Pmsm Vector Control Matlab

A Human-Vector Susceptible–Infected–Susceptible Model for Analyzing and Controlling the Spread of Vector-Borne Diseases Lorenzo Zino, Alessandro Casu, and Alessandro Rizzo Abstract— We propose an epidemic model for the spread of vector-borne diseases. The model, which is built extending the classical susceptible–infected–susceptible model, accounts for two populations —humans and vectors— and for cross- contagion between the two species, whereby humans become in- fected upon interaction with carrier vectors, and vectors become carriers after interaction with infected humans. We formulate the model as a system of ordinary differential equations and leverage monotone systems theory to rigorously characterize the epidemic dynamics. Specifically, we characterize the global asymptotic behavior of the disease, determining conditions for quick eradication of the disease (i.e., for which all trajectories converge to a disease-free equilibrium), or convergence to a (unique) endemic equilibrium. Then, we incorporate two control actions: namely, vector control and incentives to adopt protection measures. Using the derived mathematical tools, we assess the impact of these two control actions and determine the optimal control policy.

I. Introduction

In the last decade, mathematical models of epidemic diseases have gained traction within the systems and control

Community –. In Fact, The Development Of Increas-

ingly refined models has allowed to accurately predict the course of an epidemic outbreak and, ultimately, to design and assess intervention policies , , . In particular, the latest epidemiological threats, such as the outbreaks of Ebola, COVID-19, and seasonal flu have provided further motivation to pursue these studies, yielding tailored versions of these general epidemic models –.

Typical modeling setups deal with human-to-human conta- gion mechanisms –. However, according to the World Health Organization, more than 17% of all infectious dis- eases are vector-borne . This means that they are not transmitted through human-to-human interactions, but by arthropod vectors (such as mosquitoes, fleas, or ticks) that can carry pathogens and transmit them to humans .

Vector-borne diseases (including dengue, malaria, and West Nile fever) pose a significant threat to our society, being causing more than 700,000 deaths annually . Moreover, the ongoing climate change crisis exacerbates concerns on the prevention of these diseases, as vectors adapt to new habi- tats . This is the case, e.g., of Aedes aegypti (responsible for the transmission of several diseases, including dengue, Zika, and yellow fever), which is predicted to infest many regions of Europe if the temperature increases by 2°C .

Numerous mathematical models of vector-borne diseases have been proposed and studied, particularly in response to the increasing concern for dengue fever, leading to a rich body of research , . However, most of these models, developed by computational epidemiologists as complex simulation tools, offer limited analytical tractability .

Conversely, there is a scarcity of parsimonious models that efficiently balance accuracy and interpretability. Here, we fill in this gap by proposing a novel mathematical model for vector-borne diseases. Our model, grounded in dynamical systems theory, considers two interacting popula- tions of humans and vectors. Through such interactions, the pathogen is transmitted from carrier vectors to susceptible humans and from infectious humans to vectors, establishing a positive feedback loop of contagion. Formally, we cast our model as a system of nonlinear ordinary differential equa- tions (ODEs), in which we couple i) an epidemic model for humans, inspired by the Susceptible–Infected–Susceptible (SIS) model , ii) a contagion model for vectors, inspired by the Susceptible–Infected (SI) model , and iii) a vital dynamics for vectors, which is modeled using a birth-death process . We refer to the model obtained as the human- vector SIS (HV-SIS) epidemic model.

In addition to the formulation of the model, the main contribution of this paper is twofold. First, by leveraging

Monotone Dynamical Systems Theory , We Perform A

thorough analysis of the asymptotic behavior of the HV-SIS model, characterizing two regimes: one where the epidemic outbreak is quickly eradicated, leading to global convergence to a disease-free equilibrium; and one where the disease becomes endemic, and the system converges to a (unique) en- demic equilibrium. Second, we introduce two control actions: namely, vector control —which focuses on reducing the vector population of vectors (e.g., using pesticides) — and the use of personal protection measures against conta- gion . By studying the controlled HV-SIS model and formulating an optimization problem, we investigate the optimal control policies to prevent outbreaks of vector-borne diseases, as a function of the model parameters and the cost associated with implementing interventions.

Ii. Human-Vector Sis Epidemic Model

We consider a large population of humans that interact with a population of vectors. Similar to most epidemic models , we observe that the duration of an epidemic outbreak is typically negligible with respect to the life-span of humans. Hence, we approximate the size of the human population as constant. Moreover, being the population large,

Vector Dynamics

Vector Dynamics

Fig. 1: Schematic of the human-vector epidemic model. Solid arrows represent possible transitions of the state of humans (S and I for susceptible and infected, respectively) and vectors (N and C for non-carrier and carrier, respectively). Dashed arrows are associated with the vital dynamics of vectors. Dotted colored arrows indicate transitions that are triggered by interactions with humans or vectors with a specific state.

we approximate it as a continuum of individuals with total mass equal to 1 . On the contrary, the life-span of a vector is typically comparable with the infection propagation dynamics . Hence, we assume that the total quantity of vectors v(t) ≥0 (normalized with respect to the unit mass human population) evolves in continuous-time t ≥0 accord- ing to a classical ODE associated with a birth-death process, typically used in mathematical biological models :

(1)

where ω > 0 and µ > 0 are two constants representing the birth and death rate, respectively. Humans can be healthy and susceptible to the disease or infected with the disease. We assume that there is no natural immunity: after recovery, individuals are again susceptible to the disease. This is a good proxy for many diseases, e.g., dengue fever, for which natural immunity wanes quickly and it only protects against the virus serotype specific of the previous infection . We denote by x(t) ∈[0, 1] and s(t) ∈[0, 1] the fraction of infected individuals and susceptible individuals at time t ≥0, respectively. Since there is no immunity, it holds s(t) = 1 −x(t). Similarly, vectors can be either carriers of the pathogen or non-carriers.

We denote by y(t) ≥0 the number of non-carrier vectors and by z(t) ≥0 the quantity of carrier vectors. Being v(t) the total quantity of vectors at time t, then y(t)+z(t) = v(t).

We assume that the two populations are well-mixed, and we define a human-vector compartmental model that describes the evolution of the fraction of susceptible and infected individuals and the quantity of carriers and non- carriers in the two populations. The compartmental model, illustrated in Fig. 1, yields the following 3-dimensional

(2C)

with initial condition in the domain D := {(x, y, z) : x, y, z ≥0, x ≤1}. In the following paragraphs, we extensively discuss these equations.

In Eq. (2a), the fraction of infected individuals evolves ac- cording to two contrasting mechanisms: the negative contri- bution −γx(t) accounts for infected individuals who recover at a rate γ > 0; the positive contribution βh(1 −x(t))z(t) accounts for new infections, whose number is proportional to the quantity of susceptible humans, the quantity of carriers, and a parameter βh > 0 that captures the human contagion rate (i.e., the likelihood that the pathogen is transmitted from a carrier vector to a human through a human-vector inter- action). This equation resembles the classical SIS epidemic model , but here new contagions, instead of being propor- tional to the quantity of infected humans, are proportional to the quantity of carriers. For this reason, we shall refer to the model with dynamics in Eq. (2) and initial condition in D as the human-vector SIS model, abbreviated as HV-SIS model.

The other two equations, Eqs. (2b)–(2c), govern the dy- namics of vectors. In particular, the term βvx(t)y(t) captures new carriers and gives a positive contribution to the dynamics of carriers and a negative contribution to non-carriers. This term is proportional to the number of non-carrier vectors, infected humans, and a parameter βv ≥0 that captures the vector contagion rate (i.e., the likeliness that the pathogen is transmitted from a human to a vector through a human- vector interaction). The other two terms come from Eq. (1): new born vectors are not carriers of the pathogen (so the rate ω appears in Eq. (2b)), while the death rate is independent of the pathogen, since vectors are only carriers and not infected with the disease, leading to the terms −µy(t) and −µz(t), respectively.

Iii. Main Results On The Hv-Sis Epidemic Model

In this section, we present our main results on the analysis of the HV-SIS epidemic model. First, we prove that the equations are well-defined.

:=

{(x, y, z) : x, y, z ≥0, x ≤1} can be split into two domains

Μ}, Which Are

positive invariant under Eq. (2). Proof. The domain D is closed and convex and the vector field in Eq. (2) is Lipschitz-continuous. Hence, Nagumo’s Theorem can be applied . We need to verify that the vector field at the boundaries of the domain does not point towards the boundary. We immediately observe that, if any of the variables is equal to 0, then the corresponding derivative is always non-negative (hence, it does not point towards the boundary). Similarly, at x = 1, we get that Eq. (2a) is

Μ, From Summing

Eq. (2b) and Eq. (2c) we get ˙y + ˙z = 0. Hence, the vector field does not point towards any boundary, yielding the first claim. The second claim follows the same arguments, where we observe that it always holds that ˙y + ˙z ≤0 in D2.

Then, we provide a complete characterization of the asymptotic behavior of the HV-SIS model, determining its equilibria. Specifically, we will prove that, depending on the model parameters, there is always one equilibrium that is (almost) globally asymptotically stable, characterizing two distinct regimes: either the disease is eradicated and all trajectories converge to a disease-free equilibrium (DFE), or the disease becomes endemic and (almost) all trajectories converge to an endemic equilibrium (EE), where a fraction of the population is infected (and a fraction of the vectors are carriers). The phase transition between these two regimes, which is a typical phenomenon of many epidemic models , , is shaped by the value of the model parameters that determine the so-called epidemic threshold . We start our analysis by determining the equilibria of Eq. (2) and determining their (local) stability.

Proposition 1. The HV-SIS model in Eq. (2) has, at most



.

(4)

Specifically, let us define the epidemic threshold

Σ0 := Βhβvω

γµ2 .

(5)

The DFE in Eq. (3) always exists and is locally exponentially stable if σ0 < 1 and unstable if σ0 > 1. The EE in Eq. (4) exists and is distinct from the DFE if and only if σ0 > 1 and (if it exists) it is always locally exponentially stable.

Proof. First, we compute the equilibria of Eq. (2) by equat- ing the right hand sides to 0, obtaining a system of three nonlinear equations, which yields the two solutions in Eq. (3) and Eq. (4). Then, we observe that the DFE is always in the domain D. On the contrary, the EE is in the domain D if and only if the numerators of ¯x and ¯z are non-negative, i.e.,

Μ2Γ

≥1.

= 1, The Dfe And The

EE coincide, yielding the strict inequality for the existence of a second equilibrium of Eq. (2). At this stage, we compute the Jacobian matrix of Eq. (2)



.

(6)

By evaluating Eq. (6) at the DFE in Eq. (3), we get

P

γ2µ2 −2γµ3 + µ4 + 4βhβvωµ), which is negative if and only if γ2µ2 −2γµ3 + µ4 + 4βhβvωµ < (γµ + µ2)2, which

< 1. Hence The

DFE is locally exponentially stable if σ0 < 1 and unstable if σ0 > 1. Similarly, we evaluate the Jacobian matrix in Eq. (6) at

(8)

whose eigenvalues are λ1 = −µ, which is always negative, and another pair of eigenvalues, which are not reported due to their cumbersome expression. Again, by imposing that the largest of the two has negative real part, we obtain a complicated condition which can be simplified to σ0 > 1, where computations are omitted due to space constraints.

Remark 1. From the expression of the epidemic threshold in Eq. (5), we observe that, as predictable, increasing the infection rates βh and βv favors the spread of the disease.

A similar effect is observed by increasing the vector birth rate ω. On the other hand, increasing the vector death rate µ and/or the human recovery rate γ favors the eradication of the disease. Interestingly, the vector death rate µ has a larger impact, since it appears squared at the denominator, suggesting that vector control is a potentially effective strat- egy to avoid outbreaks of vector-borne diseases.

Proposition 1 characterizes the local behavior of Eq. (2) about the two equilibria of the system. In order to prove global convergence, we now leverage monotone systems theory . However, since the Jacobian of Eq. (2) in Eq. (6) is evidently not a Metzler matrix, we cannot directly apply the monotone systems theory to Eq. (2), and we need to introduce a change of variables, as detailed in the proof of the following result.

Theorem 1. Let σ0 be the epidemic threshold from Eq. (5). If σ0 ≤1, then all trajectories of the HV-SIS model in Eq. (2) converge to the DFE in Eq. (3). If σ0 > 1, then all trajectories with initial condition such that x(0)̸ = 0 or z(0)̸ = 0 converge to the EE in Eq. (4).

Proof. We operate a change of variables, where we introduce an auxiliary 3-dimensional system formed by x(t), z(t), and v(t) = y(t) + z(t), which is governed by Eq. (1). Being

(9B)

˙v(t) = ω −µv(t).

(9C)

First, from Lemma 1, we derive that the two invariant sets, written in terms of the new variables are D1 := {(x, z, v) :

Μ, Z ≤V}. Second, We Prove

that all trajectories in D2 are bounded (those in D1 are necessarily bounded, being D1 compact). From Eq. (9c),

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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