J. Nig. Soc. Phys. Sci. 1 (2019) 82–87 Journal of the Nigerian Society of Physical Sciences Original Research The Solution of a Mathematical Model for Dengue Fever Transmission Using Differential Transformation Method Felix Yakubu Eguda∗, Andrawus James, Sunday Babuba Department of Mathematics, Federal University, Dutse, Jigawa State. Abstract Differential Transformation Method (DTM) is a very effective tool for solving linear and non-linear ordinary differential equations. This paper uses DTM to solve the mathematical model for the dynamics of Dengue fever in a population. The graphical profiles for human population are obtained using Maple software. The solution profiles give the long term behavior of Dengue fever model which shows that treatment plays a vital role in reducing the disease burden in a population. Keywords: Dengue Fever, Mathematical Model, Differential Transformation Method, Ordinary Differential Equations Article History : Received: 15 June 2019 Received in revised form: 13 July 2019 Accepted for publication: 16 July 2019 Published: 03 September 2019 c©2019 Journal of the Nigerian Society of Physical Sciences. All Rights Reserved. Communicated by: T. Latunde 1. Introduction Dengue fever is an infectious vector borne disease spread- ing in tropical and subtropical countries with more than 50 mil- lion dengue fever cases per year. It is transmitted to humans by the bite of infected aedes mosquitoes. The major vector, Aedes aegypti, thrives in tropical regions, mainly in urban ar- eas, closely linked to human populations providing artificial water-holding containers as breeding sites. A second potential vector, Aedes Albopictus, resides in temperate regions (North America and Europe), where it may give rise to occasional dengue outbreaks. [1, 2]. As part of awareness campaigns, different kinds of precau- tions have been suggested towards preventing mosquito’s bite. Some of the precautions that can be taken are: to keep home, environment and surrounding clean, to remove all stagnant wa- ter and containers, to cover all containers properly to prevent ∗Corresponding Author Tel. No: +2348160559365 Email address: felyak_e@yahoo.co.uk (Felix Yakubu Eguda ) dengue mosquito breeding there, to use mosquito repellents to avoid mosquito bite, to use mosquitoes net around bed while sleeping etc. [1]. Different studies have shown the impor- tance of mathematical approaches in understanding dengue dis- ease transmission and evaluating the effectiveness and/or cost- effectiveness of control strategies [1]. In recent years, the field of public health has benefited tremen- dously from the use of mathematical models to study the spread of infectious disease. Most epidemiological models are repre- sented using systems of non-linear ordinary differential equa- tions [3]. Differential Transformation Method is a semi-analytical method of solving both linear and nonlinear system of ordi- nary differential equations (ODE) to obtain approximate series solutions. This method which was derived from the Taylor’s series expansion has been used to solve problems in Mathe- matics and Physics [4], fractional differential- algebraic equa- tions [5], fourth-order parabolic partial differential equations [6], fractional-order integrodifferential equations [7], differen- tial equation [8] and problems in epidemic models [9]. 82 Eguda et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 82–87 83 2. Materials and Methods 2.1. Model Formulation Two populations consisting of human and vector popula- tion will be considered in this work. The model sub-divides these populations into a number of mutually-exclusive com- partments, as given below. The total population of human and vectors is divided into the following mutually exclusive epi- demiological classes, namely, susceptible humans ( S H (t) ), hu- mans with dengue in latent stage ( E1 (t)), humans with dengue ( I1 (t) ), humans treated of dengue ( R1 (t) ), susceptible vec- tors ( S V (t) ), vectors with latent dengue ( EV (t) ), vectors with dengue ( IV (t) ). 2.2. Derivation of Model Equations Let NH (t) and NV (t) denote the total number of humans and vectors at time t, respectively. Hence, we have that, NH (t) = S H (t) + E1 (t) + I1 (t) + R1 (t) and NV (t) = S V (t) + EV (t) + IV (t) Susceptible humans are recruited at a rate H while the suscepti- ble vectors are recruited at a rate V. Susceptible humans contract dengue at a rate λDV = βV H (ηv Ev + Iv) NH , where ηv < 1 , this accounts for the relative infectiousness of vectors with latent dengue Iv compared to vectors in the Iv class. Susceptible vectors acquire dengue infection from infected humans at a rate λDH = βHV (ηA E1 + ηB I1) NH , where ηA < ηB this accounts for the relative infectiousness of humans with latent dengue E1 compared to humans in the I1 class [10]. The model equations for dengue disease transmission incor- porating treatment as a control measure are given below: S H = ΛH −µH S H −λDV S H E1 = λDV S H − (γ1 + µH ) E1 I1 = γ1 E1 − (τ1 + µH + δD1) I1 R1 = τ1 I1 −µH R1 S V = ΛV −µV S V −λDH S V EV = λDH S V − (γV + µV ) EV IV = γV EV − (µV + δHV ) IV 2.3. Differential Transformation Method (DTM) An arbitrary function f (t) can be expanded in Taylor series about a point t = 0 as f (t) = ∞∑ k=0 tk k! [ dk f dtk ] t=0 (1) The differential transformation of f (t) is defined as F (t) = 1 k! [ dk f dtk ] t=0 (2) Then the inverse differential transform is f (t) = ∞∑ k=0 tk F (t) (3) If y (t) and g (t) are two uncorrelated functions with t where Y (k) and G (k) are the transformed functions corresponding to y (t) and g (t) then, the fundamental mathematical operations performed by differential transform can be proved easily and are listed as follows Table 1: The Fundamental Mathematical Operations by Differential Transfor- mation Method (DTM). Source: [11, 12] Transformed Function Original Function y (t) = f (t) ± g (t) Y (k) = F(k) ± G(k) y (t) = a f (t) Y (k) = aF(k) y (t) = d f (t)dt Y (k) = (k + 1)F(k + 1) y (t) = d 2 f (t) dt2 Y (k) = (k + 1)(k + 2)F(k + 2) y (t) = d m f (t) dtm Y (k) = (k + 1)(k + 2)...(k + m)F(k + m) y (t) = 1 Y (k) = δ(k) y (t) = t Y (k) = δ(k − 1) y (t) = tm Y (k) = δ(k − m) = { 1, k = m 0, k , m y (t) = f (t) g (t) Y (k) = ∑k m=0 G(m) f (k − m) y (t) = e(λt) Y (k) = λ k k! y (t) = (1 + t)m Y (k) = (m(m−1)...(m−k+1))k! 2.4. Analytical solution of the model equations using differen- tial transformation method (DTM) In this section, the Differential Transformation Method (DTM) is employed to solve the system of non-linear differential equa- tions which describe our model for Dengue fever. Let the model equation be a function q (t), q (t) can be ex- panded in Taylor series about a point t = 0 as q (t) = ∞∑ k=0 tk k! [ dkq dtk ] t=0 , (4) where, q (t) = {sH (t) , E1 (t) , I1 (t) , R1 (t) , S v (t) , Ev (t) , Iv (t)}(5) The differential transformation of q (t) is defined as Q (t) = 1 k! [ dkq dtk ] t=0 (6) 83 Eguda et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 82–87 84 Table 2: Values for parameters used for analytical solutions Parameter Description Values Unit Reference ΛH, ΛV Recruitment rate into the population of susceptible hu- mans and vectors respectively. 500,10000000 Year−1 [14] µH,µv Natural death for humans, vectors respectively. 0.02041,0.5 Year−1 [13] βV H Effective contact rate for 0.5 dengue from vectors to humans 0.5 Year−1 [14] βHV Effective contact rate for 0.4 dengue from humans to vectors 0.4 Year−1 [14] τ1 Dengue treatment rate for I1 (0,1) Ind−1Year−1 [14] γ1 Progression rate to active dengue 0.3254 Year−1 [14] γV Progression rate to active dengue (vectors) 0.03 Year−1 [14] δD1 Disease induced death Dengue 0.365 Year −1 [13] δHV Disease induced death dengue (vectors) 0 Year−1 [14] ηV , ηA, ηB Modification parameters for Ev, E1, i1 0.4,1.2,0.5 Year−1 [13] Then the inverse differential transform is q (t) = ∞∑ k=0 tk Q (t) . (7) Using the fundamental operations of differential transformation method in Table 1, we obtain the following recurrence relation of equation (1) as S H (k + 1) = 1 k + 1 [ ΛH −µH S H (k) − βV Hηv NH k∑ m=0 S H (m) EV (k − m) − βV H NH k∑ m=0 S H (m) IV (k − m)  (8) E1 (k + 1) = 1 k + 1 βV HηvNH k∑ m=0 S H (m) EV (k − m) − βV H NH k∑ m=0 S H (m) IV (k − m) − (γ1 + µH ) E1 (k) ] (9) I1 (k + 1) = 1 k + 1 [ γ1 E1 (k) − (τ1 + µH + δD1) I1 ] (10) R1 (k + 1) = 1 k + 1 [ τ1 I1 (k) −µH R1 (k) ] (11) S V (k + 1) = 1 k + 1 Λv − βHVηANH k∑ m=0 S V (m) E1 (k − m) − βHVηB NH k∑ m=0 S V (m) I1 (k − m)  (12) EV (k + 1) = 1 k + 1 βHVηANH k∑ m=0 S V (m) E1 (k − m) − βHVηB NH k∑ m=0 S V (m) I1 (k − m) − (γv + µv) EV (k) ] (13) IV (k + 1) = 1 k + 1 [ γV EV (k) − (µV + δHV ) IV (k) ] (14) With the initial conditions S H (0) = 3503, E1 (0) = 490, I1 (0) = 390, R1 (0) = 87, S V (0) = 390, EV (0) = 100, IV (0) = 190 (15) The parameter values are NH = 4470, NV = 610, ΛH = 500, ΛV = 1, 000, 000, µH = 0.02041,µV = 0.5,βV H = 0.5,βHV = 0.4, τ1 = 0.75,γ1 = 0.3254,γV = 0.03,δD1 = 0.365, δHV = 0,ηV = 0.4,ηA = 1.2,ηB = 0.5 (16) We consider k = 0, 1, 2, 3. Cases A1 to A3 are the variation of different values of τ1 Case A1: High Dengue Treatment Rate, τ1 = 0.75 S H (1) = −237.6147983, S H (2) = 12320.67062, S H (3) = −146425.8636, S H (4) = 1347694.18, E1 (1) = −522.0979067, E1 (2) = 6637.369770, E1 (3) = −82219.53607, E1 (4) = 775491.4995, I1 (1) = −283.36390, I1 (2) = 75.92177345, I1 (3) = 691.1992607, I1 (4) = −6884.757898, R1 (1) = 290.72433, R1 (2) = −109.2283043, R1 (3) = 19.72355993, R1 (4) = 129.4992219, S V (1) = −14008.26174, S V (2) = 260942.4174, 84 Eguda et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 82–87 85 S V (3) = −3239718.545, S V (4) = 30208553.32, EV (1) = −3515.845638, EV (2) = 61669.12070, EV (3) = −717387.0983, EV (4) = 6204920.468, IV (1) = −4742.00, IV (2) = 86488.76230, IV (3) = −1051663.250, IV (4) = 9591046.752. (17) Then the closed form of the solution where k = 0, 1, 2, 3 can be written as S H (t) = ∞∑ k=0 S H (k) t k = 3503 − 237.6147983t +12320.67062t2 − 146425.8636t3 +1347694.180t4 E1 (t) = ∞∑ k=0 E1 (k) t k = 490 − 522.0979067t +6637.369770t2 − 82219.53607t3 +775491.4995t4 I1 (t) = ∞∑ k=0 I1 (k) t k = 390 − 283.36390t +75.92177345t2 + 691.1992607t3 −6884.757898t4 R1 (t) = ∞∑ k=0 R1 (k) t k = 87 + 290.72433t −109.2283043t2 + 19.72355993t3 +129.4992219t4 S V (t) = ∞∑ k=0 S V (k) t k = 390 − 14008.26174t +260942.4174t2 − 3239718.545t3 +30208553.32t4 EV (t) = ∞∑ k=0 EV (k) t k = 100 − 3515.845638t +61669.12070t2 − 717387.0983t3 +6204920.468t4 IV (t) = ∞∑ k=0 IV (k) t k = 130 − 4742.00t +86488.76230t2 − 1051663.250t3 +9591046.752t4 (18) Case A2: Moderate Dengue Treatment Rate, τ1 = 0.5 S H (1) = −237.6147983, S H (2) = 12320.67062, S H (3) = −146421.4193, S H (4) = 1347571.999, E1 (1) = −522.0979067, E1 (2) = 6637.369770, E1 (3) = −82223.98033, E1 (4) = 775613.8858, I1 (1) = −185.86390, I1 (2) = −2.662451550, I1 (3) = 720.7191613, I1 (4) = −6848.453788, R1 (1) = 193.22433, R1 (2) = −48.43782929, R1 (3) = −0.1142032264, R1 (4) = 90.09047788 S V (1) = −14008.26174, S V (2) = 260933.9106, S V (3) = −3239404.789, S V (4) = 30202664.15, EV (1) = −3515.845638, EV (2) = 61660.61400, EV (3) = −71707.62417, EV (4) = 619913.9432, IV (1) = −4742.00, IV (2) = 86488.76230, IV (3) = −1051663.335, IV (4) = 9591049.860. (19) Then the closed form of the solution where k = 0, 1, 2, 3 can be written as S H (t) = ∞∑ k=0 S H (k) t k = 3503 − 237.6147983t +12320.67062t2 − 146421.4193t3 +1347571.999t4 E1 (t) = ∞∑ k=0 E1 (k) t k = 490 − 522.0979067t +6637.369770t2 − 82223.98033t3 +775613.8858t4 I1 (t) = ∞∑ k=0 I1 (k) t k = 390 − 185.86390t −2.662451550t2 + 720.7191613t3 −6848.453788t4 R1 (t) = ∞∑ k=0 R1 (k) t k = 87 + 193.22433t −48.43782929t2 − 0.1142032264t3 +90.09047788t4 S V (t) = ∞∑ k=0 S V (k) t k = 390 − 14008.26174t +260933.9106t2 − 3239404.789t3 +30202664.15t4 EV (t) = ∞∑ k=0 EV (k) t k = 100 − 3515.845638t +61660.61400t2 − 717076.2417t3 +6199139.432t4 IV (t) = ∞∑ k=0 IV (k) t k = 130 − 4742.00t +86488.76230t2 − 1051663.335t3 +9591049.860t4 (20) Case A3: Low Dengue Treatment Rate, τ1 = 0.25 S H (1) = −237.6147983, S H (2) = 12320.67062, S H (3) = −1464169750, S H (4) = 1347571.999, E1 (1) = −522.0979067, E1 (2) = 6637.369770, E1 (3) = −82223.98033, E1 (4) = 775613.8858, I1 (1) = −88.36390, I1 (2) = −56.87167655, I1 (3) = 720.7191613, I1 (4) = −6848.453788, R1 (1) = 95.72433, R1 (2) = −12.02235429, 85 Eguda et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 82–87 86 R1 (3) = −0.1142032264, R1 (4) = 90.09047788 S V (1) = −14008.26174, S V (2) = 260925.4039, S V (3) = −3239404.789, S V (4) = 30202664.15, EV (1) = −3515.845638, EV (2) = 61652.10730, EV (3) = −71707.62417, EV (4) = 619913.9432, IV (1) = −4742.00, IV (2) = 86488.76230, IV (3) = −1051663.335, IV (4) = 9591049.860. (21) Then the closed form of the solution where k = 0, 1, 2, 3 can be written as S H (t) = ∞∑ k=0 S H (k) t k = 3503 − 237.6147983t +12320.67062t2 − 146416.9750t3 +1347450.373t4, E1 (t) = ∞∑ k=0 E1 (k) t k = 490 − 522.0979067t +6637.369770t2 − 82228.42463t3 +775735.7158t4, I1 (t) = ∞∑ k=0 I1 (k) t k = 390 − 88.36390t −56.87167655t2 + 731.9789850t3 −6805.559035t4, R1 (t) = ∞∑ k=0 R1 (k) t k = 87 + 95.72433t −12.02235429t2 − 4.657514297t3 +45.77245152t4, S V (t) = ∞∑ k=0 S V (k) t k = 390 − 14008.26174t +260925.4039t2 − 323909.2451t3 +30196827.38t4, EV (t) = ∞∑ k=0 EV (k) t k = 100 − 3515.845638t +61652.10730t2 − 717666.8033t3 +6193410.4t4, IV (t) = ∞∑ k=0 IV (k) t k = 130 − 4742.00t +86488.76230t2 − 1051663.420t3 +9591052.958t4. (22) 2.5. Numerical Simulation and Graphical Representation of the Solutions of the Model Equations The numerical simulation which illustrates the analytical solution of the Model is demonstrated using Maple software. This is achieved by using some set of parameter values given in the Table 2. The following initial conditions for the hu- man populations S H (0) = 3503, E1 (0) = 490, I1 (0) = 390, R1(0) = 87, S V (0) = 390, EV (0) = 100, IV (0) = 190 are con- sidered. Figure 1: Solution of Susceptible Population Using DTM Figure 2: Solution of Exposed Population Using DTM Figure 3: Solution of Human Population with Dengue Using DTM 3. Discussion of results The Figures 1 to 6 give the numerical profiles of the solu- tions (17), (19) and (21) using DTM. Figure 1 shows increase in the population of susceptible individuals while Figure 2 indi- cates a decreasing population of the exposed owing to the pro- gression out of exposed class to class of human with Dengue. Figure 3 implies a decrease in the population of human infected with dengue which later increases. Figure 4 implies the treated 86 Eguda et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 82–87 87 Figure 4: Solution of Treated Human Population Using DTM Figure 5: The Effect of Different Treatment Rates on Human Population with Dengue Using DTM Figure 6: The Effect of Different Treatment Rates on Treated Human Population Using DTM human population increases sharply to a point and then de- creases. Figure 5 indicates that increasing the treatment rate of individuals in the infected population leads to a reduction in the number of infected individuals which is due to progression into the treated population while Figure 6 shows that increasing the treatment rate of individuals in the infected population leads to a corresponding increase in the treated population. 4. Conclusion We formulated a compartmental model to investigate the dynamics of dengue fever in a population with treatment as a control measure. Differential Transform Method (DTM) was employed to obtain the series solution of the model. 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