Two-Scale Network Epidemic Dynamic Model for Vector Borne Diseases

Abstract

A SIRS delayed epidemic dynamic model for a vector-borne disease in a two-scale network structured dynamic population is presented. The distributed time delay represents the varying incubation period of the disease. The epidemic dynamic process is influenced by a two-scale network human mobility dynamic process. The global asymptotic stability results for the system and the significance of the results to the general disease dynamic process are presented. The results are exhibited in different special real life scenarios. The presented results are demonstrated by numerical simulation results.

Full text

Proceedings of Dynamic Systems and Applications 6 (2016) 228–232
Two-Scale Network Epidemic Dynamic Model for
Vector Borne Diseases
Divine Wanduku
Department of Mathematics, Keiser University
2400 Interstate Drive, Lakeland, Florida 33805, U.S.A. E-mail: wandukudivine@yahoo.com, dwanduku@keiseruniversity.edu
ABSTRACT: A SIRS delayed epidemic dynamic model for a vector-borne disease in a two-scale network structured dynamic
population is presented. The distributed time delay represents the varying incubation period of the disease. The epidemic dynamic
process is influenced by a two-scale network human mobility dynamic process. The global asymptotic stability results for the system
and the significance of the results to the general disease dynamic process are presented. The results are exhibited in different special
real life scenarios. The presented results are demonstrated by numerical simulation results.
AMS (MOS) Subject Classification. 35K60, 35K57
1 INTRODUCTION
Multi-group epidemic dynamic models[13, 14, 15, 16, 17, 18, 19, 21, 20, 22, 23, 24, 25] representing the dynamics of diseases
in metapopulations involving geographical subpopulation distributions have been used to investigate the dynamics of diseases in
complex mobile human populations. Moreover, some extensions and expansions of these models from the much simpler single
interaction level multi-group epidemic dynamic models into more advanced hierarchical multidimensional interaction level multi-
group epidemic dynamic models[8, 9, 10, 11, 12] have also been studied.
The inclusion of the effects of disease latency or immunity into the epidemic dynamic modeling process leads to more realistic
epidemic dynamic models. The disease latency is represented as the incubation period of the disease or as the infectivity period of
the infectious population[2, 3, 4, 5, 6, 7]. Vector borne disease dynamic models[1, 2, 3, 4, 5, 6, 7] involving the incubation period
of the disease agent in the form of temporary or distributed delays have been studied.
Recently, Divine and Ladde [9] introduced an algorithmic mathematical dynamic model for a two-scale network SIRS
(susceptible-infectious-removal-susceptible) epidemic dynamic model. In the framework of the earlier study [9] a more realistic
two-scale network vector borne epidemic dynamic model is presented. The epidemic dynamic model includes varying incubation
period of the vector borne disease which is represented as distributed delays in the epidemic dynamic model. Due to limited space
and the complex nature of the two scale network epidemic dynamic model studied in [9], only the extended vector-borne disease
dynamic model is presented in this paper, and all other definitions and notations are referred to [9].
2 Derivation of the SIRS Delayed dynamic Model
In this model, the vector borne epidemic dynamic process in a two scale network dynamic population influenced by the two scale
network human mobility process[8] exhibits a comparable structure to the general large scale two level deterministic SIRS dynamic
epidemic model in earlier study[9]. The disease transmission process is driven by a resident vector. Indeed, an infectious visitor
Ivu
ba (t−s),∀u, v ∈I(1, M ),∀a∈I(1, nu),∀b∈I(1, nv)to site su
ain region Cu, travelling from site sv
bin region Cvinteracts
with a resident susceptible vector at earlier time t−s, where s∈[t0, h], h > 0. After the incubation period s∈[t0, h]the exposed
vector becomes infectious. At time t, the infectious resident vector interacts with a susceptible human visitor Sru
ia (t)to site su
ain
region Cu, travelling from site sr
iin region Cr. The susceptible human visitor Sru
ia (t)acquires infection and enters the infectious
state Iru
ia (t). Utilizing ideas from [1, 2, 3, 4, 5, 6, 7] in addition to the assumption that the infectious vector population at time tis
proportional to the infectious human population at the earlier time t−s, the force of the infection or the incidence of the disease to
the population influenced by the two scale network human mobility process is given as follows
M
X
v=1
nv
X
b=1
βurv
aib Sru
ia Zh
t0
furv
aib (s)Ivu
ba (t−s)ds, r, u ∈I(1, M ), i ∈I(1, nr), a ∈I(1, nu),(2.1)
where the delay kernel furv
aib (s), s ∈[t0, h]represents the fraction of exposed resident vectors of site su
ain region Cuthat acquired
infection from the infectious human Ivu
ba (t−s), and that at time tare fully infectious and have successfully transmitted the infection
Dynamic Publishers, Inc.

Two-Scale Network Epidemic Dynamic Model for Vector Borne Diseases 229
to the susceptible human visitor Sru
ia (t). Furthermore,
Zh
t0
furv
aib (s)ds = 1,(2.2)
In addition, βurv
aib is the daily contact rate, i.e., the average number of contacts per infective per day between susceptible human
visitors to site su
ain region Cu, travelling from site sr
iin region Cr, and infective resident vectors of site su
ain region Cu, where the
infective resident vectors previously acquired infection from infectious human visitors Ivu
ba from site sv
bin region Cv. Furthermore,
it is assumed that sites in the two scale network population are considerably spaced apart, and vectors do not migrate. In addition,
the vectors exhibit comparable biological and physiological characteristics at all sites in the the two-scale population. Substituting
the incidence (2.1) into the large scale two level SIRS epidemic dynamic model equations of (2.15)-(2.17) in [9] gives the following
system of ordinary differential equations describing a vector borne diseases in a two scale network dynamic population.
dSrq
il =

















Br
i+Pnr
k=1 ρrr
ik Srr
ik +PM
q6=rPnq
a=1 ρrq
ia Srq
ia +ηr
iIrr
ii +αr
iRrr
ii
−(γr
i+σr
i+δr
i)Srr
ii −PM
u=1 Pnu
a=1 βrru
iia Srr
ii Rh
t0frru
iia (s)Iur
ai (t−s)dsdt, f or q =r, l =i,
σrr
ij Srr
ii +ηr
jIrr
ij +αr
jRrr
ij −(ρrr
ij +δr
j)Srr
ij −PM
u=1 Pnu
a=1 βrru
jia Srr
ij Rh
t0frru
jia (s)Iur
aj (t−s)dsdt,
for q =r, l =j, j 6=i,
γrq
il Srr
ii +ηq
lIrq
il +αq
lRrq
il −(ρrq
il +δq
l)Srq
il
−PM
u=1 Pnu
a=1 βqru
lia Srq
il Rh
t0fqru
lia (s)Iuq
al (t−s)dsdt, f or q 6=r,
(2.3)
dIr q
il =

















Pnr
k=1 ρrr
ik Irr
ik +PM
q6=rPnq
a=1 ρrq
ia Irq
ia −ηr
iIrr
ii −%r
iIrr
ii
−(γr
i+σr
i+δr
i+dr
i)Irr
ii +PM
u=1 Pnu
a=1 βrru
iia Srr
ii Rh
t0frru
iia (s)Iur
ai (t−s)dsdt, f or q =r, l =i
σrr
ij Irr
ii −ηr
jIrr
ij −%r
jIrr
ij −(ρrr
ij +δr
j+dr
j)Irr
ij +PM
u=1 Pnu
a=1 βrru
jia Srr
ij Rh
t0frru
jia (s)Iur
aj (t−s)dsdt,
for q =r, l =j, j 6=i,
γrq
il Irr
ii −ηq
lIrq
il −%q
lIrq
il −(ρrq
il +δq
l+dq
l)Irq
il
+PM
u=1 Pnu
a=1 βqru
lia Srq
il Rh
t0fqru
lia (s)Iuq
al (t−s)dsdt, f or q 6=r,
(2.4)
dRrq
il =

Pnr
k=1 ρrr
ik Rrr
ik +PM
q6=rPnq
l=1 ρrq
il Rrq
il +%r
iIrr
ii −(γr
i+σr
i+αr
i+δr
i)Rrr
ii dt, f or q =r, l =i
σrr
ij Rrr
ii +%r
jIrr
ij −(ρrr
ij +αr
j+δr
j)Rrr
ij dt, f or q =r, l =j, j 6=i,
γrq
il Rrr
ii +%q
lIrq
il −(ρrq
il +αq
l+δq
l)Rrq
il dt, f or q 6=r,
(2.5)
where i∈I(1, nr), l ∈Ir
i(1, nq); r∈I(1, M ), q ∈Ir(1, M ); all parameters are as defined before in [9]. Furthermore, for each
r∈I(1, M ), and i∈I(1, nr), we have the following initial conditions
(Sru
ia (t), Ir u
ia (t), Rru
ia (t)) = (ϕru
ia1(t), ϕru
ia2(t), ϕru
ia3(t)), t ∈[−h, t0],
ϕru
iak ∈ C([−h, t0],R+),∀k= 1,2,3,∀r, q ∈I(1, M ), a ∈I(1, nu), i ∈I(1, nr),
ϕru
iak(t0)>0,∀k= 1,2,3,(2.6)
where C([−h, t0],R+)is the space of continuous functions with the supremum norm
||ϕ||∞=Sup−h≤t≤t0|ϕ(t)|.(2.7)
3 Model Validation Results
In this section, the initial value problem associated with the delayed system (2.3)-(2.5) is shown to have a unique non-negative
solution.
Theorem 3.1. Let r, u ∈I(1, M ),i∈I(1, nr)and a∈I(1, nu). Given the initial conditions (2.6)-(2.7), there exists a
unique solution xru
ia (t) = (Sru
ia (t), Ir u
ia (t), Rru
ia (t))Tsatisfying (2.3)-(2.5), for all t≥t0. Furthermore, the solutions yru
ia =
Sru
ia +Iru
ia +Rru
ia ≥0. Moreover, ||xr u
ia (t)||1=PM
r=1 PM
u=1 Pnr
i=1 Pnu
a=1 yru
ia (t)≤1
µPM
r=1 Pnr
i=1 Br
i, for t ≥t0,
whenever ||xru
ia (t0)||1=PM
r=1 PM
u=1 Pnr
i=1 Pnu
a=1 yru
ia (t0)≤1
µPM
r=1 Pnr
i=1 Br
i,where µ= min1≤u≤M,1≤a≤nu(δu
a).
Proof: The unique global existence follows from the extension results about local solutions in [26]. The rest of the results follow
from (Lemma 3.1, Lemma 3.2, [9]).
Remark 3.1. From Theorem 3.1, one can conclude that a closed ball ¯
BR3n2(~
0; r)in R3n2under the sum norm ||.||1centered at
the origin ~
0∈R3n2, with radius r=1
µPM
r=1 Pnr
i=1 Br
iis self-invariant with regard to a two-scale network dynamics of human
epidemic process (2.3)-(2.5) that is under the influence of human mobility process[8]. That is,
¯
BR3n2(~
0; r) = ((Sru
ia , Ir u
ia , Rru
ia ) : yru
ia (t)≥0and ||x00
00||1=
M
X
r=1
M
X
u=1
nr
X
i=1
nu
X
a=1
yru
ia (t)≤1
µ
M
X
r=1
nr
X
i=1
Br
i)(3.1)

230 D. Wanduku
is a positive self-invariant set for system (2.3)-(2.5). Denote
¯
B≡1
µ
M
X
r=1
nr
X
i=1
Br
i(3.2)
4 Existence and Asymptotic Behavior of Disease Free Equilibrium
In this section, the global uniform asymptotic stability results of the disease free equilibrium are established. Due to limited space,
the results are presented. The details will appear else where.
For any r, u ∈I(1, M ),i∈I(1, nr)and a∈I(1, nu), let let (Sru∗
ia , Ir u∗
ia , Rru∗
ia ),be the equilibrium state of the delayed
system (2.3)-(2.5). One can see that the disease free equilibrium state is given by Eru
ia = (Sru∗
ia ,0,0), where
Sru∗
ia =






Br
i
Dr
i, for u =r, a =i,
Br
i
Dr
i
σrr
ij
ρrr
ij +δr
j, for u =r, a 6=i,
Br
i
Dr
i
γru
ia
ρru
ia +δu
a, for u 6=r,
(4.1)
and, Dr
i=γr
i+σr
i+δr
i−Pnr
a=1
ρrr
ia σrr
ia
ρrr
ia +δr
a−PM
u6=rPnu
a=1
ρrr
ia γru
ia
ρru
ia +δu
a>0. The asymptotic stability of Eru
ia will be established by
verifying the conditions of the Lyapunov functional technique given in [27]. The following transformation is utilized in (2.3)-(2.5)
to establish the uniform asymptotic stability results.



Uru
ia =Sru
ia −Sru∗
ia
Vru
ia =Iru
ia
Wru
ia =Rru
ia .
(4.2)
The following lemmas would be useful in establishing the stability results.
Lemma 4.1. Let V1:R3n2×R+→R+be a function defined by



V1(˜x00
00) = PM
r=1 PM
u=1 Pnr
i=1 Pnu
a=1 V(˜xru
ia ),
V1(˜xru
ia )=(Sru
ia −Sru∗
ia +Iru
ia )2+cru
ia (Iru
ia )2+ (Rru
ia )2
˜x00
00 = (Uru
ia , V ru
ia , W ru
ia )Tand cru
ia ≥0.
(4.3)
Then V1∈ C2,1(R3n2×R+,R+), and it satisfies b(||˜x00
00||)≤V1(˜x00
00(t)) ≤a(||˜x00
00||)where
b(||˜x00
00||) = min1≤r,u≤M,1≤i≤nr,1≤a≤nuncru
ia
2+cru
ia oPM
r=1 PM
u=1 Pnr
i=1 Pnu
a=1 (Uru
ia )2+ (Vru
ia )2+ (Wru
ia )2,
a(||˜x00
00||) = max1≤r,u≤M,1≤i≤nr,1≤a≤nu{cr u
ia + 2}PM
r=1 PM
u=1 Pnr
i=1 Pnu
a=1 (Uru
ia )2+ (Vru
ia )2+ (Wru
ia )2.
Proof: See ( Lemma 4.1, [9]).
Another lemma that would be useful to establish the stability results is presented. The details of the proof will appear else where.
Lemma 4.2. Assume that the hypothesis of Lemma 4.1 is satisfied. Define a Lyapunov functional V=V1+V2, where V1is defined
by (4.3), and
V2=
M
X
r=1
nr
X
i=1
M
X
u=1
nu
X
a=1
cru
ia
M
X
v=1
nv
X
b=1 βurv
aib (Sru∗
ia +¯
B) + (vurv
aib )2(Sru∗
ia +¯
B)2Zh
t0
furv
aib (s)Zt
t−s
(Vvu
ba (θ))2dθds (4.4)
Furthermore, let
Urv
ia =















PM
v=1 Pnv
b=1 ρrv
ib µrr
ii +PM
v6=rPnv
b=1
γrv
ib
µrr
ii
+Pnr
b6=i
σrr
ib
µrr
ii
+1
2(γr
i+σr
i+αr
i+%r
i+δr
i+dr
i)µrr
ii +1
2
(γr
i+σr
i+δr
i)
µrr
ii
(γr
i+σr
i+δr
i)v=r, b =i
ρrr
ib
µrr
ii
+1
2
(ρrr
ib +δr
b)
µrr
ib
+σrr
ib µrr
ii +1
2(ρrr
ib +αr
b+%r
b+δr
b+dr
b)µrr
ib
(ρrr
ib +δr
b)v=r, b 6=i
ρrv
ib
µrr
ii
+1
2
(ρrv
ib +δv
b)
µrv
ib
+γrv
ib µrr
ii +1
2(ρrv
ib +αv
b+%v
b+δv
b+dv
b)µrv
ib
(ρrv
ib +δv
b)v6=r, b ∈I(1, nv)
(4.5)
Vrv
ib =








1
2PM
v=1 Pnv
b=1 ρrv
ib µrr
ii +1
2PM
v=1 Pnv
b=1 βrrv
iib (Srr∗
ii +¯
B)+ 1
2βrrr
iii (Srr∗
ii +¯
B)+ 1
2(Vrrr
iii )2(Srr∗
ii +¯
B)2
(γr
i+σr
i+δr
i+%r
i+ηr
i+dr
i)v=r, b =i
1
2σrr
ib µrr
ii +1
2PM
u=1 Pnu
a=1 βrru
bia (Srr∗
ib +¯
B)+ 1
2βrrr
bii (Srr∗
ib +¯
B)+ 1
2(Vrrr
bii )2(Srr∗
ib +¯
B)2
(ρrr
ib +δr
b+%r
b+ηr
b+dr
b)v=r, b 6=i
1
2γrv
ib µrr
ii +1
2PM
u=1 Pnu
a=1 βvru
bia (Srv∗
ib +¯
B)+ 1
2βvrr
bii (Srv∗
ib +¯
B)+ 1
2(Vvrr
bii )2(Srv∗
ib +¯
B)2
(ρrv
ib +δv
b+%v
b+ηv
b+dv
b)v6=r, b ∈I(1, nv)
(4.6)

Two-Scale Network Epidemic Dynamic Model for Vector Borne Diseases 231
and
Wrv
ib =















1
2PM
v=1 Pnv
b=1 ρrv
ib µrr
ii +1
2PM
v6=rPnv
b=1
γrv
ib
µrr
ii
+1
2Pnv
b6=i
σrr
ib
µrr
ii
+αr
i
µrr
ii
+1
2%r
iµrr
ii
(γr
i+σr
i+αr
i+δr
i)v=r, b =i
1
2
ρrr
ib
µrr
ii
+1
2σrr
ib µrr
ii +αr
b
µrr
ib
+1
2%r
bµrr
ib
(ρrr
ib +δr
b+δr
b)v=r, b 6=i
1
2
ρrv
ib
µrr
ii
+1
2γrv
ib µrr
ii +αv
b
µrv
ib
+1
2%v
bµrv
ib
(ρrv
ib +δv
b+δv
b)v6=r, b ∈I(1, nv)
(4.7)
for some suitably defined positive number µru
ia , where µru
ia depends on δu
a, for all r, u ∈Ir(1, M ),i∈I(1, n)and a∈Ir
i(1, nr).
Assume that Uru
ia ≤1,Vru
ia <1and Wru
ia ≤1. There exist positive numbers φru
ia ,ψru
ia and ϕru
ia such that the differential operator
LV associated with Ito-Doob type stochastic system (2.3)-(2.5)satisfies the following inequality
LV (˜x00
00)≤
M
X
r=1
nr
X
i=1 −[φrr
ii (Urr
ii )2+ψrr
ii (Vrr
ii )2+ϕrr
ii (Wrr
ii )2]
−
nr
X
a6=i
[φrr
ia (Urr
ia )2+ψrr
ia (Vrr
ia )2+ϕrr
ia (Wrr
ia )2]
−
M
X
u6=r
nu
X
a=1
[φru
ia (Urr
ia )2+ψru
ia (Vru
ia )2+ϕru
ia (Wru
ia )2]
.(4.8)
Moreover,
LV (˜x00
00)≤ −cV1(˜x00
00)(4.9)
where a positive constant cis defined by
c=min1≤r,u≤M,1≤i≤nr,1≤a≤nu{φru
ia , ψru
ia , ϕru
ia }
max1≤r,u≤M,1≤i≤nr,1≤a≤nu{cru
ia + 2}(4.10)
Proof: The results are obtained by first transforming system (2.3)-(2.5) by applying (4.2), then finding the derivative of the
Lyapunov functional V=V1+V2with respect to the system (2.3)-(2.5), where, V1and V2are given in (4.3) and(4.4) respec-
tively. The process is completed by estimating the derivative using quadratic estimates, Lemma 4.1, in addition to other algebraic
manipulations and simplifications. Due to limited space, the detailed proof will appear elsewhere. The uniform asymptotic stability
theorem for the disease free equilibria are now formally stated.
Theorem 4.1. Given r, u ∈I(1, M ),i∈I(1, nr)and a∈I(1, nu). If the hypotheses of Lemma 4.2 are satisfied then the disease
free solutions Eru
ia , are globally uniformly asymptotically stable. Moreover, the solutions Eru
ia are exponentially stable.
Proof: From the application of comparison result[27, 28], the proof of the global asymptotic stability follows immediately.
Remark 4.1. The asymptotic stability results are exhibited in several real life scenarios considered in earlier study [8]. Further-
more, several simulation results are developed to illustrates the various types of two-scale network dynamic epidemic models. But
due to limited space, these results will appear elsewhere.
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