EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1533-1552 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Mathematical analysis of an immune-structured chikungunya transmission model Famane Kambiré1,∗, Elisée Gouba1, Sadou Tao1, Blaise Somé1 1 Laboratoire d’Analyse Numérique d’Informatique et de Biomathématiques, Département de Mathématiques, Université Joseph KI ZERBO, Burkina Faso Abstract. In this paper, we have formulated a new deterministic model to describe the dynamics of the spread of chikunguya between humans and mosquitoes populations. This model takes into account the variation in mortality of humans and mosquitoes due to other causes than chikungunya disease, the decay of acquired immunity and the immune sytem boosting. From the analysis, it appears that the model is well posed from the mathematical and epidemiological standpoint. The existence of a single disease free equilibrium has been proved. An explicit formula, depending on the parameters of the model, has been obtained for the basic reproduction numberR0 which is used in epidemiology. The local asymptotic stability of the disease free equilibrium has been proved. The numerical simulation of the model has confirmed the local asymptotic stability of the disease free equilbrium and the existence of endmic equilibrium. The varying effects of the immunity parameters has been analyzed numerically in order to provide better conditions for reducing the transmission of the disease. 2010 Mathematics Subject Classifications: 81T80, 92B05, 92D30, 93D20 Key Words and Phrases: Chikungunya, Differential equations, Asymptotic stability, Basic reproduction number, Modeling and numerical simulation. 1. Introduction Chikungunya is an infectious disease caused by a virus transmited by the bite of fe- male mosquitoes belonging to the genus Aedes: Aedes aegypti and Aedes albopictus. The chikungunya virus is present in sub-saharan Africa and in south-east Asia since 1952 [3]. It had been isolated for the first time in Tanzania and then spread to the rest of the world. It is estimated that chikungunya is an emerging desease in Asia. Many vaccine trials have been conducted since the 1970s but have proved ineffective [3]. Thus chikungunya disease is a major public health problem. In January 2013, the mortality caused by chikungunya was estimated at 1 per 1,000: most deaths occur in newborns, immunocompromised people and the elderly [9]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3532 Email addresses: famane.kambire@yahoo.fr (Famane Kambiré), elgouba@gmail.com (Elisée Gouba), sadoutao@yahoo.fr (Sadou Tao), some@univ-ouaga.bf (Blaise Somé) http://www.ejpam.com 1533 c© 2019 EJPAM All rights reserved. Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1534 Many mathematical models have been formulated to describe the dynamics of chikun- gunya transmission; see for example [2, 5, 8]. According to our knowledge the existing models do not simultaneously take into account the variation of the mortality due to the other causes than the disease, the decay of immunity and immune system boosting of the host. They consider a constant mortality rate and only one compartment for recovered hosts. Our model aims at correcting these inadequacies and proposing necessary condi- tions to stop the spread of the disease . In [6, 7] the authors have developed a deterministic model which describes the dynamics of malaria transmission. In their model they have used death rates linearly dependent on the total population for humans and mosquitoes. In [14], the authors have formulated a deterministic model in a populaton structured by immune status and they study waning immunity and immune system boosting. They have used three compartments of immune humans but they considered constant mortality rates and their model did not take into account vector-borne diseases. By combining these two approaches, we propose a model that best describes the transmission of chikungunya. The model we formulate is based on the principle of compartmental analysis described in [21, 22]. This paper is organized in five sections. The introduction is given in the first section. The second section is dedicated to the formulation of the model. We present the basic asumptions, the parameters and the variables used in the model. We describe the inter- actions between the different classes of the model and we formulate the model. In the third section we analyze the model. The existence, the positivity and the boudedness of the solutions is demonstrasted. The disease free equilibrium and the basic reproduction number are determined. The local asymptotic stability of the disease free equilibrium is studied. In the fourth section the theoretical results of the analysis are illustrated by a numerical simulation. In the fifth section we end the paper by a conclusion. 2. The mathematical model We devide the humans population into five copartments and the mosquitos population into two compartments. The varibles of the model are given by Table 1 and the pameters are given Table 2 . 2.1. Assumptions • Infected mosquito bites are the only ways to transmit viruses between mosquitoes and humans. • At birth, all newborns are supposed to be suseptible in both populations. • Infected mosquitoes remain infected throughout the rest of their lives. • The disease does not kill mosquitoes. • After contact with the chikungunya virus, recovered humans no longer carry viruses. So they can no longer transmit viruses to mosquitoes. Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1535 Table 1: Variables used in the model Variables Description S(t) Number of susceptible humans at time t. I(t) Number of infected humans at time t. RF (t) Number of immune humans with high level of immunity at time t. RW (t) Number of immune humans with intermediate level of immunity at time t. RC(t) Number of immune humans with critically low level of immunity at time t. NH(t) Total human population at t: NH(t) = S(t) + I(t) +RF (t) +RW (t) +RC(t). SV (t) Number of susceptible mosquitoes at time t. IV (t) Number of infective mosquitoes at time t. NV (t) Total mosquito population at time t: NV (t) = SV (t) + IV (t). • Recovered humans acquire an immunity; this immunity gradually decreases several years after recovery when a recovered human is no longer exposed to viruses. • New exposure to viruses boosts the immune system then prolong the time in which a recovered individual is immune. • The mortality induced by other causes than the desease depends on the total pop- ulation for both human and mosquitoe populations and is given by fH(NH) = dH + d2HNH and fV (NV ) = dV + d2VNV respectively for humans and mosquitoes; see [7, 15]. 2.2. The interactions Per time unit, bHNH(t) humans and bVNV (t) mosquitoes are born in the susceptible compartments S and SV respectively. Due to bites by infcted mosquitoes, βH IV NH S suscep- tible humans and βV I NH SV susceptible mosquitoes are newly infected. Then they enter the infected compartments I and IV respectively. γI infected humans are recovered and enter the compartment RF with high level of immunity. When there is no new exposure after recovery, the immunity acquired by humans decays progressively and is lost. We consider three levels of immunity and we divide human population into five compartments of humans; see Table 1. µRF humans with high level of immunity will leave RF com- partment and enter the compartment RW of intermediate level of immunity while λRW humans leave RW compartment and enter the compartment RC of critically low level of immunity and σRC humans leave RC compartment and return in the susceptible com- partment S. When there is new exposure some humans in the RW and RC compartments get their immune system boosted. Thus βV IV NH (1 − θ)RC humans of RC compartment and βH IV NH RW humans of RW return into RF compartment; βH IV NH θRC humans of RC compartment return into RW compartment. Per time unit, fV SV susceptible mosquitoes and fV IV infected mosquitoes leave the Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1536 Table 2: Parameters used in the model Parameters Meaning bH Humans birth rate. bV Mosquitoes birth rate. pvh Probability of virus transmission from infected mosquitoe to susceptible human. piv Probability of virus transmission from an infected human to a susceptible mosquito. nv Number of bites done by a mosquito on the whole human population. βH Average number of bites that cause a viruses transmission from infected mosquitoes to susceptible humans. βH = nvpvh. βV Average number of bites that cause a viruses transmission from infected humans. to susceptible mosquitoes. βV = nvpiv. dH Density independent part of the death rate for humans. d2H Density dependent part of the death rate for humans. dV Density independent part of the death rate for mosquitoes. d2V Density dependent part of the death rate for mosquitoes. dI Chikungunya induced death rate in human population γ Recovery rate. 1 γ is the average duration of the infectious period. µ Immunity decay rate from high level RF to intermediate level RW . λ Immunity decay rate from intermediate level RW to critially low level RC . σ Immunity loss rate. θ Immune system boosting probability from critially low level RC to intermediate level RW . 1− θ is Immune system boosting probability from RC to high level RF . mosquito population due to death. fHS, (fH + dI)I, fHRF , fHRW and fHRC humans die respevtively in S, I, RF , RW and RC compartments. The quantities used in the interactions are all positive. The interactions described are illustrated by Figure 1. Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1537 Figure 1: A compartmental model of the chikungunya virus transmission. The arrows with full lines represent the transfers of individuals. The arrows with dot- ted lines indicate the direction of the infection: from humans to mosquitoes and from mosquitoes to humans. The red arrows with dotted lines indicate that immune system boost is due to bites of infected mosquitoes. 2.3. The mathematical model Using the interactions illustrated by Figure 1, the dynamics of the transmission given by the system of non linear differential equations (1). dS dt = bHNH + σRC − βH IV NH S − dHS − d2HNHS dI dt = βH IV NH S − (γ + dH + dI)I − d2HNHI dRF dt = γI − (µ+ dH)RF − d2HNHRF + βH IV NH ((1− θ)RC +RW ) dRW dt = µRF − (dH + λ)RW − d2HNHRW + βH IV NH (θRC −RW ) dRC dt = λRW − (dH + σ)RC − d2HNHRC − βH IV NH RC dSV dt = bVNV − dV SV − d2VNV SV − βV I NH SV dIV dt = βV I NH SV − dV IV − d2VNV IV dNH dt = (bH − dH) 1− NH bH − dH d2H NH − dII dNV dt = (bV − dV ) 1− NV bV − dV d2V NV (1) Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1538 where βH = nvpvh, βV = nvpiv. bH − dH d2H is the carrying capacity for humans population and bV − dV d2V is the carrying ca- pacity for mosquitoes population. That is to say the maximum humans and the maximum mosquitoes the ecosystem is able to sustain indefinitely. Remark 1. The carrying capacities for human population and mostiquitoes population are strictly positve. Thus bH > dH and bV > dV . In the rest of this paper, we consider that bH > dH and bV > dV . We disignate the initial condition of the model, that is to say the initial number of indi- viduals in the different compartments, by ( S0, I0, RF0 , RW0 , RC0 , SV0 , IV0 , NH0 , NV0 )t . 3. Mathematical analysis of the model 3.1. Existence, positivity and boundedness of the solutions To prove that the model is well posed from the mathematical and epidmiological stand- point, we begin by defining the folowing sets: Ω1 = {( S, I, RF , RW , RC , SV , IV )t ∈ R7 + } , (2) Ω2 =  ( NH , NV )t 0 < NH ≤ bH − dH d2H 0 < NV ≤ bV − dV d2V  (3) and Ω = Ω1 × Ω2. Noting the elements of Ω by x = ( S, I, RF , RW , RC , SV , IV , NH , NV )t , we rewrite the model (1) as the following autonomous differential system: dxi dt = fi(x); i = 1; 2; ...; 9 (4) We can state the following result: Proposition 1. For all initial condition x0 = ( S0, I0, RF0 , RW0 , RC0 , SV0 , IV0 , NH0 , NV0 )t in Ω, the model (1) admits a unique solution defined for all time t ≥ 0. Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1539 Proof Each component fi of the function f is given by the member on the right of the i-th equation of the model (1) which is a sum of polynomials and rational functions defined on Ω. It shows that f ∈ C∞(Ω) and in particular f ∈ C1(Ω). So f is locally lips- chitzian on Ω. Thus, by the Cauchy-Lipschitz theorem [4], for any initial condition x0 = ( S0, I0, RF0 , RW0 , RC0 , SV0 , IV0 , NH0 , NV0 )t ∈ Ω, the model (1) admits a unique maximum solution defined for all time t ≥ 0 � Theorem 1. For all initial condition x0 ∈ Ω it exits, for the model (1), a unique solution that stays in Ω for all time t ≥ 0. Proof The proposition 1 guarantees the existence and the uniqueness of the solution for each initial condition. Now we must prove that Ω is positively invariant by the system (1). That is to say for S0 ≥ 0, I0 ≥ 0, RF0 ≥ 0, RW0 ≥ 0,RC0 ≥ 0, SV0 ≥ 0, IV0 ≥ 0, 0 < NH0 ≤ bH − dH d2H and 0 < NV0 ≤ bV − dV d2V , the solution verifying this initial condition also satisfies the condition S ≥ 0, I ≥ 0, RF ≥ 0, RW ≥ 0, RC ≥ 0, SV ≥ 0, IV ≥ 0, 0 < NH ≤ bH − dH d2H and 0 < NV ≤ bV − dV d2V . To show the positivity of the solutions of the model we express each differential equa- tion of the system (1) as a differential inequality and we use the technique of separation of variables. After integration of the different differential inequalities, we obtain the positivity of S, I, RF , RW , RC , SV , IV , NH and NV . This approach is also used in [1],[18]. Positivity of the number of susceptible humans S: As bHNH + σRC ≥ 0, dS dt ≥ −βH IV NH S − dHS − d2HNHS. (5) Separating the variables and integrating we obtain S ≥ S0 exp ( −βH IV NH − dH − d2HNH ) . (6) As exp ( −βH IV NH − dH − d2HNH ) > 0, S0 exp ( −βH IV NH − dH − d2HNH ) ≥ 0 for S0 ≥ 0. Thus S ≥ 0 for S0 ≥ 0. Positivity of the number of infected humans I: βH IV NH S ≥ 0. That implies that dI dt ≥ −(γ + dH + dI)I − d2HNHI (7) Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1540 By separating the variables and integrating, we have I ≥ I0 exp (−(γ + dH + dI)− d2HNH) . (8) We deduce that I ≥ 0 for I0 ≥ 0. Positivity of the number of immune humans with high level of immunity RF : As γI + βH IV NH ((1− θ)RC +RW ) ≥ 0, dRF dt ≥ −(µ+ dH)RF − d2HNHRF . (9) By separating the variables and integrating, RF ≥ RF0 exp(−(µ+ dH)− d2HNHR). (10) We deduce that RF ≥ 0 for RF0 ≥ 0. Positivity of the number of immune humans with intermediate level of immunity RW : As µRF + βH IV NH (θRC −RW ) ≥ 0, dRW dt ≥ −(dH + λ)RW − d2HNHRW . (11) By separating the variables and integrating, RW ≥ RW0 exp(−(dH + λ)− d2HNH). (12) Thus RW ≥ 0 for RW0 ≥ 0. Positivity of the number of immune humans with critically low level of immunity RC : λRW ≥ 0. That implies that dRC dt ≥ −(dH + σ)RC − d2HNHRC − βH IV NH RC . (13) By separating the variables and integrating, we obtain RC ≥ RW0 exp ( −(dH + σ)− d2HNH − βH IV NH ) . (14) Thus RC ≥ 0 for RC0 ≥ 0. Positivity of the number of susceptible mosquitoes SV : bVNV ≥ 0. That implies that dSV dt ≥ −dV SV − d2VNV SV − βV I NH SV . (15) By separating the variables and integrating, SV ≥ SV0 exp ( −dV − d2VNV − βV I NH ) . (16) Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1541 Thus SV ≥ 0 for SV0 ≥ 0. Positivity of the number of infected mosquitoes IV : βV I NH SV ≥ 0. That implies that dIV dt ≥ −dV IV − d2VNV IV . (17) By separating the variables and integrating, IV ≥ IV0 exp (−dV − d2VNV ) . (18) We deduce that IV ≥ 0 for IV0 ≥ 0. We have shown that ∀ ( S0, I0, RF0 , RW0 , RC0 , SV0 , IV0 , NH0 , NV0 )t ∈ Ω,( S, I, RF , RW , RC , SV , IV )t ∈ Ω1. Let’s show now that ∀ ( S0, I0, RF0 , RW0 , RC0 , SV0 , IV0 , NH0 , NV0 )t ∈ Ω, ( NH , NV )t ∈ Ω2. For that we must show that 0 < NH ≤ bH − dH d2H and 0 < NV ≤ bV − dV d2V . Boundedness of the solutions: NH(t) = S(t) + I(t) +RF (t) +RW (t) +RC(t). S(t), I(t), RF (t), RW (t) and RC(t) are positive. Thus NH is positive. From dNH dt = (bH − dH) 1− NH bH − dH d2H NH − dII, we obtain the inequality dNH dt ≤ (bH − dH − d2HNH)NH . (19) After separating the variables and integrating, we obtain NH(t) ≤ bH − dH d2H (1− exp(−(bH − dH)t) +NH0 exp(−(bH − dH)t)) . (20) Calculating the limit of NH(t) when t→ +∞ we finally obtain NH(t) ≤ bH − dH d2H . (21) As NV (t) = SV (t) + IV (t), NV (t) is positive. dNV dt = (bV − dV ) 1− NV bV − dV d2V NV . After integrating we obtain NV (t) = NV0(bV − dV ) d2VNV0 + (bV − dV − d2VNV0) e−(bV −dV )t . (22) Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1542 lim t→+∞ NV (t) = bV − dV d2V . (23) From the relation 22 we find dNV dt (t) = NV0 ( bV − dV d2VNV0 + (bV − dV − d2VNV0)e−(bV −dV )t )2 (bV − dV − d2VNV0) e−(bV −dV )t(24) For ( NH , NV )t ∈ Ω2, the relation 24 implies that dNV dt (t) ≥ 0. It means that NV is an increasing function. Using the relation 23, we obtain ∀t ∈ [0 ; +∞[, NV0 ≤ NV (t) ≤ bV − dV d2V . (25) Thus 0 < NV (t) ≤ bV − dV b2V . Therefore ( NH , NV )t ∈ Ω2 � Theorem 1 shows that the model (1) is well posed from a mathematical and epidemi- ological standpoint. 3.2. Disease free equilibrium and basic reproduction number R0 3.2.1. Disease free equilibrium Theorem 2. The model (1) has a unique disease free equilibrium given by Edfe = ( bH − dH d2H , 0, 0, 0, 0, bV − dV d2V , 0, bH − dH d2H , bV − dV d2V )t . (26) Proof When there is no disease, I = IV = 0. We note the disease free equilibrium by Edfe = ( Ss, 0, Rs F , R s W , R s C , S s V , 0, N s H , N s V )t where N s H and N s V are the total popu- lations respectively of humans and mosquitoes at the equilibrium. So NH 6= 0 and NV 6= 0. Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1543 By replacing I and IV by 0 in the model (1), we find Edfe after solving the sytem below: bHNH + σRC − dHSs − d2HNHS s = 0 − (µ+ dH)Rs F − d2HN s HR s F = 0 −(µ+ dH)Rs F − d2HN s HR s F = 0 µRs F − (dH + λ)Rs W − d2HN s HR s W = 0 λRs W − (dH + σ)Rs C − d2HN s HR s C = 0 bVN s V − dV Ss V − d2VN s V S s V = 0 (bH − dH) 1− NH bH − dH d2H NH = 0 (bV − dV ) 1− NV bV − dV d2V NV = 0 (27) After solving, we find N s H = bH − dH d2H , N s V = bV − dV d2V , Ss = N s H , Ss V = N s V and RF = RC = RW = 0. Thus Edfe = ( bH − dH d2H , 0, 0, 0, 0, bV − dV d2V , 0, bH − dH d2H , bV − dV d2V )t � 3.2.2. Basic reproduction number The basic reproduction number is the average number of secondary infections produced by a single infected individual in a population completely susceptible. We determine the basic reproduction number by using the next generation method described in [11, 17]. After chikungunya disease, recovered humans no longer have virus in their body. So the infected compartments are only I and IV . We define the new infections rates in the infected comparments by the relation (28) and the exchange rates of each infected compartment Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1544 with the other compartments is given by the relation (29). F(X) =  FI(X) FIV (X)  =  βH IV NH S βV I NH SV  , (28) V(X) =  VI(X) VIV (X)  =  (γ + dH + dI)I + d2HNHI dV IV + d2VNV IV  (29) With these functions, we calculate the next generation matrix FV −1 where F and V are the following matrices: F =  ∂FI ∂I (Edfe) ∂FI ∂IV (Edfe) ∂FIV ∂I (Edfe) ∂FIV ∂IV (Edfe)  =  0 βH βV N s V N s H 0  (30) V =  ∂VI ∂I (Edfe) ∂VI ∂IV (Edfe) ∂VIV ∂I (Edfe) ∂VIV ∂IV (Edfe)  =  bH + γ + dI 0 0 bV  (31) Thus the next generation matrix is given by FV −1 =  0 βH bV βV N s V N s H(bH + γ + dI) 0  (32) R0 is the spectral radius of the next generation matrix; R0 = ρ ( FV −1 ) . After detrmining the eigenvalues of FV −1, we obtain R0 = √ βHβVN s V N s H (bH + γ + dI) bV . We can also write R0 = √ R0HR0V where R0H = βH bV and R0V = βVN s V N s H (bH + dI + γ) . R0H is the average number of secondary infections produced by a single infected mosquito in a population of humans all susceptibles. R0V is the average number of secondary infec- tions produced by a single infected human in a population of mosquitoes all susceptibles. Replacing βH and βV respectively by nvpvh and nvpiv, we obtain R0H = nvpvh bV , Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1545 R0V = nvpivN s V N s H (bH + dI + γ) and R0 = nv √ pvhpivN s V N s H (bH + γ + dI) bV . Substituting N s H by bH − dH d2H and N s V by bV − dV d2V we finally obtain R0 = nv √ d2Hpvhpiv(bV − dV ) bV d2V (bH − dH)(bH + γ + dI) . (33) 3.2.3. Stability of disease free equilibrium Theorem 3. The disease free equilibrium Edfe is locally asymptoptically stable for R0 < 1 and unstable for R0 > 1. Proof The jacobian matrix of the system (1) at the disease free equilibrium is given below: J =  −bH 0 0 0 σ 0 −βH dH 0 0 J22 0 0 0 0 βH 0 0 0 γ J33 0 0 0 0 0 0 0 0 µ J44 0 0 0 0 0 0 0 0 λ J55 0 0 0 0 0 −βV 0 0 0 −bV 0 0 dV 0 βV N∗V N∗H 0 0 0 0 −bV 0 0 0 −dI 0 0 0 0 0 J88 0 0 0 0 0 0 0 −dIV 0 J99  where J22 = −(bH + γ + dI); J33 = −(bH + µ); J44 = −(bH + λ); J55 = −(bH + σ); J88 = −(bH − dH); J99 = −(bV − dV ). The characteristic polynomial of J is given by P (x) = (x+ bH)(x+ bV )(x− J33)(x− J44)(x− J55)(x− J88)(x− J99)T (x) (34) where T (x) = x2 + (bH + γ + dI + bV )x+ bV (bH + γ + dI) ( 1−R2 0 ) . (35) −bH , −bV , −(bH +µ), −(bH +λ), −(bH +σ), −(bH−dH), −(bV −dV ) are eigenvalues of J which are all negative reals. The other eigenvalues of J are the roots of the polynomial T . For R0 < 1, the coefficients of T namely 1, (bH + γ + dI + bV ) and bV (bH + γ + dI) ( 1−R2 0 ) are all strictly positive. For R0 > 1 the coefficients 1 and (bH + γ + dI + bV ) are strictly positive while the coefficients Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1546 bV (bH + γ + dI) ( 1−R2 0 ) is strictly negative. As T is a polynomial of second degree, the criterion of Routh-Hurwitz [13] allows us to affirm that the roots of T have all their real parts strictly negative for R0 < 1 and T has at least an eigenvalue whose real part is strictly positive for R0 > 1. Thus the eigenvalues of J have all their real parts strictly negative for R0 < 1 and at least an eigenvalue of J has a strictly positive real part for R0 > 1. Therefore Edfe is locally asymptotically stable for R0 < 1 and unstable for R0 > 1 � The proof of Theorem 3 is based on the analysis of the characteristic polynomial of Jacobian matrix, see [16] for a similar approach. Due to the complexity of the system (1), the existence of endemic equilibrium is not shown analytically. However, we can confirm the existence of endemic equilibrium numer- ically in the next section. 4. Numerical simulation In this section, we illustrate by the numerical simulation the theoretical results obtained from the analysis; see Theorem 3 . Our objective is to show the conformity between the theoretical results of the analysis and the numerical results. We use MATLAB ode45; see [10, 12, 19, 20]. We set the initial condition to S0 = 4950, I0 = 50, RF0 = 0, RW0 = 0, RC0 = 0, SV0 = 9900, IV0 = 100, NH0 = 5000 and NV0 = 10000. We take the same values for bH , dH , d2H , bV , dV and d2V as in [6] and we assume that an infectious human will stay infectious during 100 days. 1 γ is the average time that an infectious human will stay in the infectious compartment. By analogy 1 µ is the average time that a human of RF compartment will stay in this compartment before moving to RW compartment; 1 λ is the average time that a human of RW compartment will stay in this compartment before moving to RC compartment and 1 σ is the average time that a human of RC compartment will stay in this compartment before loosing his immunity. A recovered human acquire lasting immunity. Several years are necessary so that immunity is lost. We estimate that one year lasts 365 days. We assume that, when there is no new exposure, immunity will decay to the intermediate level after 25 years, to the critically low level after 35 years and it is lost after 45 years. When there is new exposure, we assume that the probability of immune system boosting from critically low level to intermediate level is θ = 0.6. It means that 1 − θ = 0.4. The values used for the numerical simulation are given by Table 3. With these values, we obtain Edfe = ( 133280, 0, 0, 0, 0, 12125, 0, 133280, 12125 )t . Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1547 Table 3: Values used for the numerical simualtion. Parameters Values References bH 0.04 [6] bV 0.13 [6] pvh 0.07 [6] piv 0.45 [6] dH 1.6× 10−5 [6] d2H 3× 10−7 [6] dV 0.033 [6] d2V 8× 10−6 [6] dI 0.001 [9] γ 0.01 assumed µ 1.096× 10−4 assumed λ 7.828× 10−5 assumed σ 6.088× 10−5 assumed θ 0.6 assumed 4.1. The varying effects of R0 on the behaviour of the model We fix the values of Table 3 and we choose nv appropriately so that R0 < 1 on the one hand and R0 > 1 on the other hand. For nv = 1.1408, R0 = 0.75. Then the dynamics of the transmission is given by Figure 2 for humans and Figure 3 for the mosquitoes. Figure 2: Dynamics of the transmission in the human population for R0 = 0.75 < 1. The disease will die out and all humans will be susceptible. With nv = 11.4079, we obtain R0 = 7.5 > 1. Then the dynamics of the transmission is given by the Figure 4 for humans and Figure 5 for the mosquitoes. Figure 2 and Figure 3 show that the solution of the model approaches the disease free eqilibrium whenR0 < 1. This confirms the stability of the desease free equilibrium. Figure 4 and Figure 5 show that the solution of the model does not approache the disease free Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1548 Figure 3: Dynamics of the transmission in the mosquioes population for R0 = 0.75 < 1. The disease will die out and all mosquitoes will be susceptible. Figure 4: Dynamics of the transmission in the human population for R0 = 7.5 > 1. There are infected humans every time. eqilibrium when R0 > 1. There will always be infected humans and infected mosquitoes when R0 > 1. This confirms the existence of endemic equilibrium of the system (1) when R0 > 1. 4.2. The varying effects of the immunity parameters on the behaviour of the model In this subsection we provide further simulations showing the varying effects of the immunity parameters (µ, λ, σ and θ) on the behaviour of the system (1). We analyze these effects for R0 > 1. For each parameter, we fixe the other values of table 3 and we simulate the varying effects of the parameter concerned. Figures 6, 7 and 8 show the varying effects of µ, λ and σ respectively on the dynamics of the infected humans. As for θ, its varying effects are difficult to be observed by the curve of I but the curve of RW shows clearly its varying effects; see Figure 9. Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1549 Figure 5: Dynamics of the transmission in the population of mosquitoes for R0 = 7.5 > 1. There are infected mosquitoes every time. Figure 6: Varying effects of µ. The increase of µ towards 1 favors the transmission of the disease in the human population. Figure 7: Varying effects of λ. The increase of λ towards 1 favors the transmission of the disease in the human population. Famane Kambiré et al. / Eur. J. Pure Appl. Math, 12 (4) (2019), 1533-1552 1550 Figure 8: Varying effects of σ. The increase of σ towards 1 favors the transmission of the disease in the human population. Figure 9: Varying effects of θ. The increase of θ towards 1 has a positive effect on the dynamics of the intermediate level of immunity. 5. Conclusion In this paper a deterministic model have been formulated to describe chikungunya transmission between a population of mosquitoes and a human population structured by immune status. The analysis of the model and the basic reproduction number expressed in terms of the parameters made possible to predict whether the disease will disappear (R0 < 1) or persist (R0 > 1). The numerical simulation have permitted to confirm the theoretical results obtained from the analysis. The numerical results have also shown that the decay of the immunity is a factor which favors the transmission of the disease and boosting the immune system helps to get more immune humans. Some medical researches must aim at preventing the decay of the immunity and boosting the immune system . REFERENCES 1551 References [1] Alemu Geleta Wedajo, Boka Kumsa Bole, Purnachandra Rao Koya. The Impact of Susceptible Human Immigrants on the Spread and Dynamics of Malaria Transmission. American Journal of Applied Mathematics. Vol. 6, No. 3, 2018, pp. 117-127. doi: 10.11648/j.ajam.20180603.13. [2] Anibal Munoz L., Humberto Colorado T., Carlos A. Abello M. PSIRMW Model for the Transmission of Chikunguya Virus (CHIKV) Using Markov Chains. Applied Mathematical Sciences, Vol. 12, 2018, no. 30, 1481 - 1489. [3] Bernard-Alex Gaüzère, Pierre Aubry, Infection à virus Chikungunya. Actualités 2016. Mise à jour le 11/03/2016. Diplôme de Médecine Tropicale des Pays de l’Océan Indien. [4] Daniel Sondaz, Jean-Marie Morvan, Autour du théorème de Cauchy-Lipschitz - Equa- tions différentielles. CAPES, agrégation, école d’ingénieurs. Cépadues, 2017. [5] Djamila Moulaye. Modélisation et analyse mathématique de systèmes dynamiques en épidémiologie. Application au cas du Chikungunya. Thèse de doctorat de l’université du Havre, 26 septembre 2011. [6] Ducrot A., Sirima S. B., Somé B. and Zongo P. (2009). A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host. Journal of Biological Dynamics Vol. 3, No. 6, November 2009, 574–598. [7] G.A. Ngwa and W.S. Shu. A mathematical model for endemic malaria with vari- able human and mosquito populations. Math. and Comput. Modell., 32 (2000), pp. 747–763. [8] Gilberto C. González-Parra , Diego F. Aranda, Benito Chen-Charpentier, Miguel Dı́az-Rodŕıguez and Jaime E. Castellanos. Mathematical Modeling and Characteri- zation of the Spread of Chikungunya in Colombia. Mathematical and computational Application. [9] Haute Autorité de Santé. Diagnostic biologique direct précoce du chikungunya par détection génomique du virus avec RT-PCR (transcription inverse et amplifica- tion génique par réaction de polymérisation en châıne). Rapport d’évaluation tech- nologique, Janvier 2013 [10] Jaan Kiusalaas. Numerical Methods in Engineering with MATLAB. Cambridge Uni- versity Press, 2005 [11] J. M. Heffernan, R. J. Smith and L. M. Wahl. Perspectives on the basic reproductive ratio. J. R. Soc. Interface doi :10.1098/rsif.2005.0042 [12] Lawrence F. Shampine and Mark W. Reichelt. The matlab ode suit. SIAM J. SCI. COMPUT. Vol. 18, No. 1, pp. 1-22, January 1997 REFERENCES 1552 [13] Liénard, Sur le signe de la partie réelle des racines d’une équation algébrique. Journal de mathématiques pures et appliquées 6e série, tome 10 (1914), p. 291-346. [14] M. V. Barbarossa, G. Röst, Immuno-epidemiology of a population structured by im- mune status: a mathematical study of waning immunity and immune system boosting. J. Math. Biol. (2015) 71:1737–1770. [15] N. Chitnis, J.M. Cushing and J.M. Hyman. Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model. Bull. Math. Biol., DOI 10.1007/s11538-008-9299-0, (2008). [16] O.S. Obabiyi and S. Olaniyi. Asymptotic Stability of Malaria Dynamics with Vigilant Human Compartment. International Journal of Applied Mathematics, 29(1): 127-144, (2002) [17] P. van den Dressche and J. Watmough. Reproduction numbers and sub-threshold en- demic equilibria for compartmental models of disease transmission. Math. Biosci 180: 29-48, 2002. [18] S. Olaniyi. Dynamics of zika virus Model with nonlinear incidence and optimal control strategies. Applied Mathematics and Information Sciences. 12(5): 969-982, 2018. [19] Steven T. Karris. Numerical Analysis Using MATLAB and Excel, Orchard Publica- tions, Third Edition [20] Won Young Yang, Wenwu Cao, Tae-Sang Chung and John Morris. Applied numerical methods using matlab. Wiley-interscience [21] Yves Cherruault. Biomathématique. Collection que sais-je? (n◦ 2052) P.U.F.1983. [22] Yves Cherruault. Modèles et méthodes mathématiques pour les sciences du vivant. Presse universitaire de France, Paris (1998)