Control of dengue disease: a case study in Cape Verde
Helena Sofia Rodrigues, M. Teresa T. Monteiro, Delfim F. M. Torres, Alan Zinober
aa r X i v : . [ m a t h . O C ] J un This is a preprint. The published version is:Proceedings of the 10th International Conferenceon Computational and Mathematical Methodsin Science and Engineering (CMMSE 2010),Almer´ıa (Andaluc´ıa), Spain, June 26-30 2010,
Control of dengue disease:a case study in Cape Verde
Helena Sofia Rodrigues , M. Teresa T. Monteiro , Delfim F. M.Torres and Alan Zinober School of Business Studies, Viana do Castelo Polytechnic Institute, Portugal Department of Production and Systems, University of Minho, Portugal Department of Mathematics, University of Aveiro, Portugal Department of Applied Mathematics, University of Sheffield, UK emails: [email protected] , [email protected] , [email protected] , [email protected] Abstract
A model for the transmission of dengue disease is presented. It consists of eightmutually-exclusive compartments representing the human and vector dynamics.It also includes a control parameter (adulticide spray) in order to combat themosquito. The model presents three possible equilibria: two disease-free equilibria(DFE) — where humans, with or without mosquitoes, live without the disease— and another endemic equilibrium (EE). In the literature it has been provedthat a DFE is locally asymptotically stable, whenever a certain epidemiologicalthreshold, known as the basic reproduction number , is less than one. We show thatif a minimum level of insecticide is applied, then it is possible to maintain the basicreproduction number below unity. A case study, using data of the outbreak thatoccured in 2009 in Cape Verde, is presented.
Key words: dengue, basic reproduction number, stability, Cape Verde, control.MSC 2000: 92B05, 93C95, 93D20.
Dengue is a mosquito-borne infection that has become a major international publichealth concern. According to the World Health Organization, 50 to 100 million dengueinfections occur yearly, including 500000 Dengue Haemorrhagic Fever cases and 22000deaths, mostly among children [10]. Dengue is found in tropical and sub-tropical regionsaround the world, predominantly in urban and semi-urban areas. ontrol of dengue disease: a case study in Cape Verde
There are two forms of dengue: Dengue Fever and Dengue Haemorrhagic Fever.The first one is characterized by a sudden fever without respiratory symptoms, ac-companied by intense headaches and lasts between three and seven days. The secondone has the previous symptoms but also nausea, vomiting, fainting due to low bloodpressure and can lead to death in two or three days [3].The spread of dengue is attributed to expanding geographic distribution of thefour dengue viruses and their mosquito vectors, the most important of which is thepredominantly urban species
Aedes aegypti . The life cycle of a mosquito presents fourdistinct stages: egg, larva, pupa and adult. In the case of
Aedes aegypti the first threestages take place in or near water while air is the medium for the adult stage [6]. Theadult stage of the mosquito is considered to last an average of eleven days in the urbanenvironment. Dengue is spread only by adult females, that require a blood meal forthe development of eggs; male mosquitoes feed on nectar and other sources of sugar.In this process the female acquire the virus while feeding on the blood of an infectedperson. After virus incubation for eight to ten days, an infected mosquito is capable,during probing and blood feeding, of transmitting the virus for the rest of its life.The organization of this paper is as follows. A mathematical model of the interac-tion between human and mosquito populations is presented in Section 2. Section 3 isconcerned with the equilibria of the epidemiological model and their stability. In Sec-tion 4 the results obtained in the previous section are applied to a study case. Finally,some concluding notes are given in Section 5.
Considering the work of [7], the relationship between humans and mosquitoes are nowrather complex, taking into account the model presented in [4]. The novelty in thispaper is the presence of the control parameter related to adult mosquito spray.The notation used in our mathematical model includes four epidemiological statesfor humans: S h ( t ) susceptible (individuals who can contract the disease) E h ( t ) exposed (individuals who have been infected by the parasitebut are not yet able to transmit to others) I h ( t ) infected (individuals capable of transmitting the disease to others) R h ( t ) resistant (individuals who have acquired immunity)It is assumed that the total human population ( N h ) is constant, so, N h = S h + E h + I h + R h . There are also other four state variables related to the female mosquitoes(the male mosquitoes are not considered in this study because they do not bite humansand consequently they do not influence the dynamics of the disease): A m ( t ) aquatic phase (that includes the egg, larva and pupa stages) S m ( t ) susceptible (mosquitoes that are able to contract the disease) E m ( t ) exposed (mosquitoes that are infected but are not yet ableto transmit to humans) I m ( t ) infected (mosquitoes capable of transmitting the disease to humans) odrigues, Monteiro, Torres & Zinober In order to analyze the effects of campaigns to combat the mosquito, there is alsoa control variable: c ( t ) level of insecticide campaignsSome assumptions are made in this model: • the total human population ( N h ) is constant, which means that we do not considerbirths and deaths; • there is no immigration of infected individuals to the human population; • the population is homogeneous, which means that every individual of a compart-ment is homogenously mixed with the other individuals; • the coefficient of transmission of the disease is fixed and do not vary seasonally; • both human and mosquitoes are assumed to be born susceptible; there is nonatural protection; • for the mosquito there is no resistant phase, due to its short lifetime.The parameters used in our model are: N h total population B average daily biting (per day) β mh transmission probability from I m (per bite) β hm transmission probability from I h (per bite)1 /µ h average lifespan of humans (in days)1 /η h mean viremic period (in days)1 /µ m average lifespan of adult mosquitoes (in days) µ b number of eggs at each deposit per capita (per day) µ A natural mortality of larvae (per day) η A maturation rate from larvae to adult (per day)1 /η m extrinsic incubation period (in days)1 /ν h intrinsic incubation period (in days) m female mosquitoes per human k number of larvae per human K maximal capacity of larvaeThe Dengue epidemic can be modelled by the following nonlinear time-varyingstate equations:Human Population dS h dt ( t ) = µ h N h − ( Bβ mh I m N h + µ h ) S hdE h dt ( t ) = Bβ mh ImN h S h − ( ν h + µ h ) E hdI h dt ( t ) = ν h E h − ( η h + µ h ) I hdR h dt ( t ) = η h I h − µ h R h (1) ontrol of dengue disease: a case study in Cape Verde and vector population dA m dt ( t ) = µ b (1 − A m K )( S m + E m + I m ) − ( η A + µ A ) A mdS m dt ( t ) = − ( Bβ hm I h N h + µ m ) S m + η A A m − cS mdE m dt ( t ) = Bβ hm I h N h S m − ( µ m + η m ) E m − cE mdI m dt ( t ) = η m E m − µ m I m − cI m (2)with the initial conditions S h (0) = S h , E h (0) = E h , I h (0) = I h , R h (0) = R h ,A m (0) = A m , S m (0) = S m , E m (0) = E m , I m (0) = I m . (3)Notice that the equation related to the aquatic phase does not have the controlvariable c , because the adulticide does not produce effects in this stage of the life ofthe mosquito. Let the set
Ω = { ( S h , E h , I h , A m , S m , E m , I m ) ∈ R : S h + E h + I h ≤ N h , A m ≤ kN h , S m + E m + I m ≤ mN h } be the region of biological interest, that is positively invariant under the flow inducedby the differential system (1)–(2). Proposition 1.
Let Ω be defined as above. Consider also M = − ( c ( η A + µ A ) + µ A µ m + η A ( − µ b + µ m )) .The system (1)–(2) admits, at most, three equilibrium points: • if M ≤ , there is a Disease-Free Equilibrium (DFE), called Trivial Equilibrium, E ∗ = ( N h , , , , , , ; • if M > , there is a Biologically Realistic Disease-Free Equilibrium (BRDFE), E ∗ = (cid:16) N h , , , kNh M η A µ b , kNh M µ b µ m , , (cid:17) or an Endemic Equilibrium (EE), E ∗ = ( S ∗ h , E ∗ h , I ∗ h , A ∗ m , S ∗ m , E ∗ m , I ∗ m ) . It is necessary to determine the basic reproduction number of the disease, R . Thisnumber is very important from the epidemiologistic point of view. It represents theexpected number of secondary cases produced in a completed susceptible population,by a typical infected individual during its entire period of infectiousness [5]. Following[9], we prove: Proposition 2. If M > , then the basic reproduction number associated to (1)–(2)is R = B kβ hm β mh η m ν h M µ b ( η h + µ h )( c + µ m ) ( c + η m + µ m )( µ h + ν h ) .BRDFE is locally asymptotically stable if R < and unstable if R > . odrigues, Monteiro, Torres & Zinober From a biological point of view, it is desirable that humans and mosquitoes coexistwithout the disease reaching a level of endemicity. We claim that proper use of thecontrol c can result in the basic reproduction number remaining below unity and,therefore, making BRDFE stable.In order to make effective use of achievable insecticide control, and simultaneouslyto explain more easily to the competent authorities its effectiveness, we assume that c is constant.We want to find c such that R < The simulations were carried out using the following values: N h = 480000, B = 1, β mh = 0 . β hm = 0 . µ h = 1 / (71 ∗ η h = 1 / µ m = 1 / µ b = 6, µ A = 1 / η A = 0 . η m = 1 / ν h = 1 /
4, , m = 6, k = 3, K = k ∗ N h . The initial conditionsfor the problem were: S h = m ∗ N h , E h = 216, I h = 434, R h = 0, A m = k ∗ N h , S m = m ∗ N h , E m = 0, I m = 0. The final time was t f = 84 days. The values relatedto humans describes the reality of an infected period in Cape Verde [1]. However, sinceit was the first outbreak that happened in the archipelago it was not possible to collectany data for the mosquito. Thus, for the aedes Aegypti we have selected informationfrom Brazil where dengue is already a reality long known [8, 11]. Proposition 3.
Let us consider the parameters listed above and consider c as a con-stant. Then R < if and only if c > . . For our computations let us consider c = 0 . c is crucial to prevent that an outbreak could transform an epidemiologicalepisode to an endemic disease. The computational experiences were carried out usingScilab [2].Figures 1 and 2 show the curves related to human population, with and withoutcontrol, respectively. The number of infected persons, even with small control, is muchless than without any spray campaign.The Figures 3 and 4 show the difference between a region with control and withoutcontrol.The number of infected mosquitoes is close to zero in a situation where controlis present. Note that we do not intend to eradicate the mosquitoes but instead thenumber of infected mosquitoes. It is very difficult to control or eliminate the
Aedes aegypti mosquito because it makesadaptations to the environment and becomes resistant to natural phenomena (e.g.droughts) or human interventions (e.g. control measures).During outbreaks emergency vector control measures can also include broad appli-cation of insecticides. It has been shown here that with a steady spray campaign it is ontrol of dengue disease: a case study in Cape Verde
Figure 1: Human compartments usingcontrol Figure 2:
Human compartments with no con-trol
Figure 3: Mosquito compartments usingcontrol Figure 4:
Mosquito compartments with nocontrol odrigues, Monteiro, Torres & Zinober possible to reduce the number of infected humans and mosquitoes. Active monitoringand surveillance of the natural mosquito population should accompany control effortsto determine programme effectiveness.
Acknowledgements
Work partially supported by Portuguese Foundation for Science and Technology (FCT)through the PhD Grant SFRH/BD/33384/2008 (Rodrigues) and the R&D units Algo-ritmi (Monteiro) and CIDMA (Torres).
References [1] , April 2010.[2] S. L. Campbell, J.-P. Chancelier, and R. Nikoukhah.
Modeling and simulation inScilab/Scicos . Springer, New York, 2006.[3] M. Derouich, A. Boutayeb, and E. Twizell. A model of dengue fever.
BioMedicalEngineering OnLine , 2(1):4, 2003.[4] Y. Dumont, F. Chiroleu, and C. Domerg. On a temporal model for the Chikun-gunya disease: modeling, theory and numerics.
Math. Biosci. , 213(1):80–91, 2008.[5] H. W. Hethcote. The mathematics of infectious diseases.
SIAM Rev. , 42(4):599–653, 2000.[6] M. Otero, N. Schweigmann, and H. G. Solari. A stochastic spatial dynamicalmodel for Aedes aegypti.
Bull. Math. Biol. , 70(5):1297–1325, 2008.[7] H. S. Rodrigues, M. T. T. Monteiro, and D. F. M. Torres. Optimization of DengueEpidemics: A Test Case with Different Discretization Schemes. In
NumericalAnalysis and Applied Mathematics , volume 1168 of
AIP Conference Proceedings ,pages 1385–1388. Amer. Inst. Physics, 2009. arXiv:1001.3303 [8] R. C. Thom´e, H. M. Yang, and L. Esteva. Optimal control of Aedes ae-gypti mosquitoes by the sterile insect technique and insecticide.
Math. Biosci. ,223(1):12–23, 2010.[9] P. van den Driessche and J. Watmough. Reproduction numbers and sub-thresholdendemic equilibria for compartmental models of disease transmission.
Math.Biosci.