system (see Appendix A) with a total population of size,
Figure 8 illustrates the cumulative distribution (cdf) of the attack rates for each host in the system and for the whole system. The cdf of the whole system is broad and close to that of the most abundant population host
Because of mixing, the mean attack rate
ηi=F−1i(Ai)F−1i(A0,i)=Reqv,iR0,i, | (12) |
where we have used the relation in Eq.(3) (see Section 2) to define the equivalent basic reproduction number as,
Several combinations of
heterogeneity | outbreak response | |
host 1 | host 2 | |
dilution | dilution | |
amplification | dilution | |
no effect | no effect | |
dilution | dilution | |
no effect | amplification |
• three kinds of behaviors for each host population are possible depending on the mixing and heterogeneity parameters: dilution, no effect or amplification behaviors. As shown in Table 1, the interaction between two heterogenous hosts, with at least a
• the extent to which a subsystem undergoes dilution or amplification is a function of demographic and mixing parameters with a possible transition from dilution via no effect to the amplification behaviors (and vice versa), when varying the individual
• as the proportion of recovered
Figures 9 and 10 illustrate some of the situations presented in Table 1. Figure 9 shows the coexistence of two-phase behaviors (dilution effect for a subpopulation and amplification effect for the other one), where the
The aims of this work were to define the epidemiological host heterogeneity and investigate the effect of host heterogeneity on the disease outbreak outcomes for each host in a multi-host population system, given prior knowledge of the disease epidemiology for each host population in the zero mixing situation. In other words, what is the impact of a multi-host system on the outbreak response of individual host populations involved?
We have shown that a single-host system can be canonically parametrized using two quantities, the basic reproductive number
• Heterogeneity index
• Interaction matrix: which takes into account both epidemic and demographic characteristics to structure how different hosts interact with each other. By interactions we mean that hosts have an epidemic and a demographic role in the transmission and spreading of the infection. For the two-host case presented in this analysis, the control parameter for the interaction matrix reduces to a single assortative mixing index
As minimal definition and necessary conditions, we state that the epidemiological host heterogeneity occurs in a system of epidemiologically interacting populations where each host population is characterized by a different epidemic response function. There is no host heterogeneity in the absence of interactions between populations or when interacting populations have all identical epidemic response functions.
Regarding the impacts of host heterogeneity on the outbreak outcomes, we found that they are twofold in the case of the infection transmission depending on the receiver infection susceptibility: i) -outbreak dampening, i.e., the outbreak in the heterogeneous multi-host system is always smaller than the summation of outbreaks for individual subsystems taken separately, and ii) -as summarized in Table 1, three kinds of outbreak outcomes are possible for the individual subsystem depending on the mixing and heterogeneity parameters: dilution, no effect or amplification behaviors where the outbreak responses in the multi-host system are lower, similar or higher than in the single host system, respectively, with the magnitude depending both on
Previous works, [14], have shown that, in the case of preferential mixing, like in this study (though with a different mixing pattern), the disease can invade the population when any subgroup is self-sufficient for the disease transmission (i.e.,
The previous works were largely focused on the impacts that heterogeneity may have on the global
The situation of the HPAI H5N1 outbreak in mid-February 2006 in the Dombes, France, can be analyzed within the framework of the afore outlined approach. As mentioned in the Introduction section, although the environmental conditions were conducive to the spread of the virus in the Dombes' ecosystem [31,34], the outbreak was of minor size, mainly affecting Common Pochards (Aythya ferina) and Mute Swans (Cygnus olor) [13,16,20]. It was suggested that the host heterogeneity in the response to H5N1 viral infection of different bird species was a possible explanation for the reduced size of the outbreak [13].
During the outbreak period, the situation in the Dombes was that Swans, Common Pochards and Mallards were found well mixed with a census of
To conclude, we have depicted a framework for defining the epidemiological host heterogeneity and assessing its impacts on outbreak outcomes in terms of epidemic response functions for host populations in interaction. The approach was illustrated for the case of frequency-dependent direct transmission where the infection transmission depends on the receiver infection susceptibility, (i.e.,
Stochastic simulations for the SIR model were generated using the stochastic discrete time version of the system of equations in Eq.(6), in which
{(Si,Ii,Ri)→(Si−1,Ii+1,Ri)at rate λi(t)Si[2ex](Si,Ii,Ri)→(Si,Ii−1,Ri+1)at rate αiIi with probability xi[2ex](Si,Ii,Ri)→(Si,Ii−1,Ri)at rate αiIi with probability 1−xi | (13) |
describing the transition from susceptible to infected following a Poisson process of parameter
• Single-host system: The subscript
λ(t)=pβI=[N0R0N0−R0]α×I(t)N(t), | (14) |
where
• Two-hosts system:
λi(t)=[fiN0R0,ifiN0−R0,i]αi∑j=1pij(t)Ij(t), | (15) |
where
When all infected individuals recover from infection, i.e.,
A=1−exp{−(R0N0−R0)[I(0)+AS(0)]}. | (16) |
For
A=(I(0)N0)×u+1×(1−u), | (17) |
where
u=tanh(c×e−bR0) | (18) |
where the constants
R0=F−1(A)=−1bln{−12cln[I(0)−AN0I(0)−(2−A)N0]}. | (19) |
AM is a PhD student supported by a grant from the Ministry of Education and Research of France through the Ecole Doctorale Ingénierie pour la Santé, la Cognition et l'Environnement (EDISCE) of Grenoble Alpes University.We are grateful to M. Artois for fruitful discussions. This work has benefited from the support of the Ministry of Agriculture and fisheries under the Project Cas DAR 7074.
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heterogeneity | outbreak response | |
host 1 | host 2 | |
dilution | dilution | |
amplification | dilution | |
no effect | no effect | |
dilution | dilution | |
no effect | amplification |