Modelling the spread of carrier-dependent infectious diseases with environmental effect

Modelling the spread of carrier-dependent infectious diseases with environmental effect

Mini Ghosh

^a,*

, Peeyush Chandra

^a

, Prawal Sinha

^a

, J.B. Shukla

^b

aDepartment of Mathematics, Indian Institute of Technology, Kanpur 208016, India

bCentre for Modelling, Environment and Development, MEADOW Complex, 18-Nav Sheel Dham, Kanpur 208017, India

Abstract

Many infectious diseases spread by carriers such as flies, ticks, mites, snails, etc. In this paper an SIS model for carrier-dependent infectious diseases, like cholera, diarrhea, etc. caused by direct contact of susceptibles with infectives as well as by carriers is proposed and analyzed assuming the growth of both the human and the carrier pop- ulations logistic. It is assumed further that the density of carrier population increases with the increase in the cumulative density of discharges by the human population into the environment. The mathematical model is analyzed for the following two cases: (i) the rate of cumulative environmental dischargesQis a constant, and (ii) the rate of cumulative environmental discharges Qis a function of the population density. This model is analyzed using usual theory of differential equations and computer simulation.

By computer simulation it is concluded that if the growth of carrier population caused by conducive household discharges increases, the spread of the infectious disease in- creases.

Ó 2003 Elsevier Inc. All rights reserved.

Keywords:Epidemic model; Carrier; Simulation

*Corresponding author. Present address: Department of Mathematics, University of Trento, Via Sommarive 14, 38050 Trento, Italy.

E-mail addresses: ghosh@science.unitn.it (M. Ghosh), peeyush@iitk.ac.in (P. Chandra), prawal@iitk.ac.in (P. Sinha).

0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0096-3003(03)00564-2

www.elsevier.com/locate/amc

1. Introduction

Many infectious diseases spread by carriers such as flies, ticks, mites and snails, which are present in the environment [10–12,21]. For example, air-borne carriers or bacteria spread diseases such as tuberculosis and measles; while water-borne carriers or bacteria are responsible for the spread of dysentery, gastroenteritis, diarrhea, etc. [4,19]. Various kinds of household and other wastes, discharged into the environment in residential areas of population, provide a very conducive environment for the population growth of some of these carriers [15,18]. This enhances the chance of carrying more bacteria from infectives to the susceptibles in the population leading to fast spread of carrier- dependent infectious diseases. Thus unhygienic environmental conditions in the habitat caused by humans population become responsible for the fast spread of an infectious disease.

In recent decades, there have been several investigations of infectious dis- eases using deterministic mathematical models with or without demographic change [2,3,5–9,13,14,16,20]. In particular Greenhalgh [7] has studied an in- fectious disease model with population-dependent death rate using computer simulation. Gao and Hethcote [5] analyzed an infectious disease model with logistic population growth. Zhou and Hethcote [20] have studied a few models for infectious diseases using various kinds of demographics. Hethcote [13] has discussed an epidemic model in which the carrier population is assumed to be constant. But in general the size of the carrier population varies and depends on the natural conditions of the environment as well as on various discharges into it by the human population.

Thus in this paper, the effect of variable carrier population caused by en- vironmental discharges on the spread of an infectious disease is studied.

2. The model

In this paper an SIS model with logistic growth of human population is considered so that both the birth as well as the death rates are density de- pendent in such a manner that the birth rate decreases and death rate increases as the population density increases towards its carrying capacity [5]. Here the population density NðtÞis divided into two classes: susceptibles XðtÞ and in- fectivesYðtÞ. It is assumed that all susceptibles living in the habitat are affected by a carrier population of density CðtÞ, which grows logistically with given intrinsic growth rate and carrying capacity. The growth rate of its density is further assumed to increase with the increase in the cumulative density of discharges by the human population into the environment. Keeping the above in mind and by considering simple mass action interaction, a mathematical model is proposed as follows:

X_ ¼ b

arN K

N d

þ ð1aÞrN K

XbXYkXCþmY;

Y_ ¼bXY þkXC m

þaþdþ ð1aÞrN K

Y;

N_ ¼r 1

N K

NaY;

C_ ¼sC 1

C L

dCþs1EC;

E_ ¼QðNÞ d0E;

X þY ¼N; s>d; 06a61;

ð1Þ

with initial conditions:

Xð0Þ>0; Yð0ÞP0; Nð0Þ>0; Cð0ÞP0 and Eð0Þ>0:

HereEðtÞis the cumulative density of environmental discharges conducive to the growth of carrier population;banddare the natural birth and death rates;

r¼bd >0 is the growth rate constant; K is the carrying capacity of the human population density in the natural environment;b andk are the trans- mission coefficients due to the infectives and the carrier population respec- tively;a is the disease related death rate constant and mis the recovery rate constant i.e. the rate at which individual recovers and moves to the susceptible class again from the infective class. The constantLis the carrying capacity of the carrier population in the natural environment;sis its intrinsic growth rate;

dis the death rate of carriers due to control measures, wheres>d;s₁is the per capita growth rate coefficient of the carrier population due to the cumulative environmental discharges rate QðNÞ which is human population density de- pendent (an increasing function ofN) andd0is the depletion rate coefficient of the environmental discharges. In writing the model (1), we use the term transmission coefficient in the sense as used by Anderson and May [1], which means that new cases of disease occur at the ratesbXY and kXC due to in- teraction of susceptibles with infectives and carriers respectively.

For 0<a<1, the birth rate decreases and the death rate increases as N increases to its carrying capacity K. When a¼1, the model could be called simply a logistic birth model as all of the restricted growth is due to a de- creasing birth rate and the death rate is constant. Similarly, when a¼0, it could be called a logistic death model as all of the restricted growth is due to an increasing death rate and the birth rate is constant. It is easy to note that the above model is well-posed in the region of attractionT₁given by

T₁¼ ðY;N;C;EÞ:0

6Y6N6K; 06C6L s s

dþs₁QðKÞ d0

; 06E6QðKÞ

d0

:

The model (1) is analyzed for the following two cases:

i(i) the rate of cumulative environmental dischargesQis a constant, and

(ii) the rate of cumulative environmental dischargesQis a function of the pop- ulation density.

The functionQðNÞis such that it satisfies following conditions:

Qð0Þ ¼Q₀>0; Q⁰ðNÞP0;

i.e. Q is an increasing function of N. We consider the form of QðNÞ as QðNÞ ¼Q₀þlN, wherel>0 is a constant.

2.1. Case I:Qis a constantQ_a

In this case we note from the last two equations of system (1) that

t!1lim supEðtÞ ¼Q_a d0

and lim

t!1supCðtÞ ¼L s s

dþs₁Q_a d0

¼C_m>0:

Thus to see the global behavior of the system it is reasonable to consider the following subsystem of system (1):

Y_ ¼bðNYÞYþkðNYÞCm m

þaþdþ ð1aÞrN K

Y;

N_ ¼r 1

N K

NaY;

ð2Þ

whereC_mincreases as the household discharge rateQ_aincreases.

The result of an equilibrium analysis is stated in the following theorem.

Theorem 1.There exist the following two equilibria, namely(i)E1ð0;0Þand (ii) E2ðbY;NbÞ, which exists ifmþaþd>½ðarÞ=rkCm.

Proof.Existence ofE1is obvious. The existence of second equilibrium point is shown as follows (see Fig. 1):

Setting the right-hand side of (2) to zero, we get Y ¼r

a 1

N K

N; ð3Þ

bY²hnb

ð1aÞr K

o

N ðmþaþdþkC_mÞi

Y kC_mN ¼0: ð4Þ It may be pointed out that in theN–Y plane (3) gives a parabola with vertex ðK=2;rK=4aÞand passing through the pointsð0;0ÞandðK;0Þ, while (4) gives a hyperbola with a branch in the first and fourth quadrants and passing through ð0;0Þ. The two curves will intersect at a point ðbY;NbÞ provided the slope of parabola atð0;0Þis more than that of the hyperbola branch in first quadrant at ð0;0Þ, i.e. slope of (4) atð0;0Þis less than that of slope of (3) atð0;0Þ, which gives

mþaþd>ar

r kCm: ð5Þ

Thus the condition for the existence of second equilibrium point ðYb;NbÞ is proved.

Remark 1.From (4), dY

dN

ð0;0Þ

¼ kCm

mþaþdþkCm

>0; ð6Þ

which increases asC_mincreases.

Remark 2. It is also noted from (4) that dYb=dCm>0 for Nb PK=2, which implies that equilibrium infective density increases asC_m increases.

Remark 3.It is noted that ifb¼ ð1aÞr=K, then (4) gives a parabola and in this case also there exists a unique positive root Nb in ð0;KÞ.

Fig. 1. Existence of equilibrium point.

2.1.1. Stability analysis

Now we present the stability analysis of these equilibria. The local stability results are stated in the following theorem.

Theorem 2.The equilibriumE₁ð0;0Þis unstable and the equilibriumE₂ðbY;NbÞis locally asymptotically stable provided

bYb þkN Cb _m b Y

!r

Kð2Nb KÞ þa bn

ð1aÞr K

oYb þkC_ma>0:

Proof. The variational matrix M₁ at E₁ð0;0Þ corresponding to the system of equations (2) is given by

M₁¼ kCm ðmþaþdÞ kCm

a r

:

Since one eigenvalue ofM1 is positive,E1 is unstable.

The variational matrix Mb at E2ðYb;NbÞ corresponding to the system of equations (2) is given by

b

M ¼ bYbþkN Cb _m b Y

!

bYbþkCm ð1aÞrYb K

a r

KðK2NbÞ 0

BB

@

1 CC A:

The characteristic polynomial is given by w²þ bYb

(

þkN Cb _m b Y þr

Kð2Nb KÞ )

wþ bYbþkN Cb _m b Y

!r

Kð2Nb KÞ þa bbY

(

þkCm ð1aÞrYb K

)

¼0: ð7Þ

In order that the above quadratic has roots which have negative real parts, it is necessary that

bYbþkN Cb _m b Y þ r

Kð2Nb KÞ>0 ð8Þ

and

bYb þkN Cb _m b Y

!r

Kð2Nb KÞ þa bn

ð1aÞr K

oYb þakCm>0: ð9Þ

The first inequality is obviously true in view of the equilibrium conditions i.e.

(3) and (4). Thus only the second inequality gives the condition for linear stability ofE₂ðbY;NbÞ. Hence the theorem.

Remark.It may be noted that the inequality (9) is satisfied for NbPK=2 and b>ð1aÞr=K. The conditionNbPK=2 is also compatible with the condition of dYb=dCm>0. Hence from now onward we assume the above mentioned condition.

2.1.1.1. Nonlinear analysis and simulation. Let us take the following Liapunov function:

V ¼ Y

Yb YblnY b Y

þ1

2k₁ðNNbÞ²; ð10Þ

wherek1 is to be suitably chosen. Now using the system (2), we get V_ ¼ b

þkNCm

YYb

ðYYbÞ²k1

r

KðNþNb KÞðNNbÞ² k₁a

kCm

b Y b

1a r K

ðYYbÞðNNbÞ:

After choosing k₁¼1

a kC_m

b Y

þb ð1aÞr K

;

we note that V_ is negative definite in the region K=2<N6K such that K=2<Nb6K, where it is assumed that b>ð1aÞr=K. Hence E₂ðYb;NbÞ is globally asymptotically stable in a subregion ofT₁.

To see the global behavior of the nontrivial equilibrium and to see the effects of various parameters on the spread of the disease, the system (2) is integrated by the fourth order Runge–Kutta method using following set of parameters in the simulation, which satisfies the local stability condition (9) of equilibrium E₂.

b¼0:00000031; k¼0:000000021; m¼0:012; a¼0:0005;

a¼0:3; d ¼0:0004; r¼0:0003; K ¼50000; C_m¼100000:

The equilibrium values of Yb and Nb are obtained as Yb ¼7299:645 and b

N ¼29087:309.

Simulation is performed for different initial positions 1, 2, 3, 4 as shown in Fig. 2. In this figure, the infected population is plotted against the susceptible population. From the solution curves, we observe that the system is globally stable for this set of parameters, provided that we start away from other equilibria. In Fig. 3, the infective population is plotted against time for different

C_m and from this we observe that the infective population increases as C_m increases.

2.2. Case II:Qis a variable

In this case we consider the following equivalent system of system (1) (using XþY ¼N):

Fig. 2. Variation of infective population with susceptible population.

Fig. 3. Variation of infective population with time for differentCm.

Y_ ¼bðNYÞYþkðNYÞC m

þaþdþ ð1aÞrN K

Y;

N_ ¼r 1

N K

NaY; C_ ¼sC 1

C

L

dCþs₁EC;

E_ ¼QðNÞ d0E¼Q0þlNd0E:

ð11Þ

The result of equilibrium analysis is stated in the following theorem.

Theorem 3.There exist the following five equilibria, namely ii(i) E₁ð0;0;0;Q₀=d0Þ,

i(ii) E₂ð0;K;0;Q₀=d0Þ,

(iii) E₃ðY;N;0;EÞ, which exists ifbK >ð1aÞrþmþaþd, where N¼ b 1_a^r

ð1aÞ_K^r

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 1^r_a

ð1aÞ_K^r

2

þ4_aK^brðmþaþdÞ q

2_aK^br ;

Y¼r a 1

N

K

N>0; E¼QðNÞ d0

;

(iv) E₄ð0;0;C;EÞ, where C¼L

s s

dþs1

Q₀ d0

; E¼Q₀

d0

;

i(v) E₅ðbY;Nb;C;b bEÞ, which exists if ðmþaþdÞ>kL

s

ðarÞ

r s

dþs₁Q₀ d0

:

Proof.The existence of the first four equilibria is obvious. The existence of the fifth equilibriumE5 is shown as follows. Setting the right-hand side of system (11) to zero and simplifying we get

Y ¼r a 1

N

K

N; ð12Þ

bY²hnb

ð1aÞr K

oN ðmþaþdþkCÞi

Y kCN ¼0; ð13Þ where

C¼L s s

dþs1

Q₀þlN d0

:

As before, we see that in N–Y plane (12) is a parabola and (13) is a hy- perbola unlessb¼ ð1aÞr=Kwhen it is a parabola passing through origin and a branch in the first quadrant fork>0.

From (13), we note that the slope dY

dN

¼Y k1YþkCþ^kLls_sd¹

0 ðNYÞ

ðbY²þkCNÞ

" #

>0 for Y >0; N >0;

wherek1¼b ð1aÞr=K, assumed positive.

Also dY dN

ð0;0Þ

¼

kL

s sdþ_d^s1

0Q₀

mþaþdþ^kL_sðsdÞ þ^ks_d0¹Q₀>0:

Using these aspects and plotting (12) and (13) in the first quadrant (see Fig. 4), we see that for the existence of nontrivial Yb andNb, the slope of (12) atð0;0Þ must be greater than the slope of (13) atð0;0Þ, i.e.

ðmþaþdÞ>kL s

ðarÞ

r s

dþs1

Q0

d0

; ð14Þ

which is same as (6) forQ₀¼Q_a. Thus, after knowingYb andNb, corresponding values ofCb andEb can be calculated as follows:

b C ¼L

sfsdþs₁bEg and Eb¼Q₀þlNb d0

:

Here the inequality (14) is the sufficient condition for existence of the fifth equilibrium pointE₅ðYb;Nb;C;b EbÞ.

Fig. 4. Existence of equilibrium point.

Remark. In both the cases when carrier population is absent, for disease to grow we must have a threshold condition as½bK ð1aÞr=ðmþaþdÞ>1, which is same as mentioned in [5].

2.2.1. Stability analysis

Now we discuss the linear stability of these equilibria and nonlinear stability only of the nontrivial equilibriumE5.

The local stability results of all equilibria are stated in the following theo- rem.

Theorem 4.The equilibriaE₁,E₂andE₃are unstable. The fourth equilibriaE₄is stable if

kCþmþaþd >r and ðarÞ

r kC>ðmþaþdÞ;

otherwise if

kCþmþaþd <r or ðarÞ

r kC<ðmþaþdÞ;

it is unstable and the fifth equilibriumE₅ exists. The fifth equilibrium is locally asymptotically stable provided

a₃ a₁ 1 a₂

>0 and

a3 a1 0 1 a₂ a₀ 0 a₃ a₁

>0;

wherea₀,a₁,a₂and a₃ are given in the proof of the theorem.

Proof. The variational matrices M1, M2, M3, M4 and M5 corresponding to system (11) at equilibrium pointsE1,E2,E3,E4andE5respectively are given by

M1¼

ðmþaþdÞ 0 0 0

a r 0 0

0 0 sdþs₁Q₀ d0

0

0 l 0 d0

0 BB B@

1 CC CA;

M₂¼

bK fmþaþdþ ð1aÞrg 0 kK 0

a r 0 0

0 0 sdþs₁QðKÞ

d0

0

0 l 0 d0

0 BB B@

1 CC CA;

M₃¼

m⁰₁₁ m⁰₁₂ kN 1r

a 1N K

0 a r2rN

K 0 0

0 0 sdþs₁QðNÞ d0

0

0 l 0 d0

0 BB BB BB B@

1 CC CC CC CA

;

where

m⁰₁₁¼ br aN 1

N

K

and m⁰₁₂¼nb

ð1aÞr K

or a l

N

K

N;

M4¼

ðkCþmþaþdÞ kC 0 0

a r 0 0

0 0 sC

L s₁C

0 l 0 d0

0 BB BB B@

1 CC CC CA

and

M₅¼

bYbþkNbCb Y

!

b ð1aÞr K

n o

b

Y þkCb kðNb YbÞ 0 a r2rNb

K 0 0

0 0 sCb

L s₁Cb

0 l 0 d0

0 BB BB BB BB BB

@

1 CC CC CC CC CC A :

Since the matricesM₁,M₂andM₃have positive eigenvalues so the equilibrium points corresponding to these matrices are unstable.

The characteristic polynomial corresponding to matrixM₄is ðwþd0Þ w

þsC

L

fw²þ ðkCþmþaþdrÞw rðkCþmþaþdÞ þakCg ¼0:

Clearly two roots are negative. Using the Routh–Hurwitz criteria [17], this equilibrium point is locally asymptotically stable if the following conditions are satisfied:.

kCþmþaþd >r and ðarÞ

r kC>ðmþaþdÞ; ð15Þ otherwise if

kCþmþaþd <r or ðarÞ

r kC<ðmþaþdÞ;

it is unstable and the fifth equilibrium exists as mentioned earlier in condition (14). It is clear from above that the second inequality in (15) may not be sat- isfied even if the first is satisfied, but when the first inequality is not satisfied, the second will not be satisfied. So violation of any one of the above in- equalities gives existence of the fifth equilibrium. The characteristic polynomial corresponding to matrixM5 is

w⁴þa₃w³þa₂w²þa₁wþa₀¼0;

where

a₃¼kðNbYbÞCb b

Y þ ðmþdÞ Yb

ðNbYbÞþ aYb² b

NðNbYbÞþ ð1aÞ þ rNbYb KðNbYbÞ þrNb

K þs

LCbþd0>0;

a2¼ bYbþkNbCb b Y

! r Kð2Nb n

h KÞo þs

LCbþd0

i

þr

Kð2NbKÞ s LCb n þd0

oþs

LCdb 0þahnb

ð1aÞr K

oYbþkCbi

;

a1¼ bYbþkNbCb b Y

!r

Kð2NbKÞ s LCb þd0

þr

Kð2NbKÞsCbd0

L

þ bYbþkNbCb b Y

!sCbd0

L þahnb

ð1aÞr K

oYbþkCbi sCb L þd0

!

;

a₀¼ bYbþkNbCb b Y

!r

Kð2NbKÞsCdb 0

L þasCb L d0hnb

ð1aÞr K

oYbþkCbi

þakðNb YbÞs1C l:b

By the Routh–Hurwitz criteria, conditions for local stability of the system are

a₃>0; a₃ a₁ 1 a₂

>0;

a₃ a₁ 0 1 a₂ a₀ 0 a₃ a₁

>0 and

a₃ a₁ 0 0 1 a₂ a₀ 0 0 a₃ a₁ 0 0 1 a₂ a₀

>0:

Clearly the first inequality is obvious. If the second and the third inequalities are satisfied, so is the fourth one. Hence the equilibrium point E₅ is locally asymptotically stable if the second and the third inequalities are satisfied.

Remark.It is seen that second is satisfied forNbPK=2 andb>ð1aÞr=K. So in this case only third inequality is the condition for local stability ofE5.

2.2.1.1. Nonlinear analysis and simulation. Here too we speculate that the nontrivial equilibrium pointE₅of the model (11) may be globally stable under the local stability conditions. To illustrate this and to see the effects of various parameters on the spread of the disease, the system (11) is integrated using the fourth order Runge–Kutta method by taking QðNÞ ¼Q0þlN and using the following set of parameters in the simulation, which satisfies the local stability condition mentioned above.

b¼0:00000031; k¼0:000000021; m¼0:012; a¼0:0005;

d¼0:6; d0¼0:001; r¼0:0003; d¼0:0004; a¼0:3;

K¼50000; s¼0:9; Q₀¼20; s₁¼0:000002;

l¼0:00005 and L¼100000:

All the parameters are in units of per day except the carrying capacityL, which has the same dimension asC. The equilibrium values of Yb, Nb, Cb and Eb have been found as

b

Y ¼6121:797; Nb ¼35716:819; Cb ¼38174:626; Eb ¼21785:839:

In Fig. 5, the infected population is plotted against the susceptible population and from the solution curves, it is concluded that the system appears to be globally stable for this set of parameters. In Figs. 6–11 the effects of various parameters, i.e. Q₀, L, s, s₁, r and l on the infective population have been shown. It is noted from these figures that as these parameter values increase, the infective population increases and we have similar conclusions regarding the spread of the infectious disease as discussed earlier.

Fig. 5. Variation of infective population with susceptible population.

Fig. 6. Variation of infective population with time for different cumulative environmental discharge rates.

Fig. 7. Variation of infective population with time for different carrying capacities of carrier population.

Fig. 8. Variation of infective population with time for different intrinsic growth rates of carrier population.

Fig. 9. Variation of infective population with time for different growth rate coefficients of carrier population due to the cumulative environmental discharges.

Fig. 10. Variation of infective population with time for different intrinsic growth rates of human population.

Fig. 11. Variation of infective population with time for differentl.

3. Conclusions

In this paper an SIS model for carrier-dependent infectious diseases, like cholera, diarrhea, etc. caused by direct contact of susceptibles with infectives as well as by carriers is proposed and analyzed. It is assumed that both the human and the carrier populations are growing logistically. The density of carrier population is further assumed to increase with the increase in the cumulative density of discharges by the human population into the environment. The mathematical model is analyzed for the following two cases: (i) the rate of cumulative environmental discharges Qis a constant, and (ii) the rate of cu- mulative environmental discharges Qis a function of the population density.

Equilibrium analysis is presented for both the cases and it is seen that the local stability of the nontrivial equilibria in both the cases is guaranteed only under certain conditions. By computer simulation it is shown that under local sta- bility conditions, the nontrivial equilibrium appears to be globally stable in both the cases. It is concluded from the analysis that if the growth of carrier population caused by conducive household discharges increases, the spread of the infectious disease increases. Also when the growth rate of the human population increases due to demographic changes, the infectious disease spreads even further and becomes more endemic.

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