Mathematical modeling of Avian Influenza epidemic with bird vaccination in constant population

Transcription

1 Journal of Physics: Conference Series PAPER OPEN ACCESS Mathematical modeling of Avian Influenza epidemic with ird vaccination in constant population To cite this article: M haris and Amidi 2018 J Phys: Conf Ser Related content - Mathematical Model of Seasonal Influenza with Treatment in Constant Population M haris and R Arifudin - Which is more effective for suppressing an infectious disease: imperfect vaccination or defense against contagion? azuki uga and Jun Tanimoto - Modeling microevolution in a changing environment: the evolving quasispecies and thediluted champion process Ginestra Bianconi, Davide Fichera, Silvio Franz et al View the article online for updates and enhancements This content was downloaded from IP address on 23/10/2018 at 00:34

2 Mathematical modeling of Avian Influenza epidemic with ird vaccination in constant population M haris 1* and Amidi 1 1 Department of Mathematics, Universitas Negeri Semarang, Indonesia * Corresponding author: kharismat@mailunnesacid Astract The development of the industrial world and human life is increasingly modern and less attention to environmental sustainaility causes the virus causes the epidemic has a high tendency to mutate so that the virus that initially only attack animals, is also found to have the aility to attack humans The epidemics that lasted some time were ird flu epidemics and swine flu epidemics The flu epidemic led to several deaths and many people admitted to the hospital Strain (derivatives) of H5N1 virus was identified as the cause of the ird flu epidemic while the H1N1 strain of the virus was identified as the cause of the swine flu epidemic The symptoms are similar to seasonal flu caused y H3N2 strain of the virus Outreaks of ird flu and swine flu initially only attacked animals, ut over time some people were found to e infected with the virus 1 Introduction Perdue [1], Sedyaningsih [2], and Scoones & Forster [3] suggested that ird flu virus can e transmitted to humans and can cause death so that an outreak occurs In Yang et al [4] pulished in 2009, ird flu epidemic then erupted swine flu epidemic The flu epidemic led to several deaths and many people admitted to the hospital Strain (derived) H5N1 virus preserved as the cause of the epidemic of ird flu virus H1N1 virus as a cause of swine flu epidemic Symptoms caused y seasonal flu are caused y H3N2 strain of the virus Jansen et al [5] mentioned influenza viruses responsile for the numer of deaths and people who are sick in the hospital In Widiasih [6] the sense of presence in poultry infected with this virus with a very large numer of economic terms very de Jong et al [7] mentioned that the influenza A sutype H5N1 virus with sustitution of an amino acid in neuramiside isolated from 2 patients undergoing therapy/treatment and known the virus is immune to a given drug Both of these patients died ecause of this viral infection Gooskens [8] also mentioned that there is a mutation of influenza A virus that produces a virus that is immune to oseltamivir The mutated virus is contagious pathogenic and lethal for high-risk patients The aility of the H5N1 virus to mutate is so high that it is necessary to watch out for the spread of this virus in the poultry population so that some precautions have een taken such as the destruction of infected poultry and quarantine for infected humans 2 Methods The first step to do this research was literacy study In this step, we study the fact and some assumptions from various scientific literacies After that, we complete the facts with some assumptions to uild the model The second step was uilding and analyzing mathematics model In this step, we uild the mathematics model and then analyze it to determine the equilirium points and Content from this work may e used under the terms of the Creative Commons Attriution 30 licence Any further distriution of this work must maintain attriution to the author(s) and the title of the work, journal citation and DOI Pulished under licence y IOP Pulishing Ltd 1

3 their staility The third step was making simulation with parameters value which was gotten from other paper 3 Mathematical Model From literature review we got: in Tuncer & Martcheva [9], it were stated that the avian influenza virus of sutype H5N1 can infect humans and cause death in human and ird population In Tuncer & Martcheva [9] and Bourouia [10], there was stated that vaccination in poultry are still eing implemented In Bourouia [10], it was stated that vaccinated poultry which is even free of clinical signs, should not e traded to avoid all risk of silent shedding and transmission Vemula et al [11] used several different approaches that are currently availale for diagnosis of influenza infections in humans These are used to diagnosis of influenza virus infections following natural infection and vaccination in humans In this paper, we assume that the population is constant so the death rate of infected human and infected ird were assumed have same value with natural death rate in every population We also assumed that death in infected humans and infected irds only occurs due to viral infection and the proaility of infectious contacts of ird-human and human-human are same Transfer diagram of AI epidemic is given at Figure 1 μs mi μr μn S β S N ሺI + Iሻ I γi R θr S μ N S δs β S N I I m I μ S V μ V Figure 1 Transfer diagram of AI epidemic with vaccination on susceptile ird Tale 1 The meaning of parameters Parameter The Meaning μ : Birth rate in humans is assumed same with death rate μ : Birth rate in irds is assumed same with death rate β : The proaility of infectious contact was happen in humans β : The proaility of infectious contact was happen in irds m : Death rate of infected human (assumed equal to μ) m : Death rate of infected ird (assumed equal to μ ) γ : Recovery rate of infected human θ : Immunity loss rate δ : The proportion of susceptile ird to e vaccinated 2

4 International Conference on Mathematics, Science and Education 2017 (ICMSE2017) IOP Pulishing where N is the total human population, S is total numer of susceptile person, I is total numer of infected person, R is total numer of recovered person, N is the total ird population, S is total numer of susceptile ird, I is total numer of infected ird, and V is total numer of vaccinated ird The meaning of parameter in model were given in Tale 1 From Fig 1 we construct te system of ordinary differential equation as System (1) ds = μn + θr Sሺβ I +I + μሻ dt N di dt = β S N ሺI + Iሻ ሺm + γሻi dr = γi ሺθ + μሻr dt ds = μ I dt N ሺβ + δ + μ N ሻS (1) di dt = β S I N m I dv dt = δs μ V S + I + R = N S + I + V = N We assumed that m = μ and m = μ then we get System (2) ds = μn + θr Sሺβ I +I + μሻ dt N di dt = β S N ሺI + Iሻ ሺμ + γሻi dr = γi ሺθ + μሻr dt ds = μ I dt N ሺβ + δ + μ N ሻS (2) di dt = β S I N μ I dv dt = δs μ V S + I + R = N S + I + V = N Clear that dn = 0 N = > 0, k R and dn = 0 N = L > 0, L R dt dt Hence, we get System (3) ds dt = ሺμ + θሻ θi S (β I + I + μ + θ) di dt = β S ሺI + Iሻ ሺμ + γሻi ds = μ I dt L ሺβ + δ + μ L ሻS (3) di dt = β S L I μ I Domain of System (3) is defined Γ = {P R 4 P = ሺS, I, S, I ሻ where 0 S, I < and 0 S, I < L} The existence of equilirium points of System (3) is given in Theorem 1 3

5 Theorem 1 Let r 0 = β μ + δ and R 0 = β μ + γ If r 0 < 1 and R 0 < 1 then System (3) has only one equilirium point ie non endemic equilirium μ L point P 0 = ሺS, I, S, I ሻ = P = (, 0,, 0) δ+μ 1 If r 0 < 1 and R 0 > 11 then System (3) has two equilirium ie P 0 and ሺμ + γሻ ሺμ + θሻ[β ሺμ + γሻ] μ L P 1 = ሺS, I, S, I ሻ = (,,, 0) β βሺμ + γ + θሻ δ + μ 2 if r 0 > 1 and R 0 > 1 then System (3) has Three equilirium ie P 0, P 1, and ሺμ + γሻi P 2 = ሺS, I, S, I ሻ = ( βሺi + I ሻ, I, μ L, I β ) where I = L[β ሺμ +δሻ], I = B+ B2 4AC, A = βሺθ + μ + γሻ, β B = βሺθ + μ + γሻi ሺμ + θሻ[β ሺμ + γሻ], and C = βሺμ + θሻi Proof: The equilirium points were solutions of System (4) ሺμ + θሻ θi S (β I + I + μ + θ) = 0 β S ሺI + I ሻ ሺμ + γሻi = 0 μ L (β I L + δ + μ ) S = 0 (4) β S L I μ I = 0 From the fourth equation of System (4), we get S [β L μ ] I = 0 I S = 0 β L μ = 0 I = 0 S = μ L β The case of I = 0: Sustitute the value of I to the third equation, we get S = μ L δ+μ Sustitute the value of I to the second equation, we get I = 0 S = ሺμ+γሻ β The case of I = 0: μ L For this case, we get P 0 = ሺS, I, S, I ሻ = (, 0,, 0) δ+μ The case of I 0: Clear that S = ሺμ+γሻ Sustitute to the first equation then we get I = ሺμ+θሻ[β ሺμ+γሻ] β βሺμ+γ+θሻ Clear that if R 0 = β > 1 then μ+γ I > 0 Hence, we get if R 0 > 1 and θ < 1 then μ+γ ሺμ + γሻ ሺμ + θሻ[β ሺμ + γሻ] μ L P 1 = ሺS, I, S, I ሻ = (,,, 0) β βሺμ + γ + θሻ δ + μ 4

6 The case of I 0: Clear that S = μ L Sustitute the value of S β to the third equation, we get I = L[β ሺμ +δሻ] β Clear that I > 0 if r 0 = β > 1 From the second equation, we get μ +δ S = ሺμ+γሻI β(i +I ) Sustitute to the first equation, then we get βሺθ + μ + γሻi 2 + [βሺθ + μ + γሻi ሺμ + θሻ[β ሺμ + γሻ]]i βሺμ + θሻi = 0 Let A = βሺθ + μ + γሻ, B = βሺθ + μ + γሻi ሺμ + θሻ[β ሺμ + γሻ], and C = βሺμ + θሻi Clear that A > 0 and C < 0 Hence B 2 4AC > B 2 > 0 Hence I 1 = B B2 4AC Hence, I = I 2 = B+ B2 4AC < 0 and I 2 = B+ B2 4AC > 0 as the positive root of the equation if R 0 > 1 Hence, S = ሺμ+γሻI > 0 Hence, if r β(i +I ) 0 > 1, R 0 > 1 and > 1 then there are exists μ+γ ሺμ + γሻi P 2 = ሺS, I, S, I ሻ = ( βሺi + I ሻ, I, μ L, I β ) where I = L[β ሺμ +δሻ], I = B+ B2 4AC, A = βሺθ + μ + γሻ, B = βሺθ + μ + γሻi β ሺμ + θሻ[β ሺμ + γሻ], and C = βሺμ + θሻi The Staility of equilirium points of System (3) is given in Theorem 2 Theorem 2 Let r 0 = β μ + δ and R 0 = β μ + γ 1 If r 0 < 1 and R 0 < 1 then P 0 is locally asymptotically stale 2 If r 0 < 1 and R 0 > 1 then P 0 is unstale and P 1 is locally asymptotically stale 3 if r 0 > 1 and R 0 > 1 then P 0 and P 1 are unstale, and P 2 is locally asymptotically stale θ Proof: The Jacoian matrix of System (4) was given elow βሺi + I h ሻ ሺμ + θሻ θ βs JሺPሻ = βሺi + I h ሻ 0 βs βs ሺμ + γሻ 0 βs 0 0 β I L ሺδ + μ ሻ β S L β I L β S L μ ] [ 0 0 μ L For P 0 = ሺS, I, S, I ሻ = (, 0,, 0): δ + μ The eigen values of JሺP 0 ሻ are λ 1 = ሺμ + θሻ, λ 2 = β ሺμ + γሻ, λ 3 = ሺδ + μ ሻ, and λ 4 = μ [β ሺδ + mu ሻ] δ + μ Clear that λ 1 and λ 3 are negative, λ 2 < 0 if R 0 < 1, and λ 4 < 0 if r 0 < 1 Hence, (1) P 0 is locally asymptotically stale if R 0 < 1 and r 0 < 1 and (2) P 0 is unstale if R 0 > 1 5

7 ሺμ + γሻ ሺμ + θሻ[β ሺμ + γሻ] μ L For P 1 = ሺS, I, S, I ሻ = (,,, 0): β βሺμ + γ + θሻ δ + μ This analysis was only done at R 0 > 1 The characteristics polynomial of JሺP 1 ሻ is 1 ሺμ + γ + θሻሺδ + μ ሻ {ሺλ + δ + μ ሻ[ሺδ + μ ሻλ + μ ሺδ + μ β ሻ][ሺμ + γ + θሻλ 2 + ሺβ + θሻሺμ + θሻλ + ሺμ + θሻሺμ + γ θሻሺβ μ γሻ]} = 0 From the characteristics polynomial of JሺP 1 ሻ, we got two first eigen values ie λ 1 = ሺδ + μ ሻ and λ 2 = μ ሺβ δ μ ሻ Clear that λ δ+μ 1 < 0 and λ 2 < 0 if r 0 < 1 From ሺμ + γ + θሻλ 2 + ሺβ + θሻሺμ + θሻλ + ሺμ + θሻሺμ + γ θሻሺβ μ γሻ = 0, we got the simpler equation λ 2 + ሺβ+θሻሺμ+θሻ λ + ሺμ + θሻሺβ μ γሻ = 0 ሺμ+γ+θሻ Define A = 1, B = ሺβ + θሻሺμ + θሻ, and C = ሺμ + θሻሺβ μ γሻ ሺμ + γ + θሻ Clear that B > 0 and C > 0 if R 0 > 1 Hence, λ 3 = B D 2 and λ 4 = B+ D 2 where D = B 2 4C Because C > 0 then D < B 2 Hence, Reሺλ 1 ሻ < 0 and Reሺλ 2 ሻ < 0 if R 0 > 1 Hence, (1) P 1 is locally asymptotically stale if r 0 < 1 and R 0 > 1; (2) P 1 is unstale if r 0 > 1 ሺμ + γሻi For P 2 = ሺS, I, S, I ሻ = ( βሺi + I ሻ, I, μ L, I β ): We have I = L[β ሺμ +δሻ], I = B+ B2 4AC, where A = βሺθ + μ + γሻ, B = βሺθ + μ + γሻi β ሺμ + θሻ[β ሺμ + γሻ], and C = βሺμ + θሻi This analysis was only done at r 0 > 1 and R 0 > 1 The characteristics polynomial of JሺP 2 ሻ is 1 ሺI + I h ሻL {[Lλ2 + ሺLሺδ + μ ሻ + β I ሻλ + μ β I ][A 1 λ 2 + B 1 λ + C 1 ]} = 0 where A 1 = ሺI + I h ሻ, B 1 = βi h 2 + (2βI + ሺμ + θሻ)i h + I [βi + ሺ2μ + θ + γሻ], C 1 = βሺμ + γ + θሻi h 2 + 2βI ሺμ + γ + θሻi h + I [βi ሺμ + γ + θሻ + ሺμ + θሻሺμ + γሻ] From Lλ 2 + [Lሺδ + μ ሻ + β I ]λ + μ β I = 0 we got λ 1 = ሺLሺδ + μ ሻ + β I ሻ [Lሺδ + μ ሻ + β I ] 2 4L μ β I 2L λ 2 = ሺLሺδ + μ ሻ + β I ሻ + [Lሺδ + μ ሻ + β I ] 2 4L μ β I 2L Clear that [Lሺδ + μ ሻ + β I ] 2 4L μ β I = [ሺβ I Lμ ሻ 2 + 2Lδβ I ] > 0 and [Lሺδ + μ ሻ + β I ] 2 > [Lሺδ + μ ሻ + β I ] 2 4L μ β I > 0 Hence, λ 1 and λ 2 are negative From A 1 λ 2 + B 1 λ + C 1 = 0 where A 1 = ሺI + I h ሻ, B 1 = βi h 2 + (2βI + ሺμ + θሻ)i h + I [βi + ሺ2μ + θ + γሻ], C 1 = βሺμ + γ + θሻi h 2 + 2βI ሺμ + γ + θሻi h + I [βi ሺμ + γ + θሻ + ሺμ + θሻሺμ + γሻ] and 6

8 we got λ 3 = B 1 B 2 1 4A 1 C 1 and λ 3 = B+ B2 4AC Clear that A, B, and C are all positive if r 0 > 1 and R 0 > 1 Hence, B 2 1 4A 1 C 1 < B 2 1 It caused λ 3 and λ 4 have real part Hence, P 2 is locally asymptotically stale if r 0 > 1 and R 0 > 1 4 Simulation Simulation was done for three cases like three conditions in Theorem 2 Value of some parameter followed from haris & Arifudin [12] Value of parameter were given in Tale 2 Tale 2 Value of parameters Parameter Value Parameter Value μ 0,00004 β 0 to 1 β 0 to 1 m 0,00137 m 0,00004 δ 0,7 γ 0, θ 0,037 L μ 0, Simulation for r 0 < 1 and R 0 < 1 In this case, we used the value β = 0,08 and β = 0,5 From the formula r 0 and R 0 in Theorem 1, we got r 0 = 0,713 < 1 and R 0 = 0,816 < 1 From Theorem 1, There is only one equilirium point ie P 0 = ሺS, I, S, I ሻ = ሺ6000,0,3906,0ሻ The graphs for this simulation were given on Figure 2 (a) Plane I vs I h () Plane S vs I h (c) Plane S vs I Figure 2 Vector Field in neighorhood of point P 0 at r 0 < 1 and R 0 < 1 From Figure 2, it can e seen that the solutions that is near from P 0 converge to P 0 These simulations were similar with Theorem 2 42 Simulation for r 0 < 1 and R 0 > 1 In this case, we used the value β = 0,4 and β = 0,5 From the formula r 0 and R 0 in Theorem 1, we got r 0 = 071 < 1 and R 0 = 4,08 > 1 From Theorem 1, There are two equilirium points ie P 0 = ሺS, I, S, I ሻ = ሺ6000,0,3906,0ሻ and P 1 = ሺS, I, S, I ሻ = ሺ14706,124236,3906,0ሻ The graphs for this simulation were given on Figure 3 From Figure 3, it can e seen that the solutions that is near from P 1 converge to P 1 These simulations were similar with Theorem 2 7

9 (a) Sሺtሻ () Iሺtሻ (c) S ሺtሻ (d) I ሺtሻ Figure 3 Phase portrait projection of solution at r 0 < 1 and R 0 > 1 (a) Sሺtሻ () Iሺtሻ (c) S ሺtሻ (d) I ሺtሻ Figure 4 Phase portrait projection of solution at r 0 > 1 and R 0 > 1 8

10 43 Simulation for r 0 > 1 and R 0 > 1 In this case, we used the value β = 0,4 and β = 0,8 From the formula r 0 and R 0 in Theorem 1, we got r 0 = 1,14 > 1, R 0 = 4,08 > 1 From Theorem 1, There are three equilirium points ie P 0 = ሺS, I, S, I ሻ = ሺ6000,0,3906,0ሻ, P 1 = ሺS, I, S, I ሻ = ሺ14706,124236,3906,0ሻ, and P 2 = ሺS, I, S, I ሻ = ሺ55475, , 3425, ሻ The graphs for this simulation were given on Figure 4 From Figure 4, it can e seen that the solutions that is near from P 2 converge to P 2 These simulations were similar with Theorem 2 5 Conclusion From analysis aove, we get the dynamic of mathematics model of AI-epidemic with vaccination on ird population especially for constant population We also got the formula of reproduction numer ሺr 0 and R 0 ሻ which can e used to determine whether the epidemic spread widely or vanish For the formula of r 0, we got that the proportion of vaccinated susceptile ird can change the value of r 0 If this proportion increase then r 0 decrease It means we can prevent the spreading of this epidemic in ird population y increasing the proportion of vaccinated susceptile ird For human population, we can prevent the spreading of this epidemic y reducing the proaility of infectious contact etween infected people and susceptile people It can e done y quarantine infected people For the next research, we propose to make the mathematics model for non-constant population Acknowledgments This research was funded y Ministry of Technology Research and Higher Education (emristekdikti) especially in scheme of Penelitian Produk Terapan in 2017 References [1] Perdue ML 2008 Molecular Determinants of Pathogenicity for Avian Influenza Viruses Avian Influenza ed D E Swayne (Blackwell Pulishing) [2] Sedyaningsih ER, Setiawaty V, Rif'ati L, Harun S, Heriyanto B, Nur AP, Apsari PH, Isfarandi S, Sarivadi E, Saptiawati C and Tresnaningsih E 2006 Bul Penel esehatan [3] Scoones I and Forster P 2008 The International Response to Highly Pathogenic Avian Influenza: Science, Policy and Politics, STEPS Working Paper 10 (Brighton: STEPS Centre) [4] Yang Y, Sugimoto J D, Halloran M E, Basta N E, Chao D L, Matrajt L, Potter G, enah E and Longini Jr IM 2009 Science [5] Jansen A G S C, Sanders E A M, Hoes A W, van Loon A M and Hak E 2007 Eur Respir J [6] Widiasih D A, Susetya H, Sumiarto B, Tau C R and Budiharta S 2006 J Sain Vet [7] de Jong M D, Thanh T T, hanh T H, Hien V M, Smith G J D, Chau N V, Cam B V, Qui P T, Ha D Q, Guan Y, Peiris J S M, Hien T T and Farrar J 2005 N Engl J Med [8] Gooskens J, Jonges M, Claas ECJ, Meijer A, van den Broek PJ, and roes ACM 2009 JAMA [9] Tuncer N and Martcheva M 2013 J Biol Syst [10] Bourouia L 2013 CAB Reviews 17 [11] Vemula SV, Zhao J, Liu J, Wang X, Biswas S and Hewlett I 2016 Viruses 8 96 [12] haris M and Arifudin R 2017 Int J Pure Appl Math