Content uploaded by J. Labadin
Author content
All content in this area was uploaded by J. Labadin on Nov 11, 2014
Content may be subject to copyright.
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.
REFERENCES
[1] Kathryn N.Suh, Kevin C.Kain, Jay S.Keystone, “Malaria”, CMAJ ,
May 25 2004
[2] Balbir Singh, Lee Kim Sung, Asmad Matusop, Anand Radhakrishnan,
Sunita S.G Shamsul, Janet Cox-Singh, Alan Thomas, David J Conway,
“A large focus of naturally acquired Plasmodium knowlesi infections in
human beings”, The Lancet , 2004
[3] Ross R, The prevention of malaria, London: John Murray, 1910
[4] Macdonald G, The epidemiology and control of malaria, Oxford:
Oxford University Press , 1957
[5] Dietz K, L. Molineaux, A. Thomas, “A malaria model tested in Africa
Savannah”, Bull, WHO , 1974
[6] Malaysia Ministry of Health, Annual Report 1993
[7] J. Tumwiine, J.Y.T. Mugisha, L.S. Luboobi, “A mathematical model
for the dynamics of malaria in a human host and mosquito vector with
temporary immunity”, Elsevier Applied Mathematics and
Computation, 2007
[8] C. Chiyaka, W. Garira, S. Dube, “Transmission model of endemic
human malaria in a partially immune population”, Elsevier
Mathematical and Computer Modelling, 2007
[9] WHO website, “Frequently Asked Questions, Malaria”, Global
Malaria Programme, Available : http://apps.who.int/malaria/faq.html
[10] A.P.K. deZoysa, C.Mendis, A.C.Gamage-Mendis, S.Weerasinghe,
P.R.J.Herath, K.N.Mendis, “A mathematical model for Plasmodium
vivax malaria transmission : estimation of the impact of
transmission-blocking immunity in an endemic area”, Bulletin of the
World Health Organization, 1991 pp. 725-734
[11] Heinz Mehlhorn, Philip M.Armstrong, Encyclopedic Reference of
Parasitology : Diseases, Treatment, Therapy, Springer, 2001
[12] R.M. Anderson and R.M. May, Infectious Diseases of Humans :
Dynamics and Control, Oxford University Press, Oxford, UK, 1991
[13] Nakul Chitnis, J. M. Cushing, J. M. Hyman, “Bifurcation analysis of a
mathematical model for malaria transmission”, SIAM J. APPL. MATH ,
Volume 67 , Society for Industrial and Applied Mathematics, 2006
[14] Hyun M Yang, “Malaria transmission model for different levels of
acquired immunity and temperature-dependent parameters (vector)”,
Journal of Public Health , Volume 34, 2000
[15] JC Koella, R Antia, “Epidemiological models for the spread of
anti-malarial resistance”, Malaria Journal , 2003
[16] J.Nedelman, “Inoculation and recovery rates in the malaria model of
Dietz, Molineaux and Thomas”, Mathematical Biosciences, 1984, pp.
209-233
[17] Francis E. G. Cox, Modern Parasitology, A Textbook of Parasitology,
Blackwell Publishing, 1993
[18] Cheong H Tan, Indra Vythilingam, Asmad Matusop, Seng T Chan,
Balbir Singh , Bionomics of Anopheles latens in Kapit, Sarawak,
Malaysian Borneo in relation to the transmission of zoonotic simian
malaria parasite Plasmodium knowlesi, Malaria Journal, 2008
[19] Central Intelligence Agency, “World Factbook 1993”, Libraries of the
University of Missouri- St. Louis. Available :
http://www.umsl.edu/services/govdocs/wofact93/wf940147.txt
[20] G.A. Ngwa and W.S. Shu, “A mathematical model for the endemic
malaria with variable human and mosquito populations”, Math.
Comput. Modeling, 2000, pp.747-763
[21] Bloland, P.B, Williams, H.A, Roundtable on the Demography of
Forced Migration, and Joseph L. Mailman School of Public Health,
Malaria Control during mass population movements and natural
disasters, National Academies Press, Washington, DC, 2002
[22] Jia Li, “A malaria model with partial immunity in humans”,
Mathematical Biosciences and Engineering, 2008, pp.789-801
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
[23] Doolan, D. L., Dobano, C., Baird, J. K., “Acquired Immunity to
Malaria”, Clinical Microbiology Review, Jan 2009
[24] C. Kon M.L and J. Labadin, Mathematical Modeling of the
Transmission Dynamics of Malaria, Proceedings of the 5th Asian
Mathematical Conference, Malaysia, 2009
[25] J.K Baird, “Host age as a determinant of naturally acquired immunity
to Plasmodium Falciparum”, Volume 11, Parasitology Today, 1995,
pp.105-111
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
