DYNAMICAL ANALYSIS OF CHIKUNGUNYA AND DENGUE CO-INFECTION MODEL

. The aim of this paper is to design and analyze a nonlinear mech-anistic model for chikungunya (CHIKV) and dengue (DENV) co-endemicity. The model can assess the epidemiological consequences of the spread of each disease on the co-infection transmission dynamics. Although the two diseases are diﬀerent, they exhibit similar dynamical features which show that to com-bat/control CHIKV virus (or co-infection with DENV virus) we can employ DENV control strategies and vice versa. Our analytical results show that each sub-model and the full model have two disease-free equilibria (i.e., triv- ial disease-free equilibrium (TDFE) and non-trivial disease-free equilibrium (NTDFE)). Further, qualitative analyses reveal that each of the sub-models exhibits the phenomenon of backward bifurcation (where a stable NTDFE co-exits with a stable endemic equilibrium (EE)). Epidemiologically, this implies that, in each case (CHIKV or DENV), the basic requirement of making the associated reproduction number to be less-than unity is no longer suﬃcient for the disease eradication. We further highlight that the full model, consisting of twenty-six (26) mutually exclusive compartments representing the human and mosquito dynamics, also exhibits the phenomenon of backward bifurcation. We ﬁt the full model and its sub-models using realistic data from India. Sensi- tivity analysis using the partial rank correlation coeﬃcient (PRCC) is used for ranking the importance of each parameter-output. The results suggested that the mosquito removal rates, the transmission rates, and the mosquito matu-ration rate are the top control parameters for combating CHIKV, DENV and CHIKV-DENV co-infection outbreaks.


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SALIHU SABIU MUSA, NAFIU HUSSAINI, SHI ZHAO AND DAIHAI HE spectrum of illness from asymptomatic to symptomatic or severe fatal Dengue Hemorrhagic Fever or Dengue Shock Syndrome (DHF/DSS) (the DHF/DSS has a high mortality risk with death occurring within 24hrs after the onset of shock) [17,25,29,33,35,46]. Infection with one serotype of DENV gives lifelong immunity to subsequent infection by the same serotype, but, it does not confer strong immunity against infection with other serotypes of the DENV virus, whereas CHIKV gives permanent immunity [2,26,39].
It has been reported that CHIKV virus is endemic in more than 60 countries in Asia, Africa, Europe, and America causing deaths and several clinical cases [10,39,47]. Unlike DENV, CHIKV has a low death rate, therefore it is not considered as a life-threatening disease [16]. Over 1.3 million suspected CHIKV cases and about 200 deaths have been reported in the Latin American countries, and Caribbean islands in April 2015 [47,49]. While DENV is endemic in more than 100 countries in Africa, the Americas, the Eastern Mediterranean and subtropical regions of the world, populated by over 2.5 billion people [16,26,39,48]. DENV is a deadly mosquitoborne disease, with an estimate of 390 million new infections of which about 96 million are clinical and more than 20 thousand deaths per year [2,16,30,45,49].
CHIKV and DENV are transmitted by the same vectors, that is Aedes aegypti and/or Aedes albopictus mosquitoes, and in each case, the viral infection has virtually the same signs and symptoms in the patients [16,18,26,38,39]. Therefore, it is very difficult to identify an individual infection or co-infection among the patients [39]. CHIKV-DENV co-infection cases were first reported in Thailand by Nimmannitya et al. [32], who detected many cases after diagnosing a number of patients. The fast-growing nature of the co-infection of diseases becomes a big threat to global public health and development [5,13,14,26,39]. In recent years the arboviral disease has been increased very rapidly, and in most of the cases CHIKV outbreaks are considered as a DENV due to the similarity of their primary signs and symptoms, this is especially in the DENV endemic regions. The co-infection of CHIKV and DENV and their co-circulation in the same region have made the problem more complicated and spreading very rapidly capturing many other countries [39]. Furthermore, the co-infection has been reported to occur in India, Indonesia, Sri Lanka, Malaysia, Gabon, Cameroon, Madagascar, Thailand, and Nigeria [5,2,13,26,32,39]. Also, it has been reported that simultaneous/concurrent infection of CHIKV and DENV viruses in patients occurred in many countries such as India, Yemen, Singapore, Sri Lanka, Malaysia and so on [39]. Therefore it is very important to consider the individual infection and the co-infection cases in order to reduce or eliminate the problem of the CHIKV and DENV co-infection.
Currently, both CHIKV and DENV do not have a preventive vaccine (although dengvaxia, a vaccince for the DENV, produced by Sanofi Pasteur and approved in some countries [12]), but a number of candidate vaccines are still undergoing clinical trials [1,12,45]. There is no specific treatment for the two viruses, however, the treatment could be achieved by reducing the disease symptoms [1,16,47,48].
Numerous mathematical modeling studies have been designed to show some insights into the transmission dynamics of CHIKV or DENV (see [1,10,16,26,30,39,49]). To the author's knowledge, the current study gives the first co-infection model for CHIKV and DENV with the aim to assess the impact of each disease on co-infection transmission dynamics. The model is an extension of some of the CHIKV and DENV transmission models (e.g., those in [1,16,26,49,51]) by (i) incorporating CHIKV-DENV co-infection in both human and mosquito populations, (ii) including the dynamics of aquatic (immature) stages of the mosquito life cycle (egg, larva and pupa stages) in each sub-model, (iii) using a nonlinear biting rate (constant rate was used in [1,16]), and (iv) including a transovarial/vertical transmission of CHIKV in mosquito population [22]. The paper is organized as follows. A new model for the transmission dynamics of CHIKV-DENV co-infection is formulated in Section 2. Theoretical results for the associated sub-models are reported in Section 3. The full model is analyzed in Section 4. Numerical analyses are carried out in Section 5.
2. Model formulation. The total humans population at time t, denoted by N H (t), is divided into sixteen mutually exclusive compartments as follows: susceptible individuals, who are at risk of infection of both CHIKV and DENV (S H (t)), individuals exposed to CHIKV (E C (t)), individuals exposed to DENV (E D (t)), individuals exposed (i.e., asymptomatic) to both CHIKV and DENV (E CD (t)), CHIKV-infected (only) individuals with clinical symptoms of CHIKV (I C (t)), DENV-infected (only) individuals with clinical symptoms of DENV (I D (t)), CHIKV-infected individuals with clinical symptoms of CHIKV but exposed to DENV (I CE (t)), DENV-infected individuals with clinical symptoms of DENV but exposed to CHIKV (I DE (t)), dually-infected individuals with clinical symptoms of both CHIKV and DENV (I CD (t)), individuals who recovered from CHIKV (R C (t)), individuals who recovered from DENV (R D (t)), individuals exposed to CHIKV but recovered from DENV with permanent immunity (E CT (t)), individuals exposed to DENV but recovered from CHIKV with permanent immunity (E DT (t)), CHIKV-infected individuals with clinical symptoms of CHIKV but recovered from DENV with permanent immunity (I CT (t)), DENV-infected individuals with clinical symptoms of DENV but recovered from CHIKV with permanent immunity (I DT (t)), individuals who recovered from both CHIKV and DENV with permanent immunity (T (t)), so that The total mosquitoes population at time t, denoted by N v (t), is sub-divided into sub-populations of immature mosquitoes (eggs, larvae and pupae stages), denoted by A(t), and adult mosquitoes (denoted by N I (t)), so that: where N I (t) is further divided into nine compartments as follows: adult mosquitoes susceptible to both CHIKV and DENV viruses (S v (t)), adult mosquitoes exposed to CHIKV (E vC (t)), adult mosquitoes exposed to DENV (E vC (t)), adult mosquitoes exposed to both CHIKV and DENV viruses (E M (t)), CHIKV-infected (only) adult mosquitoes (I vC (t)), DENV-infected (only) adult mosquitoes (I vD (t)), CHIKVinfected adult mosquitoes that are exposed to DENV (I vCE (t)), DENV-infected adult mosquitoes that are exposed to CHIKV (I vDE (t)), adult mosquitoes infected to both CHIKV and DENV (I vM (t)), so that The model for the CHIKV-DENV co-endemicity is given by the following deterministic system of nonlinear differential equations (a flow diagram of the model is depicted in Figure 1 and 2 and the associated variables and parameters are tabulated in Tables 1 and 2, respectively): and where, (3) In Eqns (3), β C and β D are the transmission probabilities of CHIKV and DENV respectively, and η C , η D , η CD , η vC , η vD , η M , η < 1 (and is positive) are the modification parameters accounting for the assumption that infected humans and mosquitoes are more infectious than exposed humans and mosquitoes, respectively [11], other parameters are defined in Table 2. Furthermore, b 1 (N H , N v ) is the per capita biting rate of female adult mosquito on the human host per unit time. Similarly, b 2 (N H , N v ) is the number of bites per adult mosquito per unit time. Following [6], the biting rates b 1 (N H , N v ) and b 2 (N H , N v ) are respectively given by, 2.1. Basic properties. The basic properties of the model Eqns (1)-(2) will now be explored. Let µ = min{µ A , µ v }. Consider the following equations for the rate of change of the total human and mosquito populations at time t. and Furthermore, consider the region It can be shown, by solving for N H and N v in Eqns (4)- (5), that all solutions of the system starting in the region Ω will remain in Ω for all time t (≥ 0). Thus, the region Ω is positive-invariant, and it is sufficient to consider solutions restricted in Ω. In this region, the usual existence, uniqueness and continuation results hold for the model Eqns (1)-(2) [21,43].  Table 2. 3. Analysis of sub-models. It is instructive, however to analyse the sub-models (CHIKV only model and DENV only model) from the full model Eqns (1)-(2) first of all. This is done below.
3.1. The CHIKV only sub-model. The CHIKV only sub-model (obtained by  Table 2. where, with Consider the region it can be shown, as in section 2, that the region Ω C is positively-invariant. Let R N C = Πv µv(1+µ A ) be the threshold quantity which accounts for eggs production by female adult mosquito.
The CHIKV only sub-model (6) has two disease-free equilibria, namely the trivial Population of CHIKV-infected humans with clinical symptoms of CHIKV but exposed to DENV IDE Population of DENV-infected humans with clinical symptoms of DENV but exposed to CHIKV RC Recovered CHIKV-infected humans RD Recovered DENV-infected humans ECT Population of individuals exposed to CHIKV but recovered from DENV with permanent immunity EDT Population of individuals exposed to DENV but recovered from CHIKV with permanent immunity ICT Population of CHIKV-infected individuals with clinical symptoms of CHIKV but recovered from DENV with permanent immunity IDT Population of DENV-infected individuals with clinical symptoms of DENV but recovered from CHIKV with permanent immunity T Population of individuals who recovered from both CHIKV and DENV with permanent immunity NV Total population of mosquitoes A population of immature mosquitoes (egg, lava and pupa stages) NI Total population of adult mosquitoes Sv population of adult mosquitoes susceptible to both CHIKV and DENV EvC population of adult mosquitoes exposed to CHIKV EvD population of adult mosquitoes exposed to DENV EM population of adult mosquitoes exposed to both CHIKV and DENV viruses IvC Population of CHIKV-infected (only) adult mosquitoes IvD Population of DENV-infected (only) adult mosquitoes IvCE Population of CHIKV-infected adult mosquitoes that are exposed to DENV IvDE Population of DENV-infected adult mosquitoes that are exposed to CHIKV IvM Population of adult mosquitoes infected to both CHIKV and DENV disease-free equilibrium (denoted by TDFE) and a non-trivial disease-free equilibrium (denoted by NDFE) as described below. (i) R N C ≤ 1, the CHIKV only sub-model (6) has a TDFE (where no mosquito exist) given by , the CHIKV only sub-model (6) has a disease-free (mosquito present) equilibrium (NDFE) given by:

3.2.
Asymptotic stability of disease-free equilibrium.
3.2.1. TDFE. The following Theorem 3.1 is established and the proof is given in Appendix A1.
It should be noted that the mosquito-free equilibrium,Υ 0C , is ecologically unrealistic, since mosquitoes commonly exist in the CHIKV-endemic regions of interest.
3.2.2. NDFE. The linear stability of ε 0C will be investigated using the next generation matrix method on the system (6) [44]. It follows that, the basic reproduction number of the CHIKV only sub-model is given by where Lemma 3.2. The DFE (ε 0C ), of the CHIKV only sub-model (6) is locally asymptotically stable (LAS) if R 0C < 1, and unstable if R 0C > 1.
The threshold quantity R 0C is the basic reproduction number for CHIKV only sub-model. It represents the average number of secondary cases that one infectious individuals (or mosquitoes) would generate over the duration of the infectious period if introduced into a completely susceptible population.
3.2.3. Endemic equilibria and backward bifurcation. The endemic equilibrium (EE) of the model (6) in terms of the forces of infections is given by .
We used the center manifold theory to examined the conditions on the parameter values in the model (6) that cause forward or backward bifurcation to occur [3,4].
The following Theorem is established and the proof is given in Appendix A2.
The public health significance of the phenomenon of backward bifurcation is that the classical requirement of R 0C < 1 , although necessary, is no longer sufficient for effective disease control (or elimination). In such a backward bifurcation (BB) scenario, effective disease control or elimination would depend on the initial sizes of the sub-populations (state variables) of the model (6). In other words, the presence of backward bifurcation in CHIKV only transmission dynamics makes its effective (population-wide) control more difficult.
3.3. The DENV only sub-model. The DEN only sub-model can be obtained by setting where with, Consider the region it can be shown, as in section 2, that the region Ω D is positively-invariant. It is convenient to define the threshold quantity R N D = ξΠv µv(ξ+µ A ) that accounts for eggs production by female adult mosquito.
The DENV only sub-model (9) has two disease-free equilibria, namely the trivial disease-free equilibrium (denoted by TDFE) and a non-trivial disease-free equilibrium (denoted by NDFE) as described below.
(i) If R N D ≤ 1, the DENV only sub-model (9) has a TDFE (where no mosquito exist) given by:

3.4.
Asymptotic stability of disease-free equilibrium.
3.4.1. TDFE. The following theorem is established, and the proof is given in Appendix A3.
3.4.2. NDFE. Let R N D > 1 (so that the NDFE, ε 0D , exists). The linear stability of ε 0D will be investigated using the next generation matrix method on the system (9) [44]. It follows that, the basic reproduction number of the DENV only sub-model is given by using the same approach as in the case of CHIKV only sub-model where Using theorem 2 of [44], the following result is established.

3.4.3.
Endemic equilibria and backward bifurcation. The endemic equilibrium (EE) of the model (9) in terms of the forces of infections is given by We substitute Eqn (12) into Eqn (10), and after simplification, we have and .
(14) By substituting Eqn (14) into Eqn (13), it can be shown that the non-zero equilibria of the model satisfy the following quadratic equation (in terms of λ * D ) where ]. Hence, we obtained a positive endemic equilibria of the model (9) by solving equation (12) for λ * D and substituting the positive values of λ * D into the expression in equation (15). Clearly, the coefficient a 1 , of Eqn (15), is always positive, and a 3 is positive (negative) if R 0D is less than (greater than) unity, respectively. Since a 1 > 0, the existence of the positive solutions of quadratic equation (15) will depend on the signs of a 2 and a 3 . If R 0D > 1, then a 3 < 0 and Eqn (15) has only positive solution. Thus, there is unique EE whenever R 0D > 1. If R 0D = 1 then a 3 = 0 and Eqn (15) has a unique nonzero solution of λ * D = − a2 a1 , which is positive if and only if a 2 < 0 and negative solution if a 2 > 0. Therefore, no EE exits if R 0D = 1 and a 2 > 0. The case R 0D < 1 makes a 3 > 0. If a 2 < 0, equation (15) has two positive solutions, given by Theorem 3.6. The DENV model (9) has i) a unique EE, if a 3 < 0 ⇐⇒ R 0D > 1; ii) a unique EE, if a 2 < 0 when a 3 = 0 or a 2 2 − 4a 1 a 3 = 0; iii) two EEs if, a 3 > 0, a 2 < 0 and a 2 2 − 4a 1 a 3 > 0; or iv) no EE, otherwise.
It is obvious from Theorem 3.6 (Case i) that the model has a unique EE when R 0D > 1. Further, Case (iii) indicates the possibility of backward bifurcation (where the locally-asymptotically stable DFE co-exists with a locally-asymptotically stable EE when R 0D < 1) in the model (9) when R 0D < 1. To check for this, we set the discriminant (a 2 2 − 4a 1 a 3 ) to zero and simplify for the critical value of R 0D , denoted by R c , given by Thus, the BB would occur for values of R 0D such that R c < R 0D < 1. This is illustrated by simulating the model with the following set of parameter values. Note that these parameters are chosen for illustrative purpose only, and may not necessarily be true epidemiologically: Π H = 20, Π v = 40, µ H = 0.000004, µ = 0.00099, δ D = 2.6, δ v = 0.2, σ m = 0.3, σ H = 0.011, σ v = 0.9, σ D = 0.303, τ D = 0.47, β D = 0.78 and β v = 0.398, η D = 0.266, η v = 0.2 (see Table 3, for the units of the parameters). Hence, R c = 0.8990 < 1 and R 0D = 0.9552 < 1 so that R c < R 0D < 1. Fig. 3 shows the backward bifurcation diagram of the model (9).
Although, the BB has been first shown to exist in DENV by Garba et al. [16], and the causes have been explained in detail by Gumel [19]. Thus, the analysis in this section show that the BB property of the DENV only sub-model (9) can be removed whenever R 0D = 1.   2) has two disease-free equilibria, namely the trivial disease-free equilibrium (denoted by TDFE) and a non-trivial disease-free equilibrium (denoted by NDFE) as described below: (i) R N CD ≤ 1, the full model Eqns (1)-(2) has a TDFE (where no mosquito exist) given by (ii) R N CD > 1, the full model Eqns (1)-(2) has a disease-free (mosquito present) equilibrium (NDFE) given by The following theorem is established, and the proof is given in Appendix A4.

5.
Numerical results and sensitivity Analysis.

Model fittings.
We fitted each of the sub-model and the full model to the data using the Pearson's Chi-square and the least square methods (using the R statistical software) [21,35]. Firstly, each of the sub-models (6) and (9) were fitted to the cumulative number of human cases from 2010-2017 (see Table A1 for the number of CHIKV and DENV cases) using the data obtained from National Vector Borne Disease Control Programme (NVBDCP) India [31]. The demographic time series are obtained from World Bank [41]. The demographic parameters (e.g., Π H and µ H ) are given by the average number of population in India which is given by Rates of CHIKV force of infection in humans λD

Rates of DENV force of infection in humans λvC
Rates of CHIKV force of infection in mosquitoes λvD

Rates of DENV force of infection in mosquitoes βC
Transmission probability for CHIKV to humans βD Transmission probability for DENV to humans βv Transmission probability from an infectious human to a susceptible adult mosquitoes b1 Number of bites per human per unit time b2 Number of bites per mosquitoes per unit time ξ Fraction of immature mosquitoes becoming susceptible adult α1 Modification parameter for the heterogeneity of DENV infection between susceptible humans and humans exposed to CHIKV α2 Modification parameter for the heterogeneity of CHIKV infection between susceptible humans and humans exposed to DENV ω1 Modification parameter for the heterogeneity of DENV infection between susceptible adult mosquitoes and those exposed to CHIKV ω2 Modification parameter for the heterogeneity of CHIKV infection between susceptible adult mosquitoes and those exposed to DENV σm Number of times a mosquito bites humans per unit time σH Maximum number of mosquito bites a human can receive per unit time σC Progression rate of humans from exposed state of CHIKV to the infectious state of CHIKV σD Progression rate of humans from exposed state of DENV to the infectious state of DENV σvC Progression rate of adult mosquitoes from exposed state of CHIKV to the infectious state of CHIKV σvD Progression rate of adult mosquitoes from exposed state of DEN to the infectious state of DENV γi(i = 1, 4) Progression rates of humans to active CHIKV classes γj(j = 2, 3) Progression rates of humans to active DENV classes θ1, ρ2 Progression rates of adult mosquitoes to active CHIKV classes θ2, ρ1 Progression rates of adult mosquitoes to active DENV classes τC Recovery rate of humans from infectious state of CHIKV to the recovered state of CHIKV τD Recovery rate of humans from infectious state of DENV to the recovered state of DENV η Modification parameters for the increase in infectiousness of dually-infected humans in comparison to mono-infected humans ηC, ηD, ηCD, ηvC, ηvD, ηM Modification parameters for the increase in infectiousness for the exposed classes in humans and mosquitoes, respectively δC, δD, δCE, δDE, δCD, δCT , δDT Disease-induced death rates for humans 1,143,638,692, and the average life expectancy in India is 59.4 years [40]. Therefore, the term µ −1 H = 59.4 years, implies that Π H µ H = 1, 143, 638, 692, so that Π H = 52, 748 per day. Note that all other parameters are fixed as in Table 3. Figs 4 and 5 show the fitting results of the models (6) and (9), respectively.
Secondly, the full model Eqns (1)-(2) is also fitted to the human cases from 2007-2012 (see Table A2 for the number of CHIKV-DENV co-infection cases), using the data obtained from [13]. The demographic time series are obtained from World Bank [41]. The demographic parameters (e.g., Π H and µ H ) are given by the average number of population in India which is given by 1,222,063,475, and the average life expectancy in India is 66.4 years [40]. Therefore, term µ −1 H = 66.4 years), implies that Π H µ H = 1, 222, 063, 475, so that Π H = 50, 423 per day. Note that all other parameters are fixed as in Table 3. Fig. 6 show the fitting results of the full model Eqns (1)-(2).

Sensitivity analysis.
Sensitivity analysis is used in order to establish what factors affect both sub-models and the full model outcomes. A partial rank correlation coefficient (PRCC), a method of conducting sensitivity analysis adapted from [15,29], is used for ranking the importance of each parameter-output. This gives an insight into designing a meaningful control strategy.
Firstly, 5,000 random samples are taken for each model parameter from uniform distributions using the parameter ranges of values in Table 3. Each sub-model is simulated for each random parameter values to obtain the target biological quantities (in this case, the basic reproduction number and the infection attack rate). Further, the PRCCs were computed between each parameter and target biological quantities. The PRCCs results highlighted that mosquito removal rates, i.e., µ v and µ A , the transmission rates, i.e., β v , β C , and β D , and the mosquito maturation rate, i.e., ξ, are the top control parameters for combating CHIKV, DENV and CHIKV-DENV co-infection outbreaks.  6. Discussion. A CHIKV and DENV co-infection model is constructed and used to assess the impact of the co-endemicity of these diseases on the transmission dynamics of each disease. In each of the CHIKV and DENV sub-models and the full model, it has been shown, using Lyapunov function theory and LaSalle's Invariance Principle, that the trivial disease-free equilibrium (TDFE) is globally asymptotically stable (GAS) whenever a certain associated threshold quantity (i.e., R N C , R N D , or R N CD ) is less than unity. The mosquito-present disease-free equilibrium of the full model (or its associated sub-models) is shown to be locally-asymptotically stable (LAS) whenever the associated reproduction number is less than unity, and is unstable if it exceeds unity. For the scenario where the basic reproduction number is greater than one, each of the sub-models is shown to have a unique endemic equilibrium. The CHIKV and DENV sub-models and the full model each exhibit the phenomenon of backward bifurcation, where the stable non-trivial disease-free equilibrium (NDFE) co-exists with the stable endemic equilibrium when the reproduction number is below one. The epidemiological consequences of the phenomenon of the backward bifurcation is that making the basic reproduction number to be less than one is no longer a key requirement for the disease control, therefore, it is important to consider some other features such as making the basic reproduction number even less than a critical value (i.e., R c , in the case of DENV) so that the disease eradication could be achieved. We fitted each of the sub-models and the full model with realistic data for CHIKV, DENV and CHIKV-DENV coinfection cases (obtained from, the National Vector Borne Disease Control Programme (NVBDCP) in India [31] and from [13], see Figs. 4, 5, and 6).
The sensitivity analysis results using the partial ranked correlation coefficient (PRCC) with 95% confidence interval show the top-ranked parameters of the model (i.e., µ v , µ A , β C , β D , β v , and ξ) that are sensitive to our main results, and should be prioritized in combating CHIKV, DENV and CHIKV-DENV co-infection outbreaks. The PRCCs results highlighted that, for any control strategy for combating CHIKV, DENV and CHIKV-DENV co-infection outbreaks to be effective, policymakers should focus on reducing mosquitoes in the environment (for example, spraying insecticide, proper sanitation and clearing mosquito breeding sites, etc.) and also reduce human-mosquito contact rate by taking some protection measures such as using insecticide-treated bed nets and curtains, insect repellent, etc. Finally, this study can be extended by (i) including the possibility of simultaneous transmission of both diseases; and (ii) incorporating seasonality in order to get insight of its effect on the transmissions dynamics of each disease and the co-infection.   Table 3. Since , it follows that V (Z) ≤ 0. Following the LaSalle's Invariance Principle (Theorem 6.4 of [24]), we have that, the maximal invariant set contained in V /V (Z) ≤ 0 is the T DF E Z . Thus, the transformed equilibrium, A2. The Proof of Theorem 3.3. The proof is based on using centre manifold theory [3,4]. Consider the system dx dt = f (x, ψ), where ψ is the bifurcation parameter, f is continuously differentiable at least twice in both x and ψ. The disease-free equilibrium is the line (x 0 , ψ) and the local stability of the disease-free equilibrium changes at the point (x 0 , ψ) [44]. Now it shall show that there are non trivial equilibrium near the bifurcation point (x 0 , ψ).  Table 3.
Now, consider the case when R 0C = 1. Suppose, further, that β C = β * C is chosen as a bifurcation parameter. Solving for R 0C = 1 gives By Lemma 3.2, the disease-free equilibrium E 0 is locally stable when β C < β * C and unstable when β C > β * C . Here β C = β * C is a bifurcation value. For computational convenience, we let with the associated forces of infection given by The Jacobian matrix of the system (A2.1) , evaluated at the DFE (ε 0C ) with β C = β * C (denoted by J ε 0C ), is given by where, The Jacobian, J(ε 0C ), of the linearized system has a simple zero eigenvalue (with all other eigenvalues having negative real part). Hence, the Center Manifold Theory [3,4] can be used to analyze the dynamics of the system (A2.1) around β C = β * C . In particular, a theorem 4.1 in [4]. Using the notation in [4], the following computations are carried out.
A5. CHIKV, DENV and CHIKV-DENV reported cases time series in India. The numbers of CHIKV, DENV and CHIKV-DENV cases in India used for model fitting are summarized in Tables A1 and A2, respectively.