Mathematical and Computer Modelling 49 (2009) 1148–1155 Contents lists available at ScienceDirect Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm A with-in host Dengue infection model with immune response Nuning Nuraini ∗ , Hengki Tasman 1 , Edy Soewono, Kuntjoro Adji Sidarto Industrial and Financial Mathematics Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung 40132, Indonesia article info Article history: Received 1 March 2008 Received in revised form 23 April 2008 Accepted 26 June 2008 Keywords: With-in host model Dengue viral infection Basic reproduction ratio Bifurcation diagram abstract A model of viral infection of monocytes population by Dengue virus is formulated here. The model can capture phenomena that dengue virus is quickly cleared in approximately 7 days after the onset of the symptoms. The model takes into account the immune response. It is shown that the quantity of free virus is decreasing when the viral invasion rate is increasing. The basic reproduction ratio of model without immune response is reduced significantly by adding the immune response. Numerical simulations indicate that the growth of immune response and the invasion rate are very crucial in identification of the intensity of infection. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Dengue is an infectious mosquito-borne viral disease. It is estimated that about 50 million infections occur annually in over 100 countries [21]. There is no specific treatment for curing dengue patients. Hospital treatment in general is given as supportive care which includes bed rest, antipyretics, and analgesics. Most dengue infections are asymptomatic. Few of them suffer dengue fever and dengue haemorrhagic fever, which may end up in fatality. Dengue virus is one of the most difficult arboviruses to isolate. There are four serotypes of the dengue virus, those are Den-1, Den-2, Den-3, Den-4, and each of the serotype has numerous virus strains. Infection with one dengue serotype may provide lifelong immunity to that serotype, but there is no cross-protective immunity to other serotypes [11]. Identification of the primary target cells of dengue virus replication in infected human body has proven to be extremely difficult. It is generally believed that the target cells of dengue virus are monocytes or its differentiated cells the macrophages [12]. The incubation period of the disease in an infected host is 3–14 days (average 4–7 days). At the end of the incubation period, the patient may experience a sudden onset of fever. Viraemia is the presence of virus in the blood stream. It is detected using the mosquito inoculation technique. Detectable viraemia is assumed to start on the second or the third day before the onset of symptoms and ends on the last day of illness. It usually peaks at the time of or shortly after the onset of illness [10,11,17]. Susceptible mosquitoes can be infected when they bite Dengue infected hosts during the febrile viremic stage. It is usually believed that dengue virus is quickly cleared in human body within approximately 7 days after the day of sudden onset of fever [22,17]. Naturally, this clearing process is done by the immune system which is a result of complex dynamic reactions. In this paper we try to understand the process using a mathematical model. Generally, there are two modeling approaches on mathematical epidemiology, the homogeneous and the heterogeneous approaches. In the homogeneous approach we usually use coupled ordinary differential equations which average the heterogeneous aspects, such as spatial aspects and age-structure. The heterogeneous models can give a deeper description on the spread of infectious diseases relatively with respect to homogeneous models [2,5]. ∗ Corresponding author. E-mail address: nuning@math.itb.ac.id (N. Nuraini). 1 Origin institution: Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok 16424, Indonesia. 0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.06.016