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Abstract— This paper focuses on the development of a

deterministic Malaria transmission model by considering the

recovered population with and without immunity. A

transmission model is found to be useful in providing a better

understanding on the disease and the impact towards the

human population. In this research, two possibilities were

taken into account where one possibility is that infectious

humans do not gain immunity while another possibility is that

infectious humans will gain temporary immunity. The

mathematical model is developed based on the SEIR model

which has susceptible SH, exposed EH, infectious IH and

recovered RH classes. The system of equations which were

obtained were solved numerically and results were simulated

and analyzed. The analysis includes the impact of the different

values of the average duration to build effective immunity on

the infectious humans. We observed that when the value of q,

per capita rate of building effective immunity is increased, the

maximum number of infected humans decreased. Hence, if an

effective immunity can be build in a short period of time for

those who recover from the disease, the number of cases could

be reduced.

Index Terms—mathematical modeling, malaria, transmission

model, differential equations, immunity

I. INTRODUCTION

Malaria is one of the most common infections in the world

today. It is commonly caused by four species of protozoan

parasites of the genus Plasmodium : P.falciparum, P.vivax,

P.ovale and P.malariae [1]. Malaria is transmitted through

the vectors, Anopheles mosquitoes and not directly from

human to human. The disease infects humans of all ages and

can be lethal. According to the World Health Organization

(WHO) in year 2007, about 40% of the world’s population,

mostly those living in the poorest countries, are at risk of

malaria. Of the 2.5 billion people at risk, more than 500

Manuscript received May 9, 2009. This work was supported by Universiti

Malaysia Sarawak through research grant number 02(S34)/691/2009(07) to

J. Labadin.

J. Labadin is a senior lecturer in the Department of Computational

Science and Mathematics, Faculty of Computer Science and Information

Technology, Universiti Malaysia Sarawak, 94300 Kota Samarahan,

Sarawak, Malaysia (phone: +60 82 583775; fax: +60 82 583764; e-mail:

ljane@fit.unimas.my).

C. Kon M. L. is a postgraduate student in the Department of

Computational Science and Mathematics, Faculty of Computer Science and

Information Technology, Universiti Malaysia Sarawak, 94300 Kota

Samarahan, Sarawak, Malaysia (e-mail: cynkonml@hotmail.com) . C. Kon

M. L. Master candidature was supported by Zamalah Postgraduate

UNIMAS.

S. F. S. Juan is a lecturer in the Department of Computational Science and

Mathematics, Faculty of Computer Science and Information Technology,

Universiti Malaysia Sarawak, 94300 Kota Samarahan, Sarawak, Malaysia

(e-mail: sfsjuan@fit.unimas.my).

million become severely ill with malaria every year and more

than 1 million die from the effects of the disease.

At present, malaria is endemic in most tropical countries

including America, Asia and Africa. It remains a public

health concern in many countries in South East Asia. Apart

from the four common species mentioned above, simian

malaria, P.inui, P.cynomolgi, and P.knowlesi are also known

to cause the disease in humans [2]. Cases of malaria in the

Kapit division Sarawak has been detected to be caused by

P.knowlesi [3]. This malaria parasite which infects

long-tailed and pig-tailed macaque monkeys in nature had

accounted for half of the cases studied in the Kapit division

[3].

Mathematical models for transmission dynamics of malaria

are useful in providing a better knowledge of the disease, to

plan for the future and consider appropriate control measures.

Models have played great roles in the development of the

epidemiology of the disease. The study on malaria using

mathematical modeling originated from the works of Ross

[3]. According to Ross, if the mosquito population can be

reduced to below a certain threshold then malaria can be

eradicated. MacDonald did some modification to the model

and included superinfection [4]. He showed that reducing the

number of mosquitoes have little effect on epidemiology of

malaria in areas of intense transmission. Dietz et al [5] added

two classes of humans in their mathematical model, namely

those with low recovery rate (more infections, greater

susceptibility) and high recovery rate (less infections, less

susceptibility). Compartmental models of malaria and

differential equations are constructed to model the disease

[7,8,13,14,20]. Chitnis et al [13] did a bifurcation analysis of

a malaria model. Malaria transmission model which

incorporate immunity in the human population had been

studied [7,8,14]. Epidemiological models on the spread of

anti-malarial resistance were also constructed [15].

In this paper, we present the malaria transmission model in

Section II, where we took into account two possibilities: one

is where infectious humans do not gain any immunity and the

other, who have temporary immunity. After which, the model

is simulated and the impact of changing the rate to build

effective immunity and other parameters are studied

numerically in Section III. We based our work on Malaria

cases in general and not specifically on particular parasite

genus. Finally, concluding remarks are discussed in Section

IV.

II. MODEL FORMULATION

A malaria transmission model has been produced based on

the epidemiology aspects of the disease. The compartmental

model is as shown in Fig. 1. The human population is divided

into the SEIR compartmental model which consists of four

Deterministic Malaria Transmission Model with

Acquired Immunity

J. Labadin, C. Kon M. L. and S. F. S. Juan

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol II

WCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-18210-2-7

WCECS 2009

classes: susceptible SH, exposed EH, infectious IH and

recovered RH. Blood meal taken by an infectious female

Anopheline mosquito on a susceptible individual will cause

sporozoites to be injected into the human bloodstream and

will be carried to the liver. The individual will then move to

the exposed class EH. This will decrease the susceptible

population SH. Exposed humans are those who have parasites

in them and the parasites are in asexual stages. They are

without gametocytes and are not capable of transmitting the

disease to susceptible mosquitoes. After the latent period,

humans who are exposed will be transferred to the infectious

class as they are with gametocytes in their blood stream

making them infectious to female Anopheles mosquitoes.

The infectious humans will recover after some time, gain

immunity and move to the recovered with temporary

immunity class or they can be susceptible again. This is

because continuous exposure is necessary to ensure

immunity is built [7]. Those who have recovered have

immunity against the disease for a certain period. Acquired

immunity exists but the mechanisms are yet to be fully

apprehended [22]. As the immunity is temporary, it will fade

off after a period of time [7]. Thus, the recovered humans will

return to the susceptible class. Every class of human

population is decreased by density-dependent and

independent death and emigration except for the infectious

class which has disease-induced death as an addition.

For the vector mosquitoes, the three compartments

represent susceptible SM, exposed EM, and infectious IM.

There is no recovered class for the vector as mosquitoes

never recover [13]. They are regulated by mortality [8].

Susceptible mosquitoes that feed on infectious or recovered

human would have taken gametocytes in blood meal but do

not have sporozoites in their salivary glands yet, thus this

means they are entering into the exposed class. After

fertilization, sporozoites are produced and migrate to the

salivary glands ready to infect any susceptible host, the

vector is then considered as infectious. The three

compartments for the vector mosquitoes are reduced by

density-dependent and independent death and emigration.

In the malaria model which has been constructed here, the

total number of mosquito bites is restricted by total mosquito

population whereas in [13] the total bites on humans is

dependent on both human and mosquito population. In our

model, the mosquito-human interaction follows the classic

model as mentioned in Hethcote’s review [9]. The

differential equations which describe the dynamics of malaria

in human and mosquito populations are formulated based

from the compartmental diagram described in Figure 1. The

descriptions for the variables used in the model are shown in

Table 1 and parameters in Table 2.

The following assumptions are made to characterize the

model:

(i) All newborns are susceptible to the disease.

(ii) The infectious period of mosquitoes ends when they die.

(iii) The lifespan of the mosquitoes does not depend on

infection.

(iv) Human hosts recover from infection (without immunity)

and move right back to susceptible state OR gain temporary

immunity before losing it and returning to the susceptible

class.

(v) Duration of building effective immunity is right after the

duration of recovery from the disease.

(vi) Recovered humans are still able to transmit the disease

but at a lower rate.

(vii) Duration of latent period and immune period are

constant.

(viii) Human and mosquito populations are not constant.

Figure 1: The compartmental model of the Malaria transmission of host human and vector mosquito

Following the compartmental model in Figure 1 and according to the Balance Law, the differential equations describing the

transmission of the disease are as follows:

()

HHHH

H

M

MHHH

HSNDDrIS

N

I

cRbNm

dt

dS 21 +−+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−++=

β

(1)

()

HHHH

H

M

MH

HENDDLES

N

I

dt

dE 21 +−−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

β

(2)

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol II

WCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-18210-2-7

WCECS 2009

()

HHHHHH

HINDDdIrII

rq

qr

LE

dt

dI 21 +−−−

+

−=

(3)

()

HHHH

HRNDDcRI

rq

qr

dt

dR 21 +−−

+

=

(4)

()

MMM

H

H

HMM

H

H

HMM

MSNS

N

R

S

N

I

BN

dt

dS 21

~

δδββ

+−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−=

(5)

()

MMMM

H

H

HMM

H

H

HM

MENuES

N

R

S

N

I

dt

dE 21

~

δδββ

+−−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

(6)

()

MMM

MINuE

dt

dI 21

δδ

+−=

(7)

Assumption (i) and (viii) above implies that the total

human population, NH is the summation of all the four

human population compartments in Figure 1. This means

that HHHHH RIESN +++= . Also from assumption (viii),

the total mosquito population, NM is the summation of the

three mosquito population compartments

i.e. MMMM IESN ++= . Thus, the rates of change of the

total human population and the mosquito population are

()

HHHH

HNNDDdIbNm

dt

dN 21 +−−+= ,

(8)

()

MMM

MNNBN

dt

dN 21

δδ

+−= ,

(9)

respectively.

Table I. Description of variables for transmission model

------------------------------------------------------------------------

---

Variable Description

------------------------------------------------------------------------

---

SH number of susceptible human hosts at time t

EH number of exposed human hosts at time t

IH number of infectious human hosts at time t

RH number of recovered human with temporary

immunity at time t

SM number of susceptible mosquito vectors at

time

t

EM number of exposed mosquito vectors at time t

IM number of infectious mosquito vectors at time

t

NH total human population

NM total mosquito population

Table II. Description of parameters for transmission model

------------------------------------------------------------------------

---

Parameter Description

------------------------------------------------------------------------

---

b per capita birth rate of humans (per time)

m immigration rate of humans (per time)

B per capita birth rate of mosquitoes (per time)

L per capita rate of progression of human from

exposed class to infectious class (per time)

1/L average duration of latent period in humans

u per capita rate of progressions of mosquitoes

from exposed class to infectious class (per

time)

1/u average duration of latent period in

mosquitoes

q per capita rate of building effective immunity

(per time)

1/q average duration to build effective immunity

c per capita rate of loss of immunity in human

(per time)

1/c average duration of immune period

r per capita rate of recovery (per time)

1/r average duration of recovery from disease

d per capita rate of disease-induced death in

human (per time)

e proportion of bites on man that produces an

infection (from mosquito Î human)

E proportion of bites on man that causes

infection in mosquitoes (from infectious

human Î susceptible mosquito)

E

~

proportion of bites on man that causes

infection in mosquitoes (from recovered

human Î susceptible mosquito)

F average number of bites per mosquito (per

time)

1

D density-independent death and emigration

rate

for humans (per time)

2

D density-dependent death and emigration rate

for humans (per time)

1

δ

density-independent death and emigration for

mosquitoes (per time )

2

δ

density-dependent death and emigration for

mosquitoes (per time )

All parameters of the model are assumed to be non-negative.

The total populations are assumed to be positive for t = 0.

β

MH is the average number of mosquito bites which causes

transmission of disease (from infectious mosquito

Îsusceptible human) per mosquito (per time)

β

MH = eF (10)

β

HM is the average number of mosquito bites which causes

transmission of disease (from infectious human Î

susceptible mosquito) per mosquito (per time)

β

HM = EF (11)

β

~

HM is the average number of mosquito bites which causes

transmission of disease (from recovered human Î

susceptible mosquito) per mosquito (per time)

β

~

HM = E

~ F (12)

I. MODEL ANALYSIS

The system of equations (1)-(9) is nonlinear coupled ordinary

differential equations which need to be solved numerically.

This is easily achievable given that the initial values for all

the variables are specified and the parameter values indicated

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol II

WCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-18210-2-7

WCECS 2009

appropriately. As mentioned before, there is numerous

clinical research works on Malaria was done. Thus, for the

purpose of simulating our model, we have taken the

parameters values from various sources as cited accordingly:

m = 1.217 × 10-3 [18], b = 2.417 × 10-3 [18], B = 4.227 [17],

L = 3.0438 [12], u = 3.04375 [17] , c = 8.333 × 10-2 [9],

d = 1× 10- 7[6], e = 1.2 × 10-2 [16], δ1 = 3.623 [13],

δ2 = 6.722 × 10- 7[13], E = 4.7× 10-1 [11],

E

~

= 2.35x 10-1 [10],

F = 7.609 [11], D1= 4.808 x 10-4 [19], D2 = 1.000 x 10-5[13],

q = 8.333× 102, r = 5.558× 10-2 [21],

where the unit of time is in month.

From the cases studied in [10], the number of cases with

immunity is half of the cases without immunity. Hence, we

took the probability of transmission from recovered human to

susceptible mosquito, E

as half of that of from infectious

human to susceptible mosquito, E.

Our model considered some infectious humans recover

without any gain of immunity [7, 20] and some do acquire

immunity. Here, we considered the average duration to build

effective immunity. Our analysis includes the impact of the

different values of the average duration to build effective

immunity, 1/q, on the infectious human population, IH. This

analysis is required since the actual mechanism of acquired

immunity is yet to be understood [22, 23]. The value for q is

not available and thus, we prescribed with a value which best

fit on actual cases (see figure 4 later).

To run our simulation, we have prescribed the initial

condition as SH =18000, EH = 0, IH =40, RH =35, SM = 9000,

EM =0 and IM =1000. Figure 2 shows the predicted human

populations in the susceptible, infectious and recovered cases

given the initial values above. We can see that Malaria is

contagious based from the gradient of the susceptible curve.

By the third month, the disease has infected half of the human

population. This may be a possibility if there is no

intervention to curb the spread of the disease. The outbreak

reached its peak around the fifth month and then gradually

decreases. From this simulation, we also observe that the

disease prevails, that is as time progresses the infectious

population seems to arrive to a limiting value. In order to

study whether the disease will prevail or not, the basic

reproduction number needs to be obtained which is not

covered in this paper. The numerical results are verified by

finding the steady state equilibrium points analytically for the

human populations and then compare them with the

numerical results.

The steady state equilibrium points are reached when the

differential equations do not change with time. Therefore, to

find steady state, we set all the differential equations (1-9) to

zero. That is,

=

dt

dS H=

dt

dEh=

dt

dIh=

dt

dRh=

dt

dSM=

dt

dEm=

dt

dIm

=

dt

dNH=

dt

dNM0 (13)

Having solved (13) we get the equilibrium point which is

denoted by Ee below

(

)

**,**,*,*,*,*,*, MHMMMHHHHe NNIESRIESE =,

(14)

where

Notice that the functions for NH*, RH*, EH* and SH* are

depending on IH*. Thus, to check our numerical results, we

take the value obtain for IH* numerically and substitute it into

the functions NH*, RH*, EH* and SH* in equation (14).

Together with the prescribed values of all parameters

involved, we can then calculate the value for NH*, RH*, EH*

and SH*. These values are compared with the numerical

results and we found them to be agreeable.

Numerical comparison of our model and that of Chitnis et

al. [13] had been done in our previous work [24]. The paper

[24] presented the differences and analysis between Chitnis

model and ours.

0 5 10 15 20 25

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time (mont hs)

Humans

Plot of Hum ans against Time

Infectious

Susceptible

Recovered

Figure 2: The predicted human populations for infectious, susceptible

and recovered over time in months.

As mentioned earlier, one of the parameter which is still

unknown is the average duration to build effective immunity

1/q. Figure 3 shows the sensitivity of the parameter q towards

the total infectious humans.

() ( )

()

()( )

() ( )

1

2

2

11 2

2

12

12

*,0

4*

*,

2

**

*,

***

*,

**

M

H

H

MM

M

MMH

M

HM H HM H

B

N

bD D b DdI m

ND

NI

Eu

uB N I N

SuI R

δ

δ

δδ

δδ

ββ

−

=

−+ − − −

=

+

=

++

=⎡⎤

+

⎣⎦

()

()

12

12

12

*

*,

*

**

*,

***

*.

*

H

H

H

HH

H

HHH

H

MH M

lI

RcD DN

lrd D DN I

EL

LD DN E N

SI

β

=++

++ + +

=

++

=

Proceedings of the World Congress on Engineering and Computer Science 2009 Vol II

WCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-18210-2-7

WCECS 2009

0 5 10 15 20 25

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time (months)

Infectious Humans

Plot of Infectious Humans against Time

q = 83.33

q = 0.08333

q = 0.0000083

Figure 3: The total infectious humans when the value for per capita rate

of building effective immunity per month varies.

It is observed that when the value for q is increased, the

maximum number of infected humans decreased. This shows

that if the average duration of acquiring immunity is small

then the infected population is reduced significantly. Since

this parameter is not available yet from the clinical research

for Malaria, then we will use the actual incidences of Malaria

in Malaysia to predict this value. Figure 4 depicts this result.

Here, we found that the best value for q is 83.33. This means

that the average duration to build effective immunity is

approximately 9 hours after an infectious person has

recovered from the disease.

As mentioned by Doolan et al. [23], the understanding of

the rate of onset of acquired immunity is not clear. This is

because of disagreements over relationship between

exposure to infection, antigenic polymorphism and naturally

acquired immunity [25]. However, with the understanding of

naturally acquired immunity, experimental induced

immunity against malaria can be duplicated to protect those

who are exposed to the disease.

Figure 4: Yearly cases of Malaria in Malaysia from the year 1993 to

2007 (bar chart) and the predicted values (♦)

II. CONCLUSION

A malaria transmission model had been formulated and

analyzed numerically. We can clearly see in our numerical

analysis, that if the ability to build effective immunity is fast

for those who recover from the disease, the number of cases

can be reduced. We compared our simulation results with the

actual malaria incidences in Malaysia (1993-2007) which is

taken from Vector-Borne Diseases Section, Disease Control

Division, Department of Public Health, Ministry of Health

Malaysia. We performed our simulation in the absence of

immunity and gradually increase the value of q, per capita

rate of building effective immunity to estimate the value

which is suitable. The value for q is found to be 83.333 which

means the average duration to build effective immunity is

approximately 9 hours after an infectious person has

recovered from the disease. Therefore, after a person has

recovered from the disease, safety precautions should still be

taken for the first half of the day as the person can still be

susceptible to Malaria. Also, the knowledge of the onset of

acquired immunity can contribute to the research on the

mechanism of the immunity and to develop effective

vaccines for Malaria. Hence, malaria morbidity can be

controlled by immunological means.

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Proceedings of the World Congress on Engineering and Computer Science 2009 Vol II

WCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-18210-2-7

WCECS 2009

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Proceedings of the World Congress on Engineering and Computer Science 2009 Vol II

WCECS 2009, October 20-22, 2009, San Francisco, USA

ISBN:978-988-18210-2-7

WCECS 2009