In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law whereas the minimization principle is described by a graph eikonal equation. We show that the discrete model is well-posed and the numerical examples reported confirm the validity of the proposed model and its applicability to describe real situations.
Citation: Fabio Camilli, Adriano Festa, Silvia Tozza. A discrete Hughes model for pedestrian flow on graphs[J]. Networks and Heterogeneous Media, 2017, 12(1): 93-112. doi: 10.3934/nhm.2017004
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In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law whereas the minimization principle is described by a graph eikonal equation. We show that the discrete model is well-posed and the numerical examples reported confirm the validity of the proposed model and its applicability to describe real situations.
Dengue is a vector-borne disease spread by the female mos-quitoes Aedes aegypti and Aedes albopictus. The mosquitoes Aedes aegypti were originated from Africa but have now been spread in tropical, subtropical and temperate regions of the world. The fast growth in human population, uncontrolled urbanization and inadequate waste management systems have led to abundance of mosquito breeding sites [6]. These are the main causes which bring this global distribution of mosquitoes and consequently spread of the mosquito-borne diseases. Before 1970, dengue was reported from only nine countries, now it has been widespread in the areas of North America, South America, Africa and Southeast Asia [7]. It is important to control dengue disease as more than 50 million people are affected every year [27]. Interestingly, only the female mosquitoes can transmit the virus. The female mosquitoes bite the human as they require blood for reproduction process. On biting the dengue infected person, these female mosquitoes become infected and can transmit the dengue virus to another person [7]. One of the ways to control dengue infection is to control mosquito population. Mosquito population can be chemically controlled by the use of insecticides and / or biological controlled by wolbochia [9,18,26]. The human awareness campaigns can also reduce the spread of infection.
One of the efficient ways to combat the infection is the use of sterile insect technique (SIT). In SIT, sterile male mosquitoes are released near the mosquito breeding sites. Female mosquitoes will not be able to fertilize when mating with these male mosquitoes. In this way, the mosquito population as well as the spread of infection is controlled. The idea of SIT was first conceived by American entomologist, Dr Edward F. Knipling and was successfully implemented to control the spread of screwworm fly in Florida [14]. After that, the SIT technique has been used for many flying insects by various countries [15,16]. To eradicate dengue infection, the government of China and Brazil have also released the sterile male mosquitoes to combat the dengue infection [10,11].
In literature, some mathematical models on control of vector-borne diseases are available [1,2,3,4,9,13,18,19,20,21,23,25]. Few mathematical models have also been formulated to control dengue infection by SIT [3,4,25]. In particular, Esteva and Yang proposed a mathematical model to control the mosquito density by releasing sterile male mosquitoes [3]. Optimal control analysis to control the Aedes aegypti female mosquitoes by SIT strategy has been performed by Thom
In this paper, a host-vector model has been proposed to control the primary dengue transmission using SIT strategy. In SIT, the male mosquitoes (Aedes aegypti) are sterilized in the laboratory and are released into the environment to compete with wild male mosquitoes for mating with female mosquitoes. In section 2, a network model for
Consider A be the class of mosquitoes in aquatic stage (eggs, larvae or pupae). Let
dSidt=ωi−β1iSiF4i−μiSi+n∑j=1,j≠imSijSj−n∑j=1,j≠imSjiSi | (1) |
dEidt=β1iSiF4i−(ki+μi)Ei+n∑j=1,j≠imEijEj−n∑j=1,j≠imEjiEi | (2) |
dIidt=kiEi−(αi+μi)Ii+n∑j=1,j≠imIijIj−n∑j=1,j≠imIjiIi | (3) |
dRidt=αiIi−μiRi+n∑j=1,j≠imRijRj−n∑j=1,j≠imRjiRi | (4) |
dAidt=ϕi(1−AiCi)(F2i+F4i)−(γi+di)Ai | (5) |
dF1idt=piγiAi−β2iF1iU1iU1i+U2i−β2iF1iU2iU1i+U2i−d1iF1i | (6) |
dF2idt=β2iF1iU1iU1i+U2i−β3iIiF2i−d1iF2i | (7) |
dF3idt=β2iF1iU2iU1i+U2i−d1iF3i | (8) |
dF4idt=β3iIiF2i−d1iF4i | (9) |
dU1idt=(1−pi)γiAi−d1iU1i | (10) |
dU2idt=ω1i−d1iU2i | (11) |
All model parameters are defined in Table 1 and are assumed to be non-negative. The initial conditions associated with the above system are:
Si(0)≥0,Ei(0)≥0,Ii(0)≥0,Ri(0)≥0,0≤Ai(0)≤Ci,F1i(0)≥0,F2i(0)≥0,F3i(0)≥0,F4i(0)≥0,U1i(0)≥0,U2i(0)≥0 |
Parameters | Description of parameters |
|
Human recovery rate |
|
Transmission rate of infection from female mosquitoes to human |
|
Mosquitoes mating rate |
|
Transmission rate of infection from human to female mosquito |
|
Transition rate from aquatic stage to adult mosquito |
|
Natural death rate of human |
|
Birth rate of human |
|
Constant recruitment rate of sterile male mosquito |
|
Recruitment rate for aquatic mosquito |
|
Carrying capacity for aquatic/adult mosquito |
|
Natural death rate of mosquito at aquatic state |
|
Natural death rate of mosquito |
|
Rate at which exposed human become infectious |
|
Migration rate from patch j to patch i |
|
Proportion of female mosquito |
In the next section, the dynamics of the model (1)-(11) is discussed for an isolated patch (i.e
In absence of interpatch migration, the patches are decoupled and isolated. The dynamics of each isolated patch is given below where the subscript
dSdt=ω−β1SF4−μS | (12) |
dEdt=β1SF4−kE−μE | (13) |
dIdt=kE−αI−μI | (14) |
dRdt=αI−μR | (15) |
dAdt=ϕ(1−AC)(F2+F4)−(γ+d)A | (16) |
dF1dt=pγA−β2F1U1U1+U2−β2F1U2U1+U2−d1F1 | (17) |
dF2dt=β2F1U1U1+U2−β3IF2−d1F2 | (18) |
dF3dt=β2F1U2U1+U2−d1F3 | (19) |
dF4dt=β3IF2−d1F4 | (20) |
dU1dt=(1−p)γA−d1U1 | (21) |
dU2dt=ω1−d1U2 | (22) |
Let
dXdt=Q(X(t),t),X(0)=X0≥0Q(X(t),t)=(Q1(X,t),Q2(X,t),Q3(X,t)....,Q11(X,t))⊺ | (23) |
Proposition 1. The positive cone Int(
Proof. Observe that the boundaries of non-negative cone
Proposition 2. Solutions of system (12)-(22) are bounded in the domain
Proof. Let
dNdt=ω−μN⟹lim supt→∞N(t)≤ωμ. |
For vector dynamics, it is clear that
Let
Let
dMdt=ω1+γA(t)−d1(F1(t)+F2(t)+F3(t)+F4(t)+U1(t)+U2(t))dMdt≤ω1+γC−d1M(t)⟹lim supt→∞M(t)≤ω1+γCd1. |
Therefore, the system (23) is bounded.
The equilibrium states for the system (12)-(22) are obtained by solving
Q(Xt),t)=0. |
The trivial disease-free state
˘S=ωμ,˘U2=ω1d1,˘E=0,˘I=0,˘R=0,˘A=0,˘F1=0,˘F2=0,˘F3=0,˘F4=0,˘U1=0. |
Note that, this state is without native mosquitoes. Only the susceptible human and sterile male mosquitoes survive.
Let us define the non-dimensional number
T=β2ϕpγ(β2+d1)d1(γ+d). | (24) |
It represents the average number of secondary female mosquitoes produced by single female mosquito, Esteva and Yang [3]. To maintain the mosquito population in environment,
Proposition 3. The system admits two non-trivial disease-free states
T>1and ω1<W(=(T−1)2Cγ(1−p)4T). | (25) |
Proof. Let us denote
ˉS=ωμ,ˉE=0,ˉR=0,¯F4=0and ¯U2=ω1d1. | (26) |
The other state variables
¯F1=pˉAγ(β2+d1),¯F2=β2pˉAγˉU1(ˉU1+ˉU2)(β2+d1)d1,¯F3=β2pˉAγˉU2d1(β2+d1)(ˉU1+ˉU2),¯U1=ˉAγ(1−p)d1 | (27) |
s(ˉA)=B1ˉA2+B2ˉA+B3=0B1=β2pϕγ(β2+d1)(γ+d)d1C,B2=−β2ϕpγ(β2+d1)d1(γ+d)+1,B3=ω1γ(1−p) |
By writing the quadratic
s(ˉA)=TCˉA2−(T−1)ˉA+ω1γ(1−p)=0. | (28) |
Since
ˉA±=(T−1)C2T(1±√1−ω1W); W=(T−1)2Cγ(1−p)4T. | (29) |
Accordingly, there exists two non-trivial disease-free equilibrium states
These two roots
(ˉA=)ˉA∗=(T∗−1)C2T∗ | (30) |
T∗=(1+2ω1Cγ(1−p))[1+√(1−1(1+2ω1Cγ(1−p))2)]>1 | (31) |
Therefore, the unique non-trivial disease-free equilibrium point
Let us denote the endemic states by
Here,
Remark 1. It is to be noted that for the non-trivial disease-free states
ˉA=ˆA,ˉF1=ˆF1,ˉU1=ˆU1,ˉU2=ˆU2 |
Remark 2. Simplifying the expression for
^F2=β2γpˆA^U1(^U1+^U2)(β2+d1)(β3ˆI+d1). |
Comparing it with
ˉF2>ˆF2 | (32) |
Let us define
R+0=β1β3kωˉF2+μd1(α+μ)(k+μ) and R−0=β1β3kωˉF2−μd1(α+μ)(k+μ). | (33) |
Further, for positive
β1β3kωˆF2+μd1(α+μ)(k+μ)>1 and β1β3kωˆF2−μd1(α+μ)(k+μ)>1 | (34) |
Using (32) gives the conditions for the existence of the endemic states
R+0>1 and R−0>1 | (35) |
Remark 3. It can be seen from (29) that
R+0>R−0 |
Let us define
R0=max(R+0,R−0)(>1). | (36) |
Accordingly, the following proposition is established for the existence of endemic states
Proposition 4. When condition (25) is satisfied, the two positive endemic equilibrium states
Consider the following choice of parameters [3]:
d=0.05,d1=0.0714,p=0.5,γ=0.075 |
The existence of
The basic reproduction number has been computed by next generation approach [12]. It is defined as the average number of secondary infections produced by single infected individual in susceptible population. Using the Remark 3, it is given as
R0=β1β3kωˉF2+(α+μ)d1μ(μ+k). | (37) |
For local stability, the Jacobian matrix
[−β1F4−μ0000000−β1S00β1F4−k−μ000000β1S000k−α−μ0000000000α−μ00000000000j5,50j5,70j5,9000000pγ−β2−d10000000−β3F200j7,6−d1−β3I00j7,10j7,1100000j8,60−d10j8,10j8,1100β3F2000β3I0−d1000000(1−p)γ0000−d100000000000−d1] |
j5,5=−γ−d+−ϕC(F2+F4),j7,6=β2U1U1+U2,j7,10=β2F1U2(U1+U2)2,j7,11=−β2F1U1(U1+U2)2,j8,6=β2U2(U1+U2),j8,10=−j7,10,j8,11=−j7,11j5,7=ϕ(1−AC)=j5,9 |
The eleven eigenvalues of the Jacobian matrix about the trivial disease-free state
−α−μ,−μ(multiplicity 2),−β2−d1,−d1(multiplicity 5),−μ−k,−γ−d1. |
Since all the eigenvalues have negative real part, the state
Proposition 5. The locally asymptotically stable trivial disease-free state
T<1. | (38) |
Proof. Consider the positive definite function
L(A,F1,F2,F3,F4)=C1A+C2F1+C3F2+C4F4 |
Computing its time derivative
˙L(A,F1,F2,F4)=C1˙A+C2˙F1+C3˙F2+C4˙F4˙L(A,F1,F2,F4)=C1(ϕ(1−AC)(F2+F4)−(γ+d)A)+C2(pγA−β2F1−d1F1)+C3(β2F1U1(U1+U2)−β3IF2−d1F2)+C4(β3IF2−d1F4)˙L(A,F1,F2,F4)≤−F2(C3d1−C1ϕ)−F4(C4d1−C1ϕ)−A(C1(γ+d)−C2pγ)−F1(C2(β2+d1)−C3β2) |
For the derivative of the function
d1C3>ϕC1;d1C4>ϕC1;(γ+d)C1>pγC2;(β2+d1)C2>β2C3 |
or
C3C1>ϕd1,C4C1>ϕd1,C1C2>pγ(γ+d),C2C3>β2(β2+d1) |
Now, choosing
β2ϕpγ(β2+d1)d1(γ+d)(=T)<1. |
Accordingly, the function
Proposition 6. For the local stability of non-trivial disease-free states
R0(=β1β3¯F2+kω(α+μ)μd1(μ+k))<1 | (39) |
and
k(ˉA)(=D4)=TˉA2C−ω1γ(1−p)>0 | (40) |
Proof. For the local stability of
q3(λ)=λ3+B1λ2+B2λ+B3=0 |
B1=α+2μ+d1+kB2=μ(μ+2d1)+(μ+d1)k+α(μ+d1+k)B3=(α+μ)d1(μ+k)−β1β3ˉF2kˉS |
The fourth degree polynomial is given as
q4(λ)=λ4+D1λ3+D2λ2+D3λ+D4=0. |
D1=β2+d+3d1+ˉF2ϕC+γ>0D2=(ˉF2ϕ+C(d+d1+γ))(β2+3d1)C+β2d1>0D3=d1((γ+d)(β2+2d1)+d1(β2+d1)+(γ+d)(2β2+3d1)d1(ˉU1+ˉU2)(TˉA2C−ω1d1γ(1−p)(2β2+3d1)))D4=TˉA2C−ω1γ(1−p) |
By Routh-Hurwitz criteria, all the roots of
R0(=β1β3¯F2+kω(α+μ)μd1(μ+k))<1. |
Further, all the roots of fourth degree polynomial
Di>0,i=1...4 | (41) |
D1D2D3>D23+D21D4. | (42) |
Note that
All eigenvalues have negative real part provided the conditions (39) and (40) are satisfied simultaneously and this completes the proof.
Corollary 1. The non-trivial disease-free state
Proof. For the stability of the two non-trivial disease-free states
Remark 4. The condition (40) is always satisfied for
Remark 5. When condition (39) is not satisfied, the non-trivial disease-free state
The Figure 3 is two parameter bifurcation diagram with respect to
ω=0.039,β2=0.5,k=0.1667,ϕ=0.5,μ=0.00004,d=0.1096,d1=0.0714,α=0.7,p=0.5,ω1=0.97,γ=0.7,C=500 |
In Figure 3, the curve
The local stability of
Parameters | Parameters values |
|
0.3 |
|
0.02 |
|
0.7 |
|
0.03 |
|
0.075 |
|
0.002 |
|
5 |
|
0.0000456 |
|
450 |
|
0.05 |
|
0.0714 |
|
0.1667 |
|
0.5 |
Further, the ratio between the carrying capacity to the net sterile male mosquitoes i.e. (
It is observed that the trajectories starting from the initial conditions
It is found that starting with the initial conditions
A bifurcation diagram has been drawn for the system (12)-(22) with respect to sterile male mosquitoes rate
In this paper, a host-vector model has been formulated to analyze the effect of SIT control. It is observed from the analysis that when basic offspring number
To study the effect of human migration in disease transmission, the model (1)-(11) has been analyzed in next section.
When no movement of human is considered,
R0=max(R0i);i=1,2,3...nwhere,R0i=kiβ1iβ3iωiF20i(μi+αi)(ki+μi)μid1i |
When the patches are coupled due to movement of human population, the basic reproduction number for the network model (1)-(11) is computed below:
Let us denote the susceptible human population and fertilized female population at disease free state for the
qi=β1iS0i and li=β3iF20i |
Further, define
The matrices
QE=[μ1+k1+∑1=nj=1mEj1−mE12...−mE1n−mE21μ2+k2+∑1=nj=1mEj2...−mE2n.................−mEn1.......μn+kn+∑1=nj=1mEjn] |
and
QI=[μ1+α1+∑1=nj=1mIj1−mI12...−mI1n−mI21μ2+α2+∑1=nj=1mIj2...−mI2n...................−mIn1......μn+αn+∑1=nj=1mIjn] |
The basic reproduction number is computed by next generation approach [12]. The Jacobian matrices of the system (1)-(11) for the new infections
F=[00Dq0000Dl0] |
and
Y=[QE00−DkQI000Dd1] |
FY−1=[00DqD−1d1000Q−1EQ−1IDkDlDlQ−1E0] |
The dominant eigenvalue of the next generation matrix
Rn0=ρ(Q−1EQ−1IDkDlDqD−1d1). | (43) |
Particularly, for single node
R0=k1β11β31S01F201(μ+α1)(k1+μ)d1. |
Let us perform the numerical experiments to control the disease in a network using SIT. For
Parameters | Parameters values |
|
0.5 |
|
0.001 |
|
0.7 |
|
0.001 |
|
0.075 |
|
0.029 |
|
5 |
|
0.0000456 |
|
450 |
|
0.05 |
|
0.0714 |
|
0.1667 |
|
0.5 |
Network with n=2 |
Observe that
Cases | Migration in patch-1 | Migration in patch-2 | Conclusions |
(a) | |
|
Isolated patches with |
(b) | |
|
|
(c) | |
|
Cases | Migration in patch-1 | Migration in patch-2 | Conclusions |
(d) | |
|
Isolated patches with |
(e) | |
|
|
(f) | |
|
Networkwithn=3 |
In this case, the following two cases have been considered in Table 6: From the Table 6, it is observed that the basic reproduction number for case (g) is less than one. The disease may be controlled in 3 node network model for this case. By numerically solving the network model (1)-(11) for
Cases | Migration in patch-1 | Migration in patch-2 | Migration in patch-3 | |
(g) | |
|
|
|
(h) | |
|
|
|
Keeping the results of above numerical experiments, the following conclusions can be drawn:
It is not necessary to apply SIT in the whole network. The disease can be controlled in network by applying SIT in one patch only. This is possible with suitable coupling of patches. Further, this selective way of applying SIT is more economical (cost effective). The success of SIT depends on the coupling strength of the network (migration parameters
In this paper,
We would like to thank IFCAM for visiting support for this research collaboration. Authors also would like to thank anonymous reviewers for their valuable comments. Their comments have enhanced the clarity and the quality of results and text.
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2. | Christoph Sadée, Stefano Testa, Thomas Barba, Katherine Hartmann, Maximilian Schuessler, Alexander Thieme, George M Church, Ifeoma Okoye, Tina Hernandez-Boussard, Leroy Hood, Ilya Shmulevich, Ellen Kuhl, Olivier Gevaert, Medical digital twins: enabling precision medicine and medical artificial intelligence, 2025, 25897500, 100864, 10.1016/j.landig.2025.02.004 |
Parameters | Description of parameters |
|
Human recovery rate |
|
Transmission rate of infection from female mosquitoes to human |
|
Mosquitoes mating rate |
|
Transmission rate of infection from human to female mosquito |
|
Transition rate from aquatic stage to adult mosquito |
|
Natural death rate of human |
|
Birth rate of human |
|
Constant recruitment rate of sterile male mosquito |
|
Recruitment rate for aquatic mosquito |
|
Carrying capacity for aquatic/adult mosquito |
|
Natural death rate of mosquito at aquatic state |
|
Natural death rate of mosquito |
|
Rate at which exposed human become infectious |
|
Migration rate from patch j to patch i |
|
Proportion of female mosquito |
Cases | Migration in patch-1 | Migration in patch-2 | Conclusions |
(a) | |
|
Isolated patches with |
(b) | |
|
|
(c) | |
|
Cases | Migration in patch-1 | Migration in patch-2 | Conclusions |
(d) | |
|
Isolated patches with |
(e) | |
|
|
(f) | |
|
Cases | Migration in patch-1 | Migration in patch-2 | Migration in patch-3 | |
(g) | |
|
|
|
(h) | |
|
|
|
Parameters | Description of parameters |
|
Human recovery rate |
|
Transmission rate of infection from female mosquitoes to human |
|
Mosquitoes mating rate |
|
Transmission rate of infection from human to female mosquito |
|
Transition rate from aquatic stage to adult mosquito |
|
Natural death rate of human |
|
Birth rate of human |
|
Constant recruitment rate of sterile male mosquito |
|
Recruitment rate for aquatic mosquito |
|
Carrying capacity for aquatic/adult mosquito |
|
Natural death rate of mosquito at aquatic state |
|
Natural death rate of mosquito |
|
Rate at which exposed human become infectious |
|
Migration rate from patch j to patch i |
|
Proportion of female mosquito |
Parameters | Parameters values |
|
0.3 |
|
0.02 |
|
0.7 |
|
0.03 |
|
0.075 |
|
0.002 |
|
5 |
|
0.0000456 |
|
450 |
|
0.05 |
|
0.0714 |
|
0.1667 |
|
0.5 |
Parameters | Parameters values |
|
0.5 |
|
0.001 |
|
0.7 |
|
0.001 |
|
0.075 |
|
0.029 |
|
5 |
|
0.0000456 |
|
450 |
|
0.05 |
|
0.0714 |
|
0.1667 |
|
0.5 |
Cases | Migration in patch-1 | Migration in patch-2 | Conclusions |
(a) | |
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Isolated patches with |
(b) | |
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(c) | |
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Cases | Migration in patch-1 | Migration in patch-2 | Conclusions |
(d) | |
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Isolated patches with |
(e) | |
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(f) | |
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Cases | Migration in patch-1 | Migration in patch-2 | Migration in patch-3 | |
(g) | |
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(h) | |
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