**3. Dengue fever epidemiology**

Dengue is a viral mosquito-borne infection which in recent years has become a major international public health concern. According to the estimates given by (PDVI, 2011), 3.6 billion (55% of world population) are at risk of acquiring dengue infection (see Fig. 8a)). It is estimated that every year, there are 70 − 500 million dengue infections, 36 million cases of dengue fever (DF) and 2.1 million cases of dengue hemorragic fever (DHF), with more than 20.000 deaths per year (CDC, 2011; PDVI, 2011; WHO, 2009). In many countries in Asia and South America DF and DHF has become a substantial public health concern leading to serious social-economic costs.

Fig. 8. Worldwide Dengue distribution 2010. In red Countries and areas where dengue has been reported Data source: World Health Organization (WHO) & Centers for Disease Control and Prevention (CDC). Adapted from (Gubler, 2002; Mackenzie et al., 2004).

Dengue fever is transmitted by the female domestic mosquito Aedes aegypti, although Ae. albopictus and Ae. polynesiensis can also act as transmission vector (Favier et al., 2005). Virus transmission in its simplest form involves the ingestion of viremic blood by mosquitoes and passage to a second susceptible human host. The mosquito becomes infected when taking a blood meal from a viremic person. After an extrinsic incubation period, the mosquito becomes infective and remains so during its entire life span (Rigau-Pérez et al., 1998). As the blood meal stimulates ovoposition, which undergoes at least one, often more, reproductive cycles there is an opportunity of vertical transmission to the eggs, passing the virus to the next generation of mosquitoes (CDC, 2011; Monath, 1994; Rosen et al., 1983).

There are four antigenically distinct dengue viruses, designated DEN-1, DEN-2, DEN-3, and DEN-4 (Guzmán et al., 2010; Halstead, 1994; SES, 2010; WHO, 2009). Infection by one serotype confers life-long immunity to only that serotype and a short temporary cross-immunity period to other serotypes exists. It lasts from three to nine months, when the antibody levels created during the response to that infection would be enough to protect against infection by a different but related serotype (Dejnirattisai et al., 2010; Halstead, 1994; Matheus et al., 2005; SES, 2010; WHO, 2009). Two variants of the disease exist: dengue fever (DF), a non-fatal form 10 Will-be-set-by-IN-TECH

framework for epidemiological systems, is still the SIR type model, a good and simple model for many infectious diseases. However, different extensions of the classical single-strain SIR model show a rich dynamic behavior, e.g. (Stone et al., 2007) in measles, or in generalized multi-strain SIR type models to describe the epidemiology of dengue fever (Aguiar et al.,

Dengue is a viral mosquito-borne infection which in recent years has become a major international public health concern. According to the estimates given by (PDVI, 2011), 3.6 billion (55% of world population) are at risk of acquiring dengue infection (see Fig. 8a)). It is estimated that every year, there are 70 − 500 million dengue infections, 36 million cases of dengue fever (DF) and 2.1 million cases of dengue hemorragic fever (DHF), with more than 20.000 deaths per year (CDC, 2011; PDVI, 2011; WHO, 2009). In many countries in Asia and South America DF and DHF has become a substantial public health concern leading to serious

Fig. 8. Worldwide Dengue distribution 2010. In red Countries and areas where dengue has been reported Data source: World Health Organization (WHO) & Centers for Disease Control and Prevention (CDC). Adapted from (Gubler, 2002; Mackenzie et al., 2004).

Dengue fever is transmitted by the female domestic mosquito Aedes aegypti, although Ae. albopictus and Ae. polynesiensis can also act as transmission vector (Favier et al., 2005). Virus transmission in its simplest form involves the ingestion of viremic blood by mosquitoes and passage to a second susceptible human host. The mosquito becomes infected when taking a blood meal from a viremic person. After an extrinsic incubation period, the mosquito becomes infective and remains so during its entire life span (Rigau-Pérez et al., 1998). As the blood meal stimulates ovoposition, which undergoes at least one, often more, reproductive cycles there is an opportunity of vertical transmission to the eggs, passing the virus to the next generation of

There are four antigenically distinct dengue viruses, designated DEN-1, DEN-2, DEN-3, and DEN-4 (Guzmán et al., 2010; Halstead, 1994; SES, 2010; WHO, 2009). Infection by one serotype confers life-long immunity to only that serotype and a short temporary cross-immunity period to other serotypes exists. It lasts from three to nine months, when the antibody levels created during the response to that infection would be enough to protect against infection by a different but related serotype (Dejnirattisai et al., 2010; Halstead, 1994; Matheus et al., 2005; SES, 2010; WHO, 2009). Two variants of the disease exist: dengue fever (DF), a non-fatal form

mosquitoes (CDC, 2011; Monath, 1994; Rosen et al., 1983).

2008).

**3. Dengue fever epidemiology**

social-economic costs.

of illness, and dengue hemorrhagic fever (DHF), which may evolve toward a severe form known as dengue shock syndrome (DSS).

Epidemiological studies support the association of DHF with secondary dengue infection (Guzmán et al., 2000; Halstead, 1982, 2003; Nisalak et al., 2003; Vaughn, 2000), and there is good evidence that sequential infection increases the risk of developing DHF, due to a process described as antibody-dependent enhancement (ADE), where the pre-existing antibodies to previous dengue infection cannot neutralize but rather enhance the new infection.

Fig. 9. Scheme of the immunological response on recurrent dengue infections. In (a.) the first infection with a given dengue virus serotype, in (b.) production of antibodies (Immunoglobulin M (IgM)), in (c.) inactivation of the virus and in (d.) production of antibodies (IgG class, the so called memory antibodies). In (e.) the temporary cross immunity period, that lasts between 3-9 months. After that period, the individual can get infected again with another dengue virus serotype (f.). In (g.) the IgG from the previous dengue infection binds to the new virus but do not inactivate them. In (h.) the complex antibody-virus enhances the new infection (i.). In (j.) the production of antibodies (IgM class) which is then able to inactivate the new viruses, leading to (l.), an enhanced immune response, such that hemorrhagic symptoms are observed. In (m.) production of IgG antibodies.

In the first dengue infection virus particles will be captured and processed by so-called antigen presenting cells. These viruses will be presented to T-cells causing them to become activated. And likewise B-cells will encounter their antigen free floating and become activated. B-cells produce antibodies that are used to tag the viruses to encourage their uptake by macrophages and inactivate them. In a secondary infection the antibodies from the first infection will attach to the virus particles but will not inactivate them. The antibody-virus complex suppresses innate immune responses, increasing intracellular infection and generating inflammatory citokines and chemokines that, collectively, result in enhanced disease (Dejnirattisai et al., 2010; Guzmán et al., 2010; Halstead, 1982, 1994, 2003; Mackenzie et al., 2004; WHO, 2009). Fig.9 is an scheme to illustrate the immunological response on recurrent dengue infections.

DF is characterized by headache, retro-orbital pain, myalgia, arthralgia, rash, leukopenia, and mild thrombocytopenia. The symptoms resolve after 27 days. DHF is a potentially

(*R*).

infection was caused by strain two (*I*21) or for second time infected with strain two when the first infection was caused by strain one (*I*12). Notice that infection by one serotype confers life-long immunity to that serotype. Then the individuals recover from the secondary infection

Modeling Infectious Diseases Dynamics: Dengue Fever, a Case Study 241

To capture differences in primary infection by one strain and secondary infection by another strain we consider a basic two-strain SIR-type model for the host population, which is only slightly refined as opposed to previously suggested models for dengue fever (Billings et al.,

The stochastic version of the multi-strain dengue model is now in complete analogy to the previously described SIR model, and the mean field ODE system for the multi-strain dengue model can be read from the following reaction scheme (24), describing the transitions for first infection with strain 1 and secondary infection with strain 2, and for the reverse process, where the first infection is caused by strain 2 and the secondary infection is caused by strain 1, the

> *β* −→ *I*<sup>1</sup> + *I*<sup>1</sup>

*φβ* −→ *I*<sup>1</sup> + *I*<sup>21</sup>

*β* −→ *I*<sup>12</sup> + *I*<sup>2</sup>

*φβ*

of two strains completely (for more information on the deterministic ODE system and its

The complete system of ordinary differential equations for the two strain epidemiological system is given by Eq. system (25) and the dynamics are described as follows. Susceptibles to both strains can get the first infection with strain one or strain two with force of infection *<sup>β</sup><sup>I</sup>*

is acquired via an individual in his second infection. They recover from the first infection with a recovery rate *γ*, conferring full and life-long immunity against the strain that they were exposed to, and also a short period of temporary cross-immunity *α* against the other strain, becoming susceptible to a second infection with a different strain. The susceptible with

on whom (individual on his primary or secondary infection) is transmitting the infection. Then, with recovery rate *γ*, the individuals recover and become immune against all strains. We assume no epidemiological asymmetry between strains, i.e. infections with strain one or strain two contribute in the same way to the force of infection. Here, the only relevant difference concerning disease transmissibility is that the force of infection varies accordingly to the number of previous infections the hosts have experienced. The parameter *φ* in our model, is the ratio of secondary infection contribution to the force of infection. For more

−→ *I*<sup>12</sup> + *I*<sup>12</sup>

*<sup>R</sup>*<sup>1</sup> *<sup>α</sup>* −→ *<sup>S</sup>*<sup>1</sup> (24)

−→ *S* defining the system

*<sup>N</sup>* when the infection

*<sup>N</sup>* depending

*<sup>N</sup>* or *φβ<sup>I</sup>*

*N*

same reaction scheme can be used to describe the transitions by just changing labels.

*S* + *I*<sup>1</sup>

*S* + *I*<sup>21</sup>

*S*<sup>1</sup> + *I*<sup>2</sup>

*S*<sup>1</sup> + *I*<sup>12</sup>

The demographic transitions are *<sup>S</sup>*, *<sup>I</sup>*1, *<sup>I</sup>*2, *<sup>R</sup>*1, *<sup>R</sup>*2, *<sup>S</sup>*1, *<sup>S</sup>*2, *<sup>I</sup>*12, *<sup>I</sup>*21, *<sup>R</sup> <sup>μ</sup>*

when the infection is acquired via an individual in his first infection or *φβ<sup>I</sup>*

a previous infection gets the secondary infection with force of infection *<sup>β</sup><sup>I</sup>*

parametrization, see (Aguiar et al., 2011 a)).

*I*12 *γ* −→ *R*

*I*1 *γ* −→ *R*<sup>1</sup>

2007; Ferguson et al., 1999; Schwartz et al, 2005).

deadly complication that is characterized by high fever and hemorrhagic phenomenae. DHF develops rapidly, usually over a period of hours, and resolves within 12 days in patients who receive appropriate fluid resuscitation. Otherwise, it can quickly progress to shock (CDC, 2011; WHO, 2009).

Treatment of uncomplicated dengue cases is only supportive, and severe dengue cases requires careful attention to fluid management and proactive treatment of hemorrhagic symptoms (CDC, 2011; WHO, 2009). A vaccine against dengue is not yet available, since it would have to simulate a protective immune response to all four serotypes (Stephenson, 2005), although several candidates of tetravalent vaccines are at various stages of development (WHO, 2011).

Mathematical models describing the transmission of dengue viruses appeared in the literature early as 1970 (Fischer & Halstead, 1970). More recently, mathematical models describing the transmission of dengue viruses have focused on the ADE effect and temporary cross immunity trying to explain the irregular behavior of dengue epidemics. Such models ultimately aim to be used as a predictive tool with the objective to guide the policies of prevention and control of the dengue virus transmission, including the implementation of vaccination programs when the candidate dengue fever vaccines will be accessible. In the literature the multi-strain interaction leading to deterministic chaos via ADE has been described previously, e.g. (Billings et al., 2007; Ferguson et al., 1999; Schwartz et al, 2005) but neglecting temporary cross immunity. Consideration of temporary cross immunity is rather complicated and up to now not in detail analyzed. Models formulated in (Loureço & Recker, 2010; Nagao & Koelle, 2008; Recker et al., 2009; Wearing & Rohani, 2006), did not investigate closer the possible dynamical structures. In (Aguiar & Stollenwerk, 2007; Aguiar et al., 2008, 2009, 2011 a) by including temporary cross immunity into dengue models with ADE, a rich dynamic structure including deterministic chaos was found in wider and more biologically realistic parameter regions.
