Dengue is a flavivirus, transmitted to human through the bites of infected
Aedes
aegypti
and
A. albopictus
mosquitoes. In this paper, we analyze a new system of ordinary differential equations which incorporates saturated incidence function, vector biting rate and control measures at both the aquatic and adult stages of the vector (mosquito). The stability of the system is analysed for the dengue-free equilibrium via the threshold parameter (reproduction number) which was obtained using the Next generation matrix techniques. Routh Hurwitz criterion along together with Descartes’ rule of signs change established the local asymptotically stability of the model whenever
R0
<1and unstable otherwise. Furthermore, the sensitivity analysis was carried out and the numerical simulation reveals that increasing the proportion of human antibody and putting into place a control strategy that minimize the vector biting rate are enough to reduce the infection of the disease in the population to its barest minimum.
<i>Aedes aegypti</i> <i>A. albopictus</i> Dengue Fever Reproduction Number Control Measures Sensitivity Analysis Aquatic Stage Adult Stage1. Introduction
Over the years, mathematical models and computer simulations have been known to be useful experimental tools which are used in building and examing theories, evaluating quantitative speculations, giving answers to particular questions and determining sensitivities to changes in parameter values. Understanding the epidemiology of emerging and re-emerging infectious diseases in a population produces a healthy environment for living. Mathematical models are used in likening, designing, implementing, evaluating and optimizing several detection, prevention and control plans.
Dengue fever is one of the infectious diseases that have continued to be a subject of major concern to the public health. It is known to be a mosquito-borne viral infection which is endemic in more than a hundred countries in the world [1] [2] [3] [4] , usually in a tropic and sub-tropical regions of the world [5] . In recent years, dengue transmission preponderates in urban and semi-urban areas [1] [5] where a figure of 50 to 390 million people worldwide in a year are infected which leads to half a million hospitalizations [6] [7] [8] with an approximate of 25,000 deaths [4] [6] [9] .
The dengue disease has been well known clinically for over 2 centuries, but the etiology of the disease remains unknown until year 1944 [10] [11] . It was first recognized in the Philippines in 1953 and Thailand in 1955 [10] [11] [12] . The threat of the outbreak now exists in Europe which its first local transmission was reported in France and Croatia in 2010, while cases have occurred in Florida (USA) and Yunnan (province of China) in 2013 [1] [5] .
Dengue Hemorrhagic fever being an infectious tropical disease is caused by an infective agent called dengue virus, of the family flaviviridae which has four distinguished serotypes denoted by I, II, III and IV [12] . The virus is transmitted to humans by bites of Aedes mosquitoes [Aedes aegypti and A. albopictus are the primary transistors]. The infection remains in mosquito till death [8] .
Dengue infection causes a range of illness in humans, from clinically in apparent, to severe and fatal hemorrhagic disease [11] [12] . The incubation period; which is the time between infection and appearance of the symptoms in the body is from 3 - 14 days, but often times it ranges from 4 - 7days [4] [6] [13] , and is generally observed clearly in older children and adults [11] . Dengue fever is characterized by sudden onset of fever, frontal headache, nausea, vomiting and some other symptoms.
The use of mathematics in explaining the epidemiology of dengue fever has been extensively studied by many researchers over years. Notable among these studies are [2] [6] [10] [13] [14] [15] [16] . In this study, since dengue fever is spread between two-interacting populations (human-vector), we design and analyses a mathematical compartmental model that considers the human population and the vector population (mosquito). We extended the earlier model [6] by incorporating a “Standard force of infection” with the proportion of an antibodies produced by human in response to the incidence of infection caused by mosquito and vice-versa. Also, an extension of the work is to consider some control effects or precautionary measures of the vector in the absence of vaccination. These measures includes: Larvicides for the Aquatic stage of the vector which prevents the vector from breeding, Naled and EPA-registered insects’ repellants to prevent getting bitten against the adult stage of the vectors.
2. Formulation of the Model
The formulation of dengue model requires the interaction between two-interacting populations (human-vector). The total human population at continuous-time t denoted by N h ( t ) is subdivided into six compartments namely: susceptible humans ( S h ), exposed humans ( E h ), infectious humans ( I h ), migrated population ( M h ), treatment class ( T h ), recovered humans ( R h ). Hence, the total human population N h ( t ) is given by
N h ( t ) = ( S h ) + ( E h ) + ( I h ) + ( M h ) + ( T h ) + ( R h ) (1)
Similarly, the total vector population at continuous-time t denoted by N v ( t ) is subdivided into four compartments namely: aquatic class ( A v ), susceptible mosquitoes ( S v ), exposed mosquitoes ( E v ), infectious mosquitoes ( I v ). Hence, the total vector population N v ( t ) is given by
N v ( t ) = ( A v ) + ( S v ) + ( E v ) + ( I v ) (2)
The dynamics of the dengue considered here is formulated and studied under the following assumption:
1) the model assumes a homogeneous mixing of the human and vector (mosquito) populations, so that each mosquito bite has equal chance of transmitting the virus to susceptible in the population (or acquiring infection from an infected human);
2) considering saturated incidence rate (Non-linear incidence) which incorporate the production of antibodies in response to parasites causing Dengue in both human and vector population ( υ h , υ v ) respectively.
3) the model consider the vector-aquatic class so as to investigate on the effect of the control strategies such as Larvicides at the aquatic stage;
4) that the infectious mosquitoes remain infectious until death;
5) there is loss of immunity for the recovered human population;
6) incorporating the controlling rate parameters which will monitor the effects of control strategies at the aquatic stage ( A v ) and adult stages ( S v , E v , I v ).
In summary, following the assumptions above the transmission dynamics of dengue in a population is given by the following ten compartmental system of non-linear differential equation below:
S ˙ h ( t ) = π h − b β h v S h ( t ) I v ( t ) 1 + υ h I v ( t ) − μ h S h ( t ) + ω R h ( t ) E ˙ h ( t ) = b β h v S h ( t ) I v ( t ) 1 + υ h I v ( t ) + μ 1 M h ( t ) − ( μ h + σ h ) E h ( t ) I ˙ h ( t ) = σ h E h ( t ) + μ 2 M h ( t ) − ( μ h + τ h + δ h ) I h ( t ) M ˙ h ( t ) = π m h − ( μ 1 + μ 2 + μ h ) M h ( t ) T ˙ h ( t ) = τ h I h ( t ) − ( μ h + γ 1 ) T h ( t ) R ˙ h ( t ) = γ 1 T h ( t ) − μ h R h ( t ) − ω R h ( t ) }
A ˙ v ( t ) = π v − ( γ m + μ v + C a ) A v ( t ) S ˙ v ( t ) = γ m A v ( t ) − b β v h S v ( t ) I h ( t ) 1 + υ v I h ( t ) − ( μ v + C m ) S v ( t ) E ˙ v ( t ) = b β v h S v ( t ) I h ( t ) 1 + υ v I h ( t ) − ( θ c + σ v + μ v + C m ) E v ( t ) I ˙ v ( t ) = ( θ c + σ v ) E v ( t ) − ( δ v + μ v + C m ) I v ( t ) } (3)
where a dot is representing differentiation with respect to time.
Figure 1 shows the schematic illustration of the dengue model.
Table 1 shows the description of the parameters of the model.
2.1. Basic Properties of the Model
It is important to explore the basic dynamical feature of the model. For the model (3) formulated above to be epidemiologically meaningful, it is very important to prove that all the states variables non-negative for all time (t). In other words, the solution of the model (3) with positive initial values of data will remain positive at all time t ≥ 0 .
Positivity and Boundedness of Solutions
Since model (3) describe interaction between human and vector population, it is important to state that all the parameters and variables involved are non-negative with respect to time. The dengue model (3) will be consider in the biologically-feasible region ℑ = ℑ h × ℑ v ⊂ ℜ + 6 × ℜ + 4 with
ℑ h = { S h , E h , I h , M h , T h , R h ∈ ℜ + 6 : N h ≤ π h μ h } (4)
and
Description of the parameters of dengue model (3)
ReferencesBowman, C., Gumel, A.B., van den Driessche, P., Wu, J. and Zhu, H. (2005) Mathematical Model for Assessing Control Strategies against West Nile Virus. Bulletin of 682 Mathematical Biology, 67, 1107-1133. https://doi.org/10.1016/j.bulm.2005.01.002Garba, S.M., Gumel, A.B. and Abu Bakar, M.R. (2008) Backward Bifurcations in Dengue Transmission Dynamics. Mathematical Biosciencees, 201, 11-25.https://doi.org/10.1016/j.mbs.2008.05.002Ranjit, S. and Kissoon, N. (2011) Dengue Hemorrhagic Fever and Shock Syndromes. Pediatric Critical Care Medicine, 12, Article ID: 90100. https://doi.org/10.1097/PCC.0b013e3181e911a7World Health Organization (2017) Dengue. WWW.WHO.int/denguecontrol/faq/en/index6.htmlIurii, B. (2015) A survey of Mathematical Model of Dengue Fever. Electronic Thesis & Dissertations, 1236. http://digitalcommons.georgiasouthern.edu/etd/1236Hossain, Md.S., Nayeem, J. and Podder, Dr.C. (2015) Effects of Migratory Population and Control Strategies on the Transmission Dynamics of Dengue Fever. Journal of Applied Mathematics & Bioinformatics, 5, 43-80.Whitehorn, J. and Farrar, J. (2010) Dengue. British Medical Bulletin, 95, Article ID: 161173. https://doi.org/10.1093/bmb/ldq019Wilder-Smith, A. and Schwartz, E. (2005) Dengue in Travelers. The New England Journal of Medicine, 353, 924-932. (Effect of Migratory Population and Control Strategies.)Varatharaj, A. (2010) Encephalitis in the Clinical Spectrum of Dengue Infection. Neurology India, 58, Article ID: 585591. https://doi.org/10.4103/0028-3886.68655Chowell, G., Diaz-Duenas, P., Miller, J.C., Alcazar-Velazco, A., Hyman, J.M., Fenimore, P.W. and Castillo Chavez C. (2007) Estimation of the Reproduction Number of Dengue Fever from Spatial Epidemic Data. Mathematical Biosciences, 208, 571-589. https://doi.org/10.1016/j.mbs.2006.11.011Lakshmikantham, V., Leela, S. and Martynyuk, A.A. (1989) Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc., New York and Basel.Jelinek, T. (2000) Dengue Fever in International Travelers. Clinical Infectious Diseases, 31, 144-147. https://doi.org/10.1086/313889Esteva, L. and Vargas, C. (1999) A Model for Dengue Disease with Variable Human Population. Journal of Mathematical Biology, 38, 220-240.https://doi.org/10.1007/s002850050147Coutinho, F.A.B., Burattini, M.N., Lopez, L.F. and Massad, E. (2006) Threshold Conditions for a Non-Autonomous Epidemic System Describing the Population Dynamics of Dengue. Bulletin of Mathematical Biology, 68, 2263-2282.https://doi.org/10.1007/s11538-006-9108-6Esteva, L. and Vargas, C. (1998) Analysis of a Dengue Disease Transmission Model. Mathematical Biosciences, 150, 131-151. https://doi.org/10.1016/S0025-5564(98)10003-2Esteva, L. and Vargas, C. (2000) Influence of Vertical and Mechanical Transmission on the Dynamics of Dengue Disease. Mathematical Biosciences, 167, 51-64.https://doi.org/10.1016/S0025-5564(00)00024-9Akinpelu, F.O. and Ojo, M.M. (2016) A Mathematical Model for the Dynamic Spread of Infection Caused by Poverty and Prostitution in Nigeria. International Journal of Mathematics and Physical Sciences Research, 4, 33-47.Esteva, L., Gumel, A. and Vargas, C. (2009) Qualitative Study of Transmission Dynamics of Drug-Resistant Malaria. Mathematical and Computer Modelling, 50, 611-630. https://doi.org/10.1016/j.mcm.2009.02.012Akinpelu, F.O. and Ojo, M.M. (2017) Sensitivity Analysis of Ebola Model. Asian Research Journal of Mathematics, 2, 1-10. https://doi.org/10.9734/ARJOM/2017/30642Iboi, E. and Okuonghae, D. (2016) Population Dynamics of a Mathematical Model for Syphilis. Applied Mathematical Modelling, 40, 3573-3590.https://doi.org/10.1016/j.apm.2015.09.090Hethcote, H.W. and Thieme, H.R. (1985) Stability of the Endemic Equilibrium in Epidemic Models with Sub Populations. Mathematical Biosciences, 75, 205-227.https://doi.org/10.1016/0025-5564(85)90038-0Derouich, M. and Boutayeb, A. (2006) Dengue Fever: Mathematical Modelling and Computer Simulation. Applied Mathematics and Computation, 177, 528-544.https://doi.org/10.1016/j.amc.2005.11.031Olaniyi, S. and Obabiyi, O.S. (2013) Mathematical Model for Malaria Transmission Dynamics in Human and Mosquito Populations with Nonlinear Forces of Infection. International Journal of Pure and Applied Mathematics, 88, 125-156.Takahashi, L.T., Maidana, N.A., Ferreira Jr, W.C., Pulino, P. and Yang, H.M. (2005) Mathematical Models for the Aedes aegypti Dispersal Dynamics: Travelling Waves by Wing and Wind. Bulletin of Mathematical Biology, 67, 509-528.https://doi.org/10.1016/j.bulm.2004.08.005Vanden Driessche, P. and Watmough, J. (2002) Repro-duction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 180, 29-48.https://doi.org/10.1016/S0025-5564(02)00108-6Agusto, F.B. and ELmojtaba, I.M. Optimal Control and Cost-Effective Analysis of Malaria/Visceral Leishmaniasis Co-Infection. PLoS ONE, 12, e0171102.https://doi.org/10.1371/journal.pone.0171102