Citation: Carsten Sachse. Single-particle based helical reconstruction—how to make the most of real and Fourier space[J]. AIMS Biophysics, 2015, 2(2): 219-244. doi: 10.3934/biophy.2015.2.219
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Emerging and reemerging infectious disease, for instance, the 2003 severe acute respiratory syndrome (SARS) [27,19], the 2009 H1N1 influenza epidemic [22,38,44], Ebola virus disease in west Africa [42,23] and Zika virus [2] have become major causes of mortality and morbidity in emergencies, where collapsing health service and disease control programmes, poor access to health care, interrupted supplies and logistics are often the cruel reality. Global public health systems of surveillance and response are used to fighting to curb infectious diseases by controlling it at source [30] or slowing down its development course [20,5]. An efficient way to containing a disease during an outbreak is to provide as much as possible medical resources, including sufficient medicine, health care workers, hospital beds and etc. To explore the impact of limited medical resources on the control of infectious diseases and evaluate the possible control strategies, most of the available compartmental models either assume a constant treatment rate or adopt medical resources dependent continuous functions to represent the treatment rates [41,45]. However, change of the level of available medical resources were not well modeled in these models since usually the allocation and change of medical resources do not occur in a continuous way.
As WHO statistic information system referred, in-patient beds density, as one of the available indicators, can be adopted to access the level of health service delivery [43]. Hence, the number of available hospital beds per 10000 population, termed as hospital bed-population ratio (HBPR), is always used by the public health as an approach of capturing availability of health services delivery. The variations of the level of infected cases between communities and hospitals usually make it difficult for the public health to evaluate and decide the number of hospital beds needed to control the epidemic [1]. Setting sufficient beds to cover peak demands usually leads to quite a lot of beds idle at other times while reducing hospital beds to improving average bed occupancy level results in frequent overflows and congestions within hospital. Hence, quantifying the impact of the number of hospital beds for epidemics control remains a challenge and is therefore an important issue for public health.
Recently, there have been modeling studies dedicated to exploring the impact of the number of hospital beds on the future containing of disease [35,24,33,43,34]. Some have modeled the impact by embedding a continuous function in terms of the number of hospital beds into the recovery rate [35,34]. In fact, the medical resources used to contain the disease are always limited and the policy to intervene the disease varies depending on the number of infected individuals [32]. In particular, treatment rate depends not only on the available resources of health system to the public but also on the total infected individuals seeking treatment. On one hand, the capacity of the hospital settings, especially the number of health workforce and the facilities of the hospital are vital factors for efficient treatment of infection. On the other hand, the proportion of infected individuals getting access of the medical establishment to cure may change. Therefore, a considerable issue is to model and study the impact of these factors on the control of epidemic disease.
In this paper, we choose the number of infected cases as an index to propose a threshold policy [37,44,31,28] (of course, we could adopt the size of population who are exposed to the virus or the total number of susceptible and infected individuals). In this case, the control strategy is formulated by using a piecewise smooth function which depends on the density of infected individuals. The purpose of this paper is then to formulate a novel mathematical model subject to the threshold policy to study a Filippov system [21,39,8,4]. In particular, when the number of infected individuals is below the threshold level, the per capita treatment rate will be a constant, representing the maximum per capita treatment rate; while above the level, a weaker treatment policy will be adopted by incorporating the impact of the capacity and limited resources of the health care system in terms of the HBPR and depends on the number of the infected individuals. Filippov systems have been widely investigated [7,11,3,6], especially the sliding mode bifurcations in generic Filippov systems [17,26,18,29] and the numerical methods for them [15,13]. This type of systems have gained considerable attention in recent years since they not only behave differently from smooth systems but also found very important applications in many fields, such as pest control [37], power control in circuit [9], control of ecological systems [14], evolutionary biology [12,10], forest fire [16] and mechanical systems with friction [25], and a few work found in the disease controlling[32,40]. In [32], two parameters are adopted to model the treatment proportion for selective treatment measure due to the limited medical resources while we will model the hospital bed-population ratio by a parameter in this work.
The main purpose of this paper is to explore the impact of hospital bed-population ratio (i.e. HBPR) on disease control using a Filippov system. Some natural questions to ask include how does the level of infected cases combined with the HBPR affect the existence of the sliding modes, pseudo-equilibrium and sliding bifurcation? Further, how does the threshold policy affect the course of disease transmission and control outcome? To address these questions, we will focus on the sliding mode dynamics as well as the rich sliding bifurcations with variation of the threshold level. The rest of this paper is organized as follows. In Section 2, a piecewise defined treatment programme, which is termed as switching policy in control literature, is proposed to incorporate the level of infected cases and the impact of HBPR into the epidemic model. The sliding mode as well as all possible critical points are examined in Section 3. In Section 4, we study the sliding bifurcation including local sliding bifurcation and global sliding bifurcation. In Section 5, we investigate the impact of some key parameters on the evolution of the epidemic disease. Some concluding remarks and biological interpretations will be presented in the last section.
As mentioned in the introduction, the medical resources can not always meet the increasing treatment demand of the infecteds. As WHO referred, we adopt the hospital bed-population ratio (i.e. HBPR and denoted by
We divide the population into two classes: susceptible (
$
\left\{
dSdt=Λ−μS−βSIS+I+H(I,ϵ)I,dIdt=βSIS+I−μI−νI−H(I,ϵ)I
\right.
$
|
(1) |
with
$ H(I, \epsilon) = \epsilon H_1(b, I)+(1-\epsilon)h_1 $ | (2) |
and
$
\epsilon = \left\{
0,σ(S,I,Ic)<0,1,σ(S,I,Ic)>0,
\right.
$
|
(3) |
where
In particular, the function
$
H(I, \epsilon) = \left\{ h1,I<Ic,h0+(h1−h0)bb+I,I>Ic. \right.
$
|
Model system (1) with (2) and (3) is a piecewise smooth dynamical system (PWS)[3]. In particular, it subjects to a threshold value and is indeed a so-called Filippov system [21], i.e. systems of ordinary differential equations (ODES) with non-smooth right-hand sides.
Denote
$X1(S,I)=(Λ−μS−βSIS+I+h1I,βSIS+I−μI−νI−h1I)T˙=(f11(S,I),f12(S,I))T,X2(S,I)=(Λ−μS−βSIS+I+[h0+(h1−h0)bb+I]I,βSIS+I−μI−νI−[h0+(h1−h0)bb+I]I)T˙=(f21(S,I),f22(S,I))T,
$
|
then system (1) with (2) and (3) can be written as the following Filippov system
$
\frac{\mathrm{d}Z}{\mathrm{d}t} = \left\{
X1(S,I),σ(I,Ic)<0X2(S,I),σ(I,Ic)>0.
\right.
$
|
(4) |
The attraction region of system (4) is
$\Sigma = \{(S, I):I = I_c, (S, I)\in R^2_+\}, $ |
which splits
$G1(S,I)={(S,I):I<Ic,(S,I)∈R2+},G2(S,I)={(S,I):I>Ic,(S,I)∈R2+}.
$
|
For convenience, system (4) restricted in region
Definition 2.1. Regular equilibrium of system (4) refers to those equilibria located in the region
A real equilibrium
$X_1(Z_*) = 0, \ \sigma(Z_*)<0$ |
or
$X_2(Z_*) = 0, \ \sigma(Z_*)>0.$ |
A virtual equilibrium
$X_1(Z_*) = 0, \ \sigma(Z_*)>0$ |
or
$X_2(Z_*) = 0, \ \sigma(Z_*)<0.$ |
Definition 2.2. A boundary equilibrium
$X_1(Z_*) = 0, \ \sigma(Z_*) = 0$ |
or
$X_2(Z_*) = 0, \ \sigma(Z_*) = 0.$ |
Definition 2.3. A tangency point
$X_i\sigma(Z) = \langle\sigma(Z), X_i(Z)\rangle.$ |
We examine the dynamics of the subsystems
Dynamics of subsystem
$J_1(S, I) = \left(
−μ−βI2(S+I)2−βS2(S+I)2+h1βI2(S+I)2βS2(S+I)2−r1
\right).$
|
If we denote
$\delta_1 = (\mbox{tr}(J_1(S_1, I_1)))^2-4\mbox{det}(J_1(S_1, I_1))$ |
and the dynamics of system
Theorem 2.4. There is a disease-free equilibrium
$S_1 = \frac{r_1I_1}{\beta-r_1}, \ I_1 = \frac{(\beta-r_1)\Lambda}{\mu r_1+(\mu+\nu)(\beta-r_1)}, $ |
which is a stable node for
Dynamics of subsystem
$a_0 = \beta\mu+\nu(\beta-r_0), a_1 = b\beta(\mu+\nu)+\Lambda(r_0-\beta)-b\nu r_1, \ a_2 = b\Lambda(r_1-\beta)$ |
and
$C_0 = [\beta b(\mu+\nu)-\Lambda(r_0-\beta)-b\nu r_1]^2+4\beta\mu b\Lambda(h_0-h_1).$ |
One can verify that there are two possible endemic equilibria of system
$S_i = \frac{\Lambda-(\mu+\nu)I_i}{\mu}, \ I_2 = \frac{-a_1+\sqrt{C_0}}{2a_0}, \ I_3 = \frac{-a_1-\sqrt{C_0}}{2a_0}.$ |
Only one endemic equilibrium
$S∗=Λ−(μ+ν)I∗μ,I∗=−a12a0,S4=Λ−(μ+ν)I4μ,I4=bΛ(β−r1)bν(r0−r1)+Λ(r0−β).
$
|
We address the existence of equilibria for system
Theorem 2.5. The disease-free equilibrium
Range of parameter values | Existence of endemic equilibria | |
Nonexistence | ||
Nonexistence | ||
Nonexistence | ||
Nonexistence |
The disease-free equilibrium
$J_2(S, I) = \left(
−μ−βI2(S+I)2−βS2(S+I)2+(h1−h0)b2(b+I)2+h0βI2(S+I)2βS2(S+I)2−(h1−h0)b2(b+I)2−r0
\right).$
|
For any endemic equilibrium
$tr(J2(¯S,¯I))=(−β+r0−μ)¯I2+2b(r1−β−μ)¯I+(−μ−β+r1)b2(b+¯I)2,det(J2(¯S,¯I))=μ{[(β−r0)Λ+νb(r1−r0)]¯I2+2Λb(β−r1)¯I+Λb2(β−r1)}(Λ−ν¯I)(b+¯I)2. $
|
One can get that the unique endemic equilibrium
For the coexistence of
$\mbox{sgn}\Big(\mbox{det}\big(J_2(\overline{S}, \overline{I})\big)\Big) = \mbox{sgn} \Big(\mathcal{A}\overline{I}^2+ 2\Lambda b(\beta-r_1)\overline{I}+\Lambda(\beta-r_1)b^2\Big)$ |
with
$\mathcal{A} = \big((\beta-r_0)\Lambda+\nu b(\beta-r_0)-\nu b(\beta-r_1)\big)>0, $ |
then
$\mbox{det}\big(J_2(S_2, I_2)\big)>0, \ \mbox{det}\big(J_2(S_3, I_3)\big)<0. $ |
This indicates that
$\mbox{tr}\big(J_2(S_2, I_2)\big) =\frac{(\nu+h_0-\beta)I_2^2+2b(\nu+h_1-\beta)I_2+(\nu+h_1-\beta)b^2} {(b+I_2)^2}.$ |
It follows that
$sgn(tr(J2(S2,I2)))=sgn((ν+h0−β)I22+2b(ν+h1−β)I2+(ν+h1−β)b2)=sgn{[a1(2a0bp1−a1p0)+2a0(a2p0−a0b2p1)]−(2a0bp1−a1p0)√C0},
$
|
where
$\mathcal{B} = a_1\big(2a_0bp_1-a_1p_0\big)+2a_0\big(a_2p_0-a_0b^2p_1\big)$ |
and
$\delta_2 = \Big[\mbox{tr}(J_2(S_2, I_2))\Big]^2-4\mbox{det}(J_2(S_2, I_2)).$ |
Then one can verify that
(ⅰ)
(ⅱ)
For the existence of endemic equilibrium
$det(J2(S4,I4))=μ(β−r1)C(bν+Λ)2(β−r0)2(h1−h0)>0,tr(J2(S4,I4))=(β−r0−μ)−[bν(h0−h1)+Λ(r0−β)]2(h1−h0)(Λ+bν)2<0,
$
|
where
$\mathcal{C} = -\mu\nu\beta b^2(h_1-h_0)^2+2b\mu\beta\Lambda(r_0-\beta)(h_1-h_0)-\Lambda^2(r_0-\beta)^3, $ |
we conclude that
Theorem 2.6. (i) If
(ⅱ)If there are two endemic equilibria
(ⅲ) If only one endemic equilibrium
By implementing similar discussion as in [35], we obtain the rich dynamics of system
Theorem 2.7. (i) If there are two endemic equilibria
$\nu+h_0<\beta<\nu+h_1, \displaystyle\frac{p_1}{p_0}>\frac{a_1}{2a_0b}, (2a_0bp_1-a_1p_0)^2C_0 = \mathcal{B}^2.$ |
(ⅱ) Bogdanov-Takens bifurcation occurs if
$\nu+h_0<\beta<\nu+h_1, \displaystyle\frac{p_1}{p_0}>\frac{a_1}{2a_0b}, (2a_0bp_1-a_1p_0)^2C_0 = \mathcal{B}^2.$ |
To establish the dynamics given by the Filippov system (4), the first step is to rigorously define the local trajectory through a point
Existence of sliding mode. Indeed, no attracting sliding mode region exists for Filippov system (4) due to the mechanisms of our model. Solving
$S_{c1} = \frac{(\mu+\nu+h_1)I_c}{\beta-(\mu+\nu+h_1)}, \ S_{c2} = \frac{[(b+I_c)(\mu+\nu+h_0)+b(h_1-h_0)]I_c}{(b+I_c) (\beta-(\mu+\nu+h_0))-b(h_1-h_0)}. $ |
The repulsing sliding mode region takes the form
$\Sigma_e = \{(S, I_c):S_{c2}\leq S\leq S_{c1}\}\dot{ = }\Sigma_{e1}$ |
for
$\Sigma_e = \{(S, I_c):S\geq S_{c2}\}\dot{ = }\Sigma_{e2}$ |
for
$ \frac{\mathrm{d}S}{\mathrm{d}t} = -\mu\bigg[ S-\frac{\Lambda-(\mu+\nu)I_c}{\mu}\bigg]\dot{ = }F_s(S, I_c). $ | (5) |
Equation (5) defines a one dimensional dynamical system on
We will define the type of equilibrium on the repulsing sliding mode
Definition 3.1. A point
The following definition gives a more detailed characterization of a tangency point.
Definition 3.2. A point
(a)
(b)
Furthermore, if
For case (a), we call a fold tangency point visible (invisible) if the trajectories of subsystem
An equilibrium
Note that
$S_s-S_{c1} = \frac{(I_1-I_c)\big[\beta\mu+\nu(\beta-r_1)\big]} {\mu(\beta-r_1)}, $ |
we have
$S_s\leq S_{c1}\Longleftrightarrow I_c\geq I_1$ |
for
$S_s-S_{c2} = \frac{\big[\beta\mu+\nu(\beta-r_0)\big]I_c^2+ \big[b\beta(\mu+\nu)-\Lambda(\beta-r_0)-b\nu r_1\big]I_c-b\Lambda(r_1-\beta)} {-\mu(\beta-r_0)(I_c-I_{c0})}$ |
that
Then we conclude that the pseudo-equilibrium
(a)
(b)
(c)
Equilibria. We address all possible critical points for system (4) as follows.
Regular Equilibrium. In terms of Definition 2.1, the disease-free equilibrium
Boundary Equilibrium. There are up to five equilibria of Filippov system (4) colliding with the switching boundary
$E_1^b = \bigg(\frac{r_1I_c}{\beta-r_1}, I_{c1}\bigg), \ E_i^b = \bigg(\frac{\Lambda-(\mu+\nu)I_c}{\mu}, I_{ci}\bigg), i = 2, 3, 4, 5$ |
if
Pseudo-equilibrium. Only a pseudo-equilibrium
(a)
(b)
(c)
Furthermore, the pseudo-equilibrium
Tangency point. By Definition 2.3, two tangency points coexist for system (4) if
$E_1^t = (S_{c1}, I_c), E_2^t = (S_{c2}, I_c), $ |
and there is one tangency point
According to Definition 3.2, fold tangency points can be classified as visible and invisible ones, so we examine the type of tangency points
For
$\left\{
sgn(X2σ(Et1))=sgn(βSc1Sc1+Ic−b(h1−h0)b+Ic−r0)=1,X21σ(Et1)=βI2c(Sc1+Ic)2f11(Sc1,Ic)+[βS2c1(Sc1+Ic)2−r1]f12(Sc1,Ic),
\right.
$
|
that
$ \mbox{sgn}\big(X_1^2\sigma(E_1^t)\big) = \mbox{sgn}\Bigg( \frac{\Lambda(\beta-r_1)-\mu r_1I_{c}-(\mu+\nu)(\beta-r_1)I_{c}}{\beta-r_1}\Bigg), $ |
so if
For the tangency point
$\left\{
sgn(X1σ(Et2))=sgn((β−r1)Sc2−r1Ic)=−1,X22σ(Et2)=βI2c(Sc2+Ic)2f21(Sc2,Ic)+[βS2c2(Sc2+Ic)2−(h1−h0)b2(b+Ic)2−(μ+ν+h0)]f22(Sc2,Ic),
\right.$
|
so
$
sgn(X22σ(Et2))=sgn{−[μβ+ν(β−r0)]I2c+[Λ(β−r0)−b(βμ+βν−r1ν)]Ic+Λb(β−r1)(β−r0)Ic+b(β−r1)}.
$
|
(6) |
There are two possibilities according to whether
$I_3 < I_c<I_2\Rightarrow X_2^2\sigma(E_2^t)>0$ |
according to (6), so
Once we have examined the sliding dynamics as well as all possible critical points for system (4), we turn to discuss the discontinuity-induced bifurcation in the following section, which is special for Filippov system and refer to a type of bifurcations involving structural changes in the sliding mode domain.
In this section, we focus our attention on sliding bifurcation, in which some sliding segment on the switching boundary is involved. According to [29,17], all sliding bifurcations can be classified as local and global bifurcations. The local sliding bifurcations include boundary equilibrium bifurcation and double tangency bifurcation. Those sliding bifurcations are called global if they involve nonvanishing cycles. Global bifurcations include grazing (touching) bifurcation, sliding homoclinic bifurcation to pseudo-saddle, or pseudo-saddle-node or saddle, and the sliding heteroclinic bifurcation between pseudo-saddles, etc. In the following, we choose the threshold value
We initially examine the local sliding bifurcation for system (4), which include boundary equilibrium bifurcation and double tangency bifurcation. There are four distinguished boundary equilibrium bifurcations, i.e. boundary node (BN), boundary focus (BF), boundary saddle (BS) and boundary saddle-node bifurcation (BSN) [29,26]. Further, if the boundary equilibrium bifurcation involves changes of the real/virtual character, it is also termed as real/virtual equilibrium bifurcation. Double tangency bifurcation is a type of bifurcations triggered by the collision of two tangency points. Since the tangency points
Definition 4.1. A boundary equilibrium bifurcation occurs at a critical point
When a boundary equilibrium bifurcation occurs, we can observe two cases. One is persistence, where a real equilibrium becomes a pseudo-equilibrium; the other is non-smooth fold, where a real equilibrium disappears after colliding together with a pseudo-equilibrium. Implementing a similar discussion as in [17], one gets the conditions to distinguish between the two possible types of unfolding of a boundary equilibrium as the parameter
Definition 4.2. For system (4), if
$
det(J1(Sic1,Ic1))≠0∂σ(Ic1,Ic)∂Ic−J3(Sic1,Ic1)J−11(Sic1,Ic1)J4(Sic1,Ic1)≠0J3(Sic1,Ic1)J−11(Sic1,Ic1)(X2(Sic1,Ic1)−X1(Sic1,Ic1))≠0,
$
|
a non-smooth fold is observed for
$J_3(S_{ic1}, I_{c1})J_1^{-1}(S_{ic1}, I_{c1})\big(X_2(S_{ic1}, I_{c1})-X_1(S_{ic1}, I_{c1})\big)<0$ |
while persistence is derived at the boundary equilibrium bifurcation point for
$J_3(S_{ic1}, I_{c1})J_1^{-1}(S_{ic1}, I_{c1})\big(X_2(S_{ic1}, I_{c1})-X_1(S_{ic1}, I_{c1})\big)>0, $ |
where
The similar condition can be presented for other boundary equilibria and we omit them here.
For the case that only one endemic equilibrium
$det(J1(Eb1))=μr1I1S1+I1+(μ+ν)βI21(S1+I1)2,det(J1(Eb2))=β(μ+ν)I22(S2+I2)2+μ[βS2I2S2+I2+(h1−h0)I2b+I2],det(J2(Eb1))=[ν(r1+β)2+βμ(β−r1)][(β−r1)(Λ+bν)+bβμ]2−Cβ[(β−r1)(Λ+bν)+bβμ]2 $
|
with
$\mathcal{C} = \beta\mu(r_1-r_0)\big[(\beta-r_1)^2(\Lambda^2+2b\nu\Lambda)+ 2b\beta\mu\Lambda(\beta-r_1)\big], $ |
so a boundary focus (or node) bifurcation occurs for
For boundary equilibrium
$
sgn(J3(Sic1,Ic1)J−11(Sic1,Ic1)(X2(Sic1,Ic1)−X1(Sic1,Ic1)))=sgn{(h1−h0)Ic1b+Ic1[h1−μ−βI2c1(Sic1+Ic1)2−βS2ic1(Sic1+Ic1)2]}=sgn{(h1−μ−β)r21+2(h1−μ)r1(β−r1)+(h1−μ−β)(β−r1)2(β−r1)2}=sgn(−2r21+β(r1+ν+2h1−β)).
$
|
Thus, a non-smooth fold occurs if
$ -2r_1^2+\beta(r_1+\nu+2h_1-\beta)>0, $ | (7) |
while persistence is derived if
$ -2r_1^2+\beta(r_1+\nu+2h_1-\beta)<0. $ | (8) |
Implementing the similar process for boundary equilibrium
$sgn(J3(Sic2,Ic2)J−12(Sic2,Ic2)(X1(Sic2,Ic2)−X2(Sic2,Ic2)))=sgn{(h1−h0)Ic2b+Ic2[−h0+μ−(h1−h0)b2(b+Ic2)2+βI2c2(Sic2+Ic2)2+βS2ic2(Sic2+Ic2)2]}=sgn{(μ−h0)−b2(h1−h0)(b+Ic2)2+β[Λ−(μ+ν)Ic2]2+βμ2I2c2(Λ−νIc2)2}=sgn{(Λ−νIc2)2[(−h0+β+μ)(b+Ic2)2−b2(h1−h0)]−2μβIc2(b+Ic2)2[Λ−(μ+ν)Ic2]},
$
|
where
$ \frac{2\beta(b+I_{c2})^2}{b^2(h_1-h_0)-(b+I_{c2})^2(\beta+\mu-h_0)} <\frac{(\Lambda-\nu I_{c2})^2}{\mu I_{c2}\big[\Lambda-(\mu+\nu)I_{c2}\big]} $ | (9) |
while a non-smooth fold appears for
$ \frac{2\beta(b+I_{c2})^2}{b^2(h_1-h_0)-(b+I_{c2})^2(\beta+\mu-h_0)} >\frac{(\Lambda-\nu I_{c2})^2}{\mu I_{c2}\big[\Lambda-(\mu+\nu)I_{c2}\big]}. $ | (10) |
If we choose the parameters as
The pseudo-saddle
Note that the endemic equilibrium
$sgn(J3(Sc3,Ic3)J−12(Sc3,Ic3)(X1(Sc3,Ic3)−X2(Sc3,Ic3)))=sgn{(Λ−νIc3)2[(h0−β−μ)(b+Ic3)2+b2(h1−h0)]+2μβIc3(b+Ic3)2[Λ−(μ+ν)Ic3]},
$
|
where
It follows from Theorem 2.2 that a unique endemic equilibrium
For other cases with the existence of endemic equilibrium
Now we turn to examine the bifurcations of limit cycles of (4). Basically, there are three types of limit cycles for Filippov system (4) [29], i.e. standard periodic cycles, sliding periodic cycles and crossing periodic cycles. Standard periodic solutions refer to the cycles lying entirely in region
Grazing bifurcation. The bifurcation that occurs once the standard piece of a periodic cycle touches the switching boundary
It is worth noting that local sliding bifurcation happens at the critical threshold value
Bifurcation of a sliding homoclinic orbit to pseudo-saddle. If a pseudo-equilibrium of system (4) is a pseudo-saddle, it can have a sliding trajectory which initiates and returns back to it at certain threshold parameter values. This is indeed the so-called sliding homoclinic orbit. If a sliding cycle collides with such a pseudo-saddle, a sliding homoclinic bifurcation occurs as shown in Fig. 4(e)-(g). It follows from Fig. 4(e) that a sliding cycle, two pieces of stable manifolds and unstable manifolds of the pseudo-saddle
It follows from Fig. 4 that as the threshold level increases from 4 to 6.8556, Filippov system (4) exhibits the interesting local and global sliding bifurcations sequentially, i.e. boundary saddle bifurcation
One of our purposes in this study is to seek better strategies for curbing the spread of diseases, and to examine when and how to implement the control measure to contain the number of infected individuals to be less than some acceptable level if it is almost impossible to eradicate an infectious disease. An efficient way is to reduce the level of infected cases in steady state as small as possible. To realize this goal, we will examine the impact of parameters associated with interventions on the dynamics, especially the equilibrium level of infected individuals of Filippov system (4) in this section. Due to the biological significance of each parameter and their effect on disease control, we focus only on the impact of threshold parameter
Impact of threshold level
It is clear that the pseudo-equilibria and sliding modes are sensitive to the threshold level
It follows from the above discussion that we can increase the threshold level
Impact of HBPR parameter
The above discussion demonstrates that only endemic equilibrium of free system exists for Filippov system (4) by increasing the HBPR (i.e.
Note that here increasing
Impact of the maximum and minimum treatment rate. In this section, we focus on the impact of maximum and minimum per capita treatment rate (i.e.
Fig. 7 is to show the variation of the number of infected individuals in steady states as the maximum per capita treatment rate
$h12=4βμbΛ−2D1−√(4βμbΛ−2D1)2−4b2ν2(D21+4βμbr0Λ)2b2ν2−(μ+ν),D1=bν[Λ(r0−β)−βb(μ+ν)],
$
|
respectively. Denote
$L1={(S,I)∈R2+: S=h11},L2={(S,I)∈R2+: S=h12}.
$
|
Then
It follows from Fig. 7 that
By choosing the minimum per capita treatment rate
$h01=2D2Λ−4βμbΛ−√(4βμbΛ−2D2Λ)2−4Λ2(D22−4βμbΛr1)2Λ2−(μ+ν),D2=βΛ−bνr1+bβ(μ+ν).
$
|
Denote
$L4={(S,I)∈R2+: S=h01}.
$
|
According to Fig. 8(a), the equilibrium number of infected cases for system
It has been observed that treatment programme plays a significant role in controlling the emerging and reemerging infectious diseases, such as HIV [36] and Ebola [23], and usually the control programme is implemented only when the number of infected individuals reaches or exceeds the threshold level
Based on the dynamics of two subsystems of the Filippov system, we investigate the long-term dynamical behavior, which reveals much more complex dynamics compared to those for the continuous counterpart. In particular, we have examined the sliding mode, pseudo-equilibrium, multiple attractors in Section 3 and Section 4, respectively. As the threshold value varies, bifurcation analysis of piecewise smooth systems [17,29] on the proposed Filippov epidemic model yields the following local sliding bifurcations theoretically, i.e. regular/virtual equilibrium bifurcation, boundary equilibrium bifurcation including BN (see Fig. 1), BF, BS (see Fig. 2) and BSN bifurcations (see Fig. 3), and global sliding bifurcations including grazing bifurcation (see Fig. 4) and sliding homoclinic bifurcation (see Fig. 4). Our results demonstrate that variety of threshold level can give rise to diversity of long-term dynamical behavior.
It is worth emphasizing that understanding of impact of threshold policy will lead to the development of effective control programmes for public health, so we have examined the impact of some key parameters related to the control measure in Section 5. According to Fig. 5(a), if the basic reproduction number is greater than unity, we can choose proper threshold level such that the real equilibrium of system
In most cases, it is impossible to eradicate an epidemic disease from the population. We have investigated how the maximum and minimum treatment rate affect the equilibrium level of infected individuals in Section 5. Fig. 7 and Fig. 8 show that the number of infected individuals at the steady state is closely related to the maximum and minimum per capita treatment rate. The maximum and minimum per capita treatment rate remain effective for reducing transmission during the outbreak. In particular, if
It is worth noting that we choose the size of infected individuals as an index. In fact, the number of susceptible individuals can also affect on the implementation of control measures, especially on those immunization policies, so it is more appropriate for modeling the impact of limited resources on vaccination policies. Moreover, it is more natural to choose the total number of susceptible individuals and infected individuals as an index, especially for the combined control measure based on treatment and vaccination policy, which may be difficult and interesting, and we leave this for future work.
In conclusion, we propose a Filippov epidemic model to study the impact of HBPR on the disease control by incorporating a piecewise defined treatment programme in this paper. The main results illustrate the significant role of switching treatment programme in response to the call of controlling the epidemic disease. Indeed, to illustrate the main idea, we formulate a simple SIS model without considering the effect of hospitalized infection. Generally, those hospitalized has no contact with susceptible individuals, so no transmission occurs. An SIH (susceptible-infective-hospitalized) model of three dimension could then be more natural, which we will consider in the future work.
The first author was supported by Scientific research plan projects of Shaanxi Education Department (16JK1047). Xiao was supported by the National Natural Science Foundation of China (NSFC, 11571273 and 11631012) and Fundamental Research Funds for the Central Universities(GK 08143042). Zhu was supported by NSERC of Canada. This research was finished when Wang visited Lamps and Department of Mathematics and Statistics, York University.
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Range of parameter values | Existence of endemic equilibria | |
Nonexistence | ||
Nonexistence | ||
Nonexistence | ||
Nonexistence |
Range of parameter values | Existence of endemic equilibria | |
Nonexistence | ||
Nonexistence | ||
Nonexistence | ||
Nonexistence |