J.Natn.Sci.Foundation Sri Lanka 2019 47 (2): 185 - 198 
DOI: 

Mathematical model for malaria with mosquito-dependent 

1 2* 3 4 
1 Department of Mathematical Sciences, Baba Ghulam Shah Badshah, University, Rajouri, India. 
2 Department of Mathematics, Lovely Professional University, Jalandhar, Punjab, India.
 Department of Mathematics, Indian Institute of Technology Roorkee, India.

4 Department of Mathematics, Govt. Post Graduate College, Rajouri, India.

* shoketali87@gmail.com  https://orcid.org/0000-0003-0383-7773)

This license permits use, distribution and reproduction, commercial and non-commercial, provided that the original work is 
properly cited and is not changed in anyway.

dynamics of malaria was proposed. In the model, mosquito 
population acts as the vector population and depends upon the 
human population for its growth and survival. It was shown 
that the environmental factors are conducive to the spread 
of malaria disease. It was also found that effective control 
programming against the spread of malaria is helpful in 
reducing the transmission dynamics of the disease. Sensitivity 
analysis was performed to show that spraying of pesticides and 
proper drainage system will effectively control the disease.

Human and vector population, stability analysis, 
transmission dynamics.

Malaria has become a worldwide problem and the 
 

109 countries. Every year there are approximately 250 
million cases of malaria reported, which result in one 
million deaths (WHO report 2013). 

transmission and several mathematical models have been 
formulated to investigate the transmission dynamics of 
malaria (Gupta et al

et al
et al et al

Erin et al et al., 
2013). Ngwa and Shu (2000) analysed a deterministic 
differential equation model for endemic malaria 
involving variable human and mosquito populations. 
Ngwa (2004) provided a solution by analysing the 
problem through the mathematical model. In the study 
the sample mathematical model has been considered and 
used a perturbation analysis and found that the death rate 

et al
2012) gave a model for infectious agents where they 
found the effect of immigration and disease induced 
death rates. The study of malaria transmission dynamics 
through mathematical modelling is a vast and fascinating 
area wherein the developed model depends on several 

a few areas like optimisation of malaria curing cost, role 
of awareness and follow-up cases, need to be researched. 

dependent on density. Mathematical models of the 
transmission of infectious agents in human communities 

Modelling of malaria epidemic has been carried out since 
et al. 

(2014) developed a mathematical model for malaria 
transmission among birds by changing vector behaviour. 
Fatmawati and Tasman (2015) studied a mathematical 
model for malaria transmission by considering the 
resistance of malaria parasites to the anti-malarial drugs 
and also incorporated mass treatment and insecticides 



186 Ram Singh et al.

June 2019 Journal of the National Science Foundation of Sri Lanka 47(2)

model for assessing the impact of anti-malaria drugs 
on the transmission dynamics of malaria was designed 
and qualitatively analysed by Forouzannia and Gumel 
(2015). Wang et al
model was developed to investigate the cause of the 
large abundance of mosquitoes in Guangdong province 

et al. (2017) developed a malaria model 
with an asymptomatic class in human population and 
exposed classes in both human and vector populations. 
The model assumes that asymptomatic individuals can 
get re-infected and move to the symptomatic class. Some 
of the developed models include density dependent death 
rates, environmental and other factors. However, in these 

dependent on the density of mosquitoes has not been 
considered. In this study, authors developed a susceptible, 

humans and a susceptible, exposed and infectious (SEI) 
model for mosquitoes and concluded that only human 

 Many mathematical models have been developed 
earlier by many researchers on transmission dynamics 
of malaria disease, however the exposed class has not 
been taken into consideration. The exposed class is a 

where 
the disease has been transmitted to the human, but its 
symptoms have not yet appeared. We have incorporated 
an exposed class of humans in our model and analysed 
its effects on dynamics. Our model also considered an 
exposed class for mosquito population which is also an 
important component of the study. This has not been 
studied earlier studied (Singh et al

In this model we assumed that there are two types 
of populations: (i) human population and (ii) vector 
(mosquito) population. The total number of individuals 
in the human population is denoted by N

h
 which is further 

subdivided into the following four subclasses:
(i) Susceptible human (S

h
) 

(ii) Exposed human (E
h
) 

(iii) Infected human (I
h
) 

h
). 

 It is assumed that people enter the susceptible class. 
When an infected mosquito bites the susceptible human, 

there may be more infected cases with some different 

moves to the recovered class. Once infected population 
is recovered, they return to the susceptible class. In this 
case total population is given as N

h
 = S

h
 + E

h
 + I

h h
.

 Similarly, the total vector population is denoted by 
N

v
 and subdivided into three classes viz. (i) susceptible 

mosquito (S
v
), (ii) exposed mosquito (E

v
) and (iii) 

infected mosquito (I
v
). Susceptible mosquitoes are 

recruited at a rate 2
the exposed class and after some time there will be more 
infected moves from exposed class to the infected class. 
In this case N

v
 = S

v
 + E

v
 + I

v 
.The transmission diagram 

of malaria is depicted in Figure (1). 

 For the mathematical formulation of the model, the 
following notations are used: 1 : recruitment rate i.e. 

2 : recruitment 
rate of mosquito population which also includes newly 

A
1 : transmission probability rate from susceptible 

2 : transmission 
probability rate from susceptible human to infected 

3 : transmission probability rate 

4 : transmission probability rate from susceptible vector 
h : progression rate from 

exposed class hE  to infected class hI v : progression rate 
from exposed class vE  to infected class vI : recovered 

: recovery rate for 
)( 21 dd : natural death rate of human 

population (mosquito population). 

d

d

d

d

d2

d2

d2

Rh Iv

Ih

Ev

Sv

Eh

Sh

h

1+ A

2

v

1ShEv

+ 2ShIv 3SvEh

+ 4SvIh



Journal of the National Science Foundation of Sri Lanka 47(2) June 2019

The following two cases have been taken into account to 
formulate the mathematical model. We will analyse the 
model under two different cases mentioned below:

 The transmission probability, i.e. the rate 1  at 
which infected mosquito bites the susceptible 
human is assumed to be constant

 Transmission probability 1  is a function of 
density of the mosquito population. It takes 
the form 0101 bNbb v  and is a positive 
constant.

The model is governed by the following system of 
differential equations

hhvvh
h SdSIERA

dt

dS
1211 )()(

hhhvv
h EdSIE

dt

dE
)()( 121

hhh
h IdE

dt

dI
)( 1

hh
h RdI

dt

dR
)( 1

11 h
h NdA

dt

dN

 ...(01)

vvhh
v SdSIE

dt

dS
2432 )(

vvvhh
v EdSIE

dt

dE
)()( 243

vvv
v IdE

dt

dI
2

v
v Nd

dt

dN
22

The developed mathematical model can be discussed in 
the following two cases:

 When 001  is a constant

Since hhhhh NRIES  and vvvv NIES  
then the system of equation (1) can be rewritten in the 
following form 

hhhhhhvv
h EdRIENIE

dt

dE
)())()(( 120

hhh
h IdE

dt

dI
)( 1

hh
h RdI

dt

dR
)( 1

11 h
h NdA

dt

dN

  
 ...(02)

vvvvvhh
v EdIENIE

dt

dE
)())()(( 243

vvv
v IdE

dt

dI
2

v
v Nd

dt

dN
22

The region of attraction of the above system is 

0:),,,,,,{(1 hhhvvvhhhh RIENIENRIE

}0, vvvvhh NNIENN

where
1

1suplim and
d

A
NN h

t
h

2

2suplim
d

NN v
t

v . 

Since we know that at equilibrium state all the derivatives 
vanish

i.e.  ,0
dt

dN

dt

dI

dt

dE

dt

dN

dt

dR

dt

dI

dt

dE vvvhhhh  

then the system of equations (2) becomes

0)())()(( 120 hhhhhhvv EdRIENIE

0)( 1 hhh IdE

0)( 1 hh RdI

011 hNdA

0)())()(( 243 vvvvvhh EdIENIE

02 vvv IdE

022 vNd
 ...(03)



188 Ram Singh et al.

June 2019 Journal of the National Science Foundation of Sri Lanka 47(2)

The following are three physically as well as biologically 
relevant equilibrium points:

(i)  Disease free equilibrium for only the human 

population is 0,0,0,,0,0,0
1

1
1 d

A
E

(ii)  Disease free equilibrium for both human and 

mosquito populations 
2

2

1

1
2 ,0,0,,0,0,0

dd

A
E

(iii)  Endemic equilibrium point is ,
~

,
~

,
~

3 hhh RIEE  

where,
~

,
~

,
~

,
~

vvvh NIEN

,

)(
~

))((
1

~

1
1112

2
0

2

2
0

hv
hhv

hv
v

h

dE
dddd

NE
d

E

 ...(4)

,
~

~

1d

E
I hh
h ,

))((

~
~

11 dd

E
R hh

h ,
~

1

1

d

A
Nh

    ...(5)

,

)(
~

1

~

~

2
11

4
3

1

4
3

vh
vh

vh
h

v

dE
dd

NE
d

E

,
~

~

2d

E
I vv
v .

~

2

2

d
Nv   ...(7)

 The equilibrium point 1E  is stable if 
all latent roots are negative. The equilibrium 2E  is 

stable if 0)4,2,1(ipi  and ,)( 4
2
13213 pppppp  and 

otherwise unstable. The equilibrium 3E  is stable if 

0)5,4,3,1(ii
 and ))(( 4

2
1

2
3

2
1321541 qqqqqqqqqq  

.)( 2
51

2
3215 qqqqqq

 The general variational matrix J corresponding to the system (2) is

2

2

4343243

20

43

1

1

1

202020120

00

0

)()()()(

000

000

000

0))(())((

0000

0000

00))(())((

000

0)(0

00)(

)()()()()(

d

d

IEIEdIE

RIENRIEN

IENIEN

d

d

d

IEIEIEdIE

J

v

hhhhvhh

hhhhhhhh

vvvvvv

h

vvvvvvhvv

0,0,0,,0,0,0
1

1
1 d

A
E , the variational matrix 

0J  is given by

2

2

2

1

1

1

1

1
2

1

1
01

0

000000

00000

00)(0000

000000

0000)(0

00000)(

0000)(

d

d

d

d

d

d
d

A

d

A
d

J

v

v

h

h



Journal of the National Science Foundation of Sri Lanka 47(2) June 2019

The characteristics polynomial of the Jacobian matrix is 
given by 0IJ

)()()(())(( 111
2

21 hddddd

 0))( 1 vd

Hence all these roots are negative. Thus the equilibrium 
point 1E  is stable.

2

2

1

1
2 ,0,0,,0,0,0

dd

A
E , 

the variational matrix 1J  is given by

2

2

2
2

2
4

2

2
3

1

1

1

1

1
2

1

1
01

1

000000

00000

00)(00

000000

0000)(0

00000)(

0000)(

d

d

d
dd

d

d

d
d

A

d

A
d

J

v

v

h

h

The characteristics polynomial of the Jacobian matrix is 
given by

0IJ

0}){)()(( 432
2

1
34

121 ppppadd

    ...(8)

where

,24321 daaap

,)( 3124324342322 vhNNdaaaaaaaaap

)( 4124342324323 vhh NNdaaaaaaaaap

 
,)( 312231 vhvvh NNNNda

()( 42221243423224324 vhhddaaaaaadaaap

 
,)2322231 vhv NNada

and

)(),(),(),( 24131211 vh dadadada

 The three roots of the polynomial given by equation 

equation give all roots with negative real part. Hence 
0)4,2,1(ipi  

and .)( 4
2
13213 pppppp  For these conditions the 

equilibrium points 2E is locally asymptotically stable.

vvhhhh IENRIEE ,
~

,
~

,
~

,
~

,
~

,
~

3

vN
~ ,  the variational matrix 2J  is given by

2

2

4343243

20

43

1

1

1

202020120

2

00

0

)
~~

()
~~

()()
~~

(

000

000

000

0))
~~~

(
~

())
~~~

(
~

(

0000

0000

00))
~~

(
~

())
~~

(
~

(

000

0)(0

00)(

~~
()

~~
()

~~
()()

~~
(

d

d

IEIEdIE

RIENRIEN

IENIEN

d

d

d

IEIEIEdIE

J

v

hhhhvhh

hhhhhhhh

vvvvvv

h

vvvvvvhvv



190 Ram Singh et al.

June 2019 Journal of the National Science Foundation of Sri Lanka 47(2)

The characteristics polynomial of the Jacobian matrix is 
given by

0IJ

0}){)(( 543
2

2
3

1
45

21 qqqqqdd

  ...(9)
where

,26543211 daaaaaaq

))(())(( 264532162642 daaaaaaadaaq v

 ( 32513121 aaaaaaaa ,) 5318752 aaaaa h

()))((( 321626453213 aaaadaaaaaaq v

 () 513121521 aaaaaaaaa ))( 2645232 daaaaaa

 )( 541872121121387 aaaaadaa hv

 ),( 26415 daaaah

)))((( 626452325131214 vadaaaaaaaaaaaaq

 ))(( 264521321 daaaaaaaa )( 2645h daaa

 ()( 21221138721121487 v dadaaaadaa

 121aa )(() 264152221 hvv daaaaaa

 ))( 6264 vadaa

)(()))((( 264562645213215 hv daaaadaaaaaaaaq

 () 2114876 v daaaa () 212138712 v daaaaa

 ))(() 6121645221 vhv aadaaaaaa

and

),(),(),(),( 24131211 vh dadadada

),(),( 436205 hhvv IEaIEa

)).(()),(( 87 vvvhhhh IENaRIENa

of the above equation give all roots with 

conditions are:  0)5,4,3,1(ii  and  )( 541

.)()( 2
51

2
32154

2
1

2
3

2
1321  For 

these conditions the equilibrium points 3E  is locally 
asymptotically stable.

 When 0101 bNbb v  and 1b  is a constant

Since hhhhh NRIES  and vvvv NIES   
the system of equation (1) can be rewritten in the 
following form
 

vhhhhv
h ERIENNbb

dt

dE
))()(( 10

 hhhhhhv EdRIENI )())(( 12

hhh
h IdE

dt

dI
)( 1

hh
h RdI

dt

dR
)( 1

11 h
h NdA

dt

dN

vvvvvhh
v EdIENIE

dt

dE
)())()(( 243

vvv
v IdE

dt

dI
2

v
v Nd

dt

dN
22

 ...(10)

 The region of attraction of the system given by 
equation (10) is
 

0:),,,,,,{(2 hhhvvvhhhh RIENIENRIE

 }0, vvvvhh NNIENN

where  
1

1suplim
d

A
NN h

t
h   and 

2

2suplim
d

NN v
t

v
.

Since we know that at equilibrium state all the derivatives 
vanish



Journal of the National Science Foundation of Sri Lanka 47(2) June 2019

i.e.  ,0
dt

dN

dt

dI

dt

dE

dt

dN

dt

dR

dt

dI

dt

dE vvvhhhh  

then the system of equations (10) becomes

(())()(( 210 hhvvhhhhv ENIERIENNbb

0)()) 1 hhhh EdRI

0)( 1 hhh IdE

0)( 1 hh RdI

011 hNdA

0)())()(( 243 vvvvvhh EdIENIE

02 vvv IdE

022 vNd ...(11)

The following are three physically as well as biologically 
relevant equilibrium points:

(i) When the exposed human population is infected 
then the disease free equilibrium only for human 

population is 0,0,0,,0,0,
1

1
1 d

A
EP h

(ii) When the exposed human population is infected only 

then the disease free equilibrium for both human and 

mosquito populations is 
2

2

1

1
2 ,0,0,,0,0,

dd

A
EP h

(iii) Endemic equilibrium point is ,
~

,
~

,
~

,
~

3 hhhh NRIEP  

,
~

,
~

,
~

vvv NIE  where

,

)(
~

))((
1

~

1
1112

2

2

2
10

1

1

2

2

2

2
10

hv
hhv

v
v

h

dE
ddddd

bb

d

A
E

dd
bb

E

 ...(12)

,

)(
~

))((
1)(

~
)(

~

1
1112

2
10

2

2
10

hv
hhv

v

hv
v

v

h

dE
dddd

Nbb

NE
d

Nbb

E

 ...(13)

,
~

~

1d

E
I hh
h ,

~
,

))((

~
~

1

1

11 d

A
N

dd

E
R h

hh
h

 ...(14)

,

)(
~

1

~

~

2
11

4
3

1

4
3

vh
vh

vh
h

v

dE
dd

NE
d

E
  

  ...(15)

.
~

,
~

~

2

2

2 d
N

d

E
I v

vv
v

 The equilibrium point 
1P  is stable if all 

latent roots are negative. The equilibrium 
2P  is stable if 
,)(and0)4,2,1( 4

2
13213 mmmmmmimi otherwise 

unstable and the equilibrium 3P  is stable if

)5,4,3,1(isi  0  and )( 541 sss ( 2
3

2
1321 sssss

.)() 2
51

2
32154

2
1 ssssssss



192 Ram Singh et al.

June 2019 Journal of the National Science Foundation of Sri Lanka 47(2)

The general variational matrix M corresponding to the system (10) is:

000

000

0))(())((

000

)(0

0)(

)()()()(

43

1

1

2102101210

vvvvvv

h

vvvvvvhvvv

IENIEN

d

d

IENbbIENbbdIENbb

M

2

2

4343243

1

210210

000

00

)()()()(0

000

0000

0000

0))(())()(()(

d

d

IEIEdIE

d

RIENRIENNbbIENbb

v

hhhhvhh

hhhhhhhhvvvv

0,0,0,,0,0,
1

1
1 d

A
EP h , the variational matrix 0M  is given by

2

2

3323

1

1

1

1

1
2

1

1
01

0

000000

00000

)(0000

000000

0000)(0

00000)(

0000)(

d

d

EEdE

d

d

d

E
d

A
E

d

A
bd

M

v

hhvh

h

hhh

The characteristics polynomial of the Jacobian matrix is given by

 
0IJ

0]))())[(()()()()(( 322311121 vhvhh EddEddddd

0])(

}){()[)()()()((

3223

223
2

11121

vhvh

vhh

EddE

ddEddddd

and the other two roots are obtained by the quadratic 
equations.

2

2

1

1
2 ,0,0,,0,0,

dd

A
EP h , 

the variational matrix 1M  is given by



Journal of the National Science Foundation of Sri Lanka 47(2) June 2019

The characteristics polynomial of the Jacobian matrix is 
given by

0IJ

){)()(( 2
2

1
34

121 mmadd

 0}43 mm  ...(17)

where,

,24321 daaam

2322244342322 dadadaaaaaaam

 
)( 32432 hv Edaaa

 
),)(( 103 hhvv ENNbbN

)( 24342324323 daaaaaaaaam

 
)()(()( 10223332 vvhv NbbdaNEaa

 ))( hh EN  )(32 hhvv ENN

 
 

),)(( 104 hhvvh ENNbbN

32233224324 vhv NdaEaadaaam

 
))(( 23210 vvhhv NaENNbb

 )( hh EN ))(( 1024 hhvvh ENNbbNd

 ),(4 hhvvh ENN

and

),(),(),( 131211 dadada h

).( 234 dEa vh

 The three roots of the polynomial given by 

biquadratic equation give all roots with negative 
are:  

)(and0)4,2,1( 4
2
13213 mmmmmmimi . For 

these conditions the equilibrium point 
2P  is locally 

asymptotically stable.

vhhhh ENRIEP ,
~

,
~

,
~

,
~

,
~

3
, 

vvv NI
~

,
~

,  the variational matrix 2M  is given by

43

1

1

2102101210

2

000

000

0))
~~

(
~

())
~~

(
~

(

000

)(0

0)(

~~
)

~
(

~~
)

~
()(

~~
)

~
(

IENIEN

d

d

IENbbIENbbdIENbb

M

vvvvvv

h

vvvvvvhvvv

2

2

3323
2

2
4

2

2
3

1

1

1

1

1
2

1

1

2

2
101

1

000000

00000

)(00

000000

0000)(0

00000)(

0000)(

d

d

EEdE
dd

d

d

d

E
d

A
E

d

A

d
bbd

M

v

hhvh

h

hhh



194 Ram Singh et al.

June 2019 Journal of the National Science Foundation of Sri Lanka 47(2)

2

2

4343243

1

210210

000

00

)
~~

()
~~

()()
~~

(0

000

0000

0000

0))
~~~

(
~

())
~~~

(
~

)(
~

(
~~

)
~

(

d

d

IEIEdIE

d

RIENRIENNbbIENbb

v

hhhhvhh

hhhhhhhhvvvv

)),((),( 7436 hhhhhh RIENaIEa

.)),(( 1098 vvvv NbbaIENa

above equation give all roots with negative real part. 

0)5,4,3,1(isi  and )( 541 sss ( 2
3

2
1321 sssss  

.)() 2
51

2
32154

2
1 ssssssss

 

 Under these conditions the equilibrium point 
3P  is 

locally asymptotically stable.

 The global stability of the above equilibriums can be 
computed by constructing the Lyapunov function.  

The numerical simulation is presented to validate the 
analytical results discussed previously. For this purpose, 

the sensitivity analysis. The dataset for case 1 was as 
follows:

  ;0.00000029 =0    ;0.00000021 = 2    ;0.00000015 =3  
;0.00000009 =4  0.00012; =1d    0.0085; = 2d  

  0.00012; =1  0.0085; = 2  0.00146; =     0.0085;=  
0.012; =1 12; =A 0.015; =2

and initial values were:

14000; = 20000; =209.2; = 4181.8; =500; = vhhhh IENRIE

963400. =30000; = vv NI ,

which are based on Gosh et al. (2005) and Hazarika and 

 Similarly, for case 2, the parameters values are taken 
as follows:

The characteristics polynomial of the Jacobian matrix is 
given by

0IJ

0}){)(( 543
2

2
3

1
45

21 sssssdd

   ...(18)

where

,26543211 daaaaaas

)(())(( 532162642 aaaaadaas v

 ())( 32513121264 aaaaaaaadaa

 ,) 5398752 aaaaaa h

)))((( 626453213 adaaaaaas v

 ()( 513121521321 aaaaaaaaaaaa

 
())( 293872645232 daaadaaaaaa

 ) 5498729291 aaaaaaaa hv

 ),( 26415 daaaah

))((( 26452325131214 daaaaaaaaaaaas

 ))(() 2645213216 v daaaaaaaaa

 
()( 91294872645h aadaaadaaa

 () 9212922913872 v aaadaadaaaa

 
)(() 264152221 hvv daaaaaa

 ))( 6264 vadaa

)))((( 62645213215 vadaaaaaaaas

 ())(( 29148762645 vh daaaaadaaa
 
 

)() 221292138712 vv aadaaaaaa

 ))(( 6121645 vh aadaaaa

and

),(),(),( 131211 hdadada

,)(),( 210524 vvvv IENbbada



Journal of the National Science Foundation of Sri Lanka 47(2) June 2019

0.00012;  =0b    ;0.000000061b    ;0.00000021  = 2  
  0.000024;  =3  000000009; =4  0.00012; =1d  

  0.0085; = 2d    0.00012; =1  0.00085; = 2 0.000146; =    
0.012;=    0.012; =1  A 0.015; =2  10; =A 

and initial values were:

500; =hE  4181.8; =hI  209.2; = hR  20000; =hN  
14000; = vE   30000; =v NI  963400. =vN

which are based on Gosh et al. (2005) and Hazarika and 

The effect of progression rate h , on infected human 
population (I

h
) is depicted in Figure 2. It is noted that 

initially when h  is low the infected human population 
is also less, but as time increases the effect of h  on I

h
 is 

reversed. 

the I
h
 decreases. 

0 100 200 300 400 500 600 700 800 900 1000
0

5

10

15

20

25

30

35

40

45

50

Time t

h= 0.012

h= 0.008

h= 0.004

 Variation in infective human population I
h
 with time 

for various values of progression rate h , when the 

Time t

0 50 100 150 200 250 300 350 400 450 500
0

1000

2000

3000

4000

5000

6000

7000

Time t

= 0.0085

= 0.0135

= 0.0185

 Variation in infective human population I
h
 with time for 

Time t

0 50 100 150 200 250 300 350 400 450 500
1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Time t

= 0.00085

= 0.00135

= 0.00185

= 0.00235

Variation in infective human population I
h
 with time for 

various values of the rate of immunity loss of recovered 

humans 

is a constant

Time t

0 50 100 150 200 250 300 350 400 450 500
1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

Time t

A= 9

A= 14
A = 19
A = 24

 Variation in infective human population I
h
 with time 

for various values of immigration constant A, when the 

Time t



196 Ram Singh et al.

June 2019 Journal of the National Science Foundation of Sri Lanka 47(2)

h
 is shown in 

h
 increases with 

h
 increases with the increase in 

rate of immunity loss of recovered humans , which is 
illustrated in Figure 5.
 

h
 is plotted against the various values 

of 
1

susceptible humans and exposed mosquitoes. Increase 
in 

1
, increases the I

h
. Whereas in Figure 7, I

h
 is plotted 

against the various values of 
2
 which is the interaction 

mosquitoes. 
2
 also follows the same trend as 

1
 with 

respect to I
h
. 

 The time-dependent patterns of infection and 
immunity are illustrated in Figure 8. With the increase 

0 50 100 150 200 250 300 350 400 450 500
2000

3000

4000

5000

6000

7000

8000

Time t

1=0.00000025

1=0.0000008

1=0.0000004

 Variation in infective human population I
h
 with time for 

1
, between 

susceptible humans and exposed mosquitoes, when the 

0 50 100 150 200 250 300 350 400 450 500
2000

3000

4000

5000

6000

7000

8000

9000

Time t

2=0.00000025

2=0.0000008

2=0.0000004

 Variation in infective human population I
h
 with time for 

2
, between 

susceptible humans and infective mosquitoes, when the 

Time t

Time t 0 500 1000 1500 2000 2500 3000
-50

0

50

100

150

200

Time t

v= 0.025

v= 0.015

v= 0.005

 Variation in infective human population I
h
 with time 

for various values of progression rate v , when the 

Time t

0 50 100 150 200 250 300 350 400 450 500
0

2000

4000

6000

8000

10000

12000

14000

16000

Time t

Ih(t)

Rh(t)

 Variation in infective population and recovered population 

humans is a constant.

Time t



Journal of the National Science Foundation of Sri Lanka 47(2) June 2019

and in the gradually decreases, due to the presence of 
non-linearity terms in the developed model. On the other 
hand, the proportion of recovered humans increases 
because of the fact that when some medicinal treatment 
is given to the infected human, its recovery rate grows 

Figures 10 and 11, show the variation in I
h
 with respect to 

b
0
 and b

1

population of humans increases with the increase in b
0
 

and b
1
. Variation in I

h
 for two values of N

v
(t) (888.8, 

1188.8) is given in Figure 12. It is noted that if N
v
 is high 

then I
h
 is also high.

proposed by considering both human and mosquito 
populations. The exposed class of both human and 
mosquito populations were included. The major factor 
which is responsible for the spread of malaria is the 

and the infected mosquito. Numerical results revealed 

with immunity loss  increases, then the number of 
infected humans increases, which are clearly illustrated 
in Figures 4 and 5. It has been concluded that effective 
control strategies against the malaria vector play an 
important role in the outburst of the disease. Proper 
drainage systems and insecticides can lead to elimination 
of mosquito population.     

Infectious Diseases of 
Humans. Oxford University Press. Oxford, UK. 

The Mathematical Theory of Epidemics, 
1
 = b

0
 + 

b
1
N

v
 is shown in Figures 9 to 12. In Figure 9 it is observed 

v
 decreases the I

h
 increases. 

0 500 1000 1500 2000 2500 3000
-50

0

50

100

150

200

250

Time t

b0=0.00012

b0=0.00008

b0=0.00004

 Variation in infective human population I
h
 with time for 

various values of b
0

humans is a variable

Time t

0 500 1000 1500 2000 2500 3000
-50

0

50

100

150

200

250

300

Time t

b1=0.00000024

b1=0.00000018

b1=0.00000010

 Variation in infective human population I
h
 with time for 

various values of b
1

humans is a variable

Time t

0 50 100 150 200 250 300 350 400 450 500
0

50

100

150

200

250

300

350

Time t

Nv(t)= 888.8

Nv(t)= 1188.8

 Variation in infective human population I
h
 with time for 

various values of N
v

humans is a variable

Time t



198 Ram Singh et al.

June 2019 Journal of the National Science Foundation of Sri Lanka 47(2)

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