Dynamical analysis of chikungunya and dengue co-infection model

Salihu Sabiu Musa; Nafiu Hussaini; Shi Zhao; He Daihai

doi:10.3934/dcdsb.2020009

Article Contents

2020, Volume 25, Issue 5: 1907-1933. Doi: 10.3934/dcdsb.2020009

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Dynamical analysis of chikungunya and dengue co-infection model

1.
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China
2.
Department of Mathematics, Kano University of Science and Technology, Wudil, Nigeria
3.
Department of Mathematical Sciences, Bayero University Kano, Nigeria
4.
School of Nursing, Hong Kong Polytechnic University, Hong Kong, China

^* Corresponding authors: nhussaini.mth@buk.edu.ng; and daihai.he@polyu.edu.hk
^* Corresponding authors: nhussaini.mth@buk.edu.ng; and daihai.he@polyu.edu.hk

Received: January 31, 2019

Revised: June 30, 2019

Early access: December 2019

Published: May 2020

The authors acknowledge the helpful comments by the editor and the reviewers

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Abstract

The aim of this paper is to design and analyze a nonlinear mechanistic 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 combat/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., trivial 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. Sensitivity 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 maturation rate are the top control parameters for combating CHIKV, DENV and CHIKV-DENV co-infection outbreaks.

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Mathematics Subject Classification: Primary: 34K18, 34K25; Secondary: 92D30.

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Figure 1. Schematic diagram of the model Eqn (1). The yellow line represent infection of CHIKV from the DENV recovered class, while the orange line represent infection of DENV from the CHIKV recovered class. The green and the blue lines represent CHIKV and DENV new infection as well as the recovery of individuals, all the parameters are defined in Table 2

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Figure 2. Schematic diagram of the model Eqns(2). The yellow line represent the vertical transmission of CHIKV virus from aquatic stage. The green and blue lines represent new infection of CHIKV and DENV, all the parameters are defined in Table 2

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Figure 3. Backward bifurcation diagram of the model (9)

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Figure 4. Fitting result of the CHIKV only sub-model (6). We used the parameter values from Table 3 and the following initial conditions: $ S_H(0) = 1.3\times 10^9 $, $ E_C(0) = 1.2\times 10^5 $, $ I_C(0) = 48176 $, $ R_C(0) = 4000 $, $ A(0) = 8\times 10^7 $, $ S_v(0) = 5\times 10^7 $, $ E_{vC}(0) = 2\times 10^5 $ and $ I_{vC}(0) = 10^4 $. The vertical axes indicate the cumulative number of CHIKV cases in India from 2010 to 2017

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Figure 6. Fitting result of the full model of Eqns (1)-(2). We used the parameter values from Table 3 and the following initial conditions: $ S_H(0) = 1179681900 $, $ E_C(0) = 1200 $, $ E_D(0) = 1200 $, $ E_{CD}(0) = 10 $, $ I_C(0) = 8 $, $ I_D(0) = 8 $, $ I_{CE}(0) = 8 $, $ I_{DE}(0) = 8 $, $ I_{CD}(0) = 6 $, $ E_{CT}(0) = 12 $, $ E_{DT}(0) = 12 $, $ I_{CT}(0) = 8 $, $ I_{DT}(0) = 8 $, $ R_{C}(0) = 8 $, $ R_{D}(0) = 8 $, $ T(0) = 3 $, $ A(0) = 8\times 10^{10} $, $ S_{v}(0) = 5\times 10^{10} $, $ E_{vC}(0) = 2\times 10^5 $, $ E_{vD}(0) = 2\times 10^5 $, $ E_{M}(0) = 2\times 10^5 $, $ I_{vC}(0) = 10^4 $, $ I_{vD}(0) = 10^4 $, $ I_{vCE}(0) = 100 $, $ I_{vDE}(0) = 10^4 $, and $ I_{vM}(0) = 10^4 $. The vertical axes indicate the cumulative number of DENV cases in India since 2007 to 2012

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Figure 5. Fitting result of the DENV only sub-model(9). We used the parameter values from Table 3 and the following initial conditions: $ S_H(0) = 1.3\times 10^9 $, $ E_D(0) = 1.1\times 10^5 $, $ I_D(0) = 28292 $, $ R_D(0) = 3000 $, $ A(0) = 8\times 10^6 $, $ S_v(0) = 5\times 10^6 $, $ E_{vD}(0) = 2\times 10^5 $ and $ I_{vD}(0) = 10^5 $. The vertical axes indicate the cumulative number of DENV cases in India from 2010 to 2017

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Figure 7. The PRCCs (of the CHIKV only sub-model(6)) of basic reproduction number (panel (a)) and infection attack rate (panel (b)) with respect to the model parameters. $ m_1 $ denotes the mosquito to human ratio. The blue dots are the estimated correlations and the bars represent the 95% CIs. The ranges of parameters are given in Table 3

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Figure 8. The PRCCs (of the DENV only sub-model(9)) of basic reproduction number (panel (a)) and infection attack rate (panel (b)) with respect to the model parameters. $ m_2 $ denotes the mosquito to human ratio. The blue dots are the estimated correlations and the bars represent the 95% CIs. The ranges of parameters are given in Table 3

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Table 1. Interpretation of the compartmental variables of the model Eqns (1)-(2)

Variable	Interpretation/Description
$ N_{H} $	Total population of humans
$ S_{H} $	Population of susceptible humans
$ E_{C} $	Population of asymptomatic CHIKV individuals
$ E_{D} $	Population of asymptomatic DENV individuals
$ E_{CD} $	Population of humans exposed to both CHIKV and DENV parasite
$ I_{C} $	Population of CHIKV-infected (only) humans with clinical symptoms of CHIKV
$ I_{D} $	Population of DENV-infected (only) humans with clinical symptoms of DENV
$ I_{CD} $	Population of dually-infected humans with symptoms of both CHIKV and DENV
$ I_{CE} $	Population of CHIKV-infected humans with clinical symptoms of CHIKV but exposed to DENV
$ I_{DE} $	Population of DENV-infected humans with clinical symptoms of DENV but exposed to CHIKV
$ R_{C} $	Recovered CHIKV-infected humans
$ R_{D} $	Recovered DENV-infected humans
$ E_{CT} $	Population of individuals exposed to CHIKV but recovered from DENV with permanent immunity
$ E_{DT} $	Population of individuals exposed to DENV but recovered from CHIKV with permanent immunity
$ I_{CT} $	Population of CHIKV-infected individuals with clinical symptoms of CHIKV but recovered from DENV with permanent immunity
$ I_{DT} $	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
$ N_{V} $	Total population of mosquitoes
$ A $	population of immature mosquitoes (egg, lava and pupa stages)
$ N_{I} $	Total population of adult mosquitoes
$ S_{v} $	population of adult mosquitoes susceptible to both CHIKV and DENV
$ E_{vC} $	population of adult mosquitoes exposed to CHIKV
$ E_{vD} $	population of adult mosquitoes exposed to DENV
$ E_{M} $	population of adult mosquitoes exposed to both CHIKV and DENV viruses
$ I_{vC} $	Population of CHIKV-infected (only) adult mosquitoes
$ I_{vD} $	Population of DENV-infected (only) adult mosquitoes
$ I_{vCE} $	Population of CHIKV-infected adult mosquitoes that are exposed to DENV
$ I_{vDE} $	Population of DENV-infected adult mosquitoes that are exposed to CHIKV
$ I_{vM} $	Population of adult mosquitoes infected to both CHIKV and DENV

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Table 2. Interpretation of the parameters of the model Eqns (1)-(2)

Parameter	Interpretation/Description
$ \Pi_H,\Pi_V $	Recruitment rate of humans and mosquitoes, respectively
$ \mu_{H} $	Natural death rate of humans
$ \mu_A $	Death rate of immature mosquitoes
$ \mu_v $	Death rate of adult mosquitoes
$ \lambda_{C} $	Rates of CHIKV force of infection in humans
$ \lambda_{D} $	Rates of DENV force of infection in humans
$ \lambda_{vC} $	Rates of CHIKV force of infection in mosquitoes
$ \lambda_{vD} $	Rates of DENV force of infection in mosquitoes
$ \beta_{C} $	Transmission probability for CHIKV to humans
$ \beta_{D} $	Transmission probability for DENV to humans
$ \beta_{v} $	Transmission probability from an infectious human to a susceptible adult mosquitoes
$ b_{1} $	Number of bites per human per unit time
$ b_{2} $	Number of bites per mosquitoes per unit time
$ \xi $	Fraction of immature mosquitoes becoming susceptible adult
$ \alpha_1 $	Modification parameter for the heterogeneity of DENV infection between susceptible humans and humans exposed to CHIKV
$ \alpha_2 $	Modification parameter for the heterogeneity of CHIKV infection between susceptible humans and humans exposed to DENV
$ \omega_1 $	Modification parameter for the heterogeneity of DENV infection between susceptible adult mosquitoes and those exposed to CHIKV
$ \omega_2 $	Modification parameter for the heterogeneity of CHIKV infection between susceptible adult mosquitoes and those exposed to DENV
$ \sigma_{m} $	Number of times a mosquito bites humans per unit time
$ \sigma_{H} $	Maximum number of mosquito bites a human can receive per unit time
$ \sigma_{C} $	Progression rate of humans from exposed state of CHIKV to the infectious state of CHIKV
$ \sigma_{D} $	Progression rate of humans from exposed state of DENV to the infectious state of DENV
$ \sigma_{vC} $	Progression rate of adult mosquitoes from exposed state of CHIKV to the infectious state of CHIKV
$ \sigma_{vD} $	Progression rate of adult mosquitoes from exposed state of DEN to the infectious state of DENV
$ \gamma_i(i=1,4) $	Progression rates of humans to active CHIKV classes
$ \gamma_j(j=2,3) $	Progression rates of humans to active DENV classes
$ \theta_1,\rho_2 $	Progression rates of adult mosquitoes to active CHIKV classes
$ \theta_2,\rho_1 $	Progression rates of adult mosquitoes to active DENV classes
$ \tau_{C} $	Recovery rate of humans from infectious state of CHIKV to the recovered state of CHIKV
$ \tau_{D} $	Recovery rate of humans from infectious state of DENV to the recovered state of DENV
$ \eta $	Modification parameters for the increase in infectiousness of dually-infected humans in comparison to mono-infected humans
$ \eta_{C},\eta_{D},\eta_{CD},\eta_{vC},\eta_{vD},\eta_{M} $	Modification parameters for the increase in infectiousness for the exposed classes in humans and mosquitoes, respectively
$ \delta_{C},\delta_{D},\delta_{CE},\delta_{DE},\delta_{CD},\delta_{CT},\delta_{DT} $	Disease-induced death rates for humans

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Table 3. Values and ranges of the parameters of the model Eqns (1)-(2)

Parameter	Baseline; (Range)	Unit	Source(s)
[0.5ex] $ \mu_H $	$ 3.9\times10^{-5};\; (3.6, 4.0)\times10^{-5} $	$ \text{day}^{-1} $	[1,34]
$ \mu_v $	$ 0.05714;\; (0.01, 0.1) $	$ \text{day}^{-1} $	[37]
$ \mu_A $	$ 0.174;\; (0.0143, 0.33) $	$ \text{day}^{-1} $	[23,50]
$ \beta_C $	$ 0.375;\; (0.001, 0.54) $	$ \text{day}^{-1} $	[8,42]
$ \beta_D $	$ 0.75;\; (0.1, 0.95) $	$ \text{day}^{-1} $	[26]
$ \beta_v $	$ 0.375;\; (0.1, 0.5) $	$ \text{day}^{-1} $	[8,9,27,36]
$ \sigma_m $	$ 0.5;\; (0.33, 1) $	$ \text{day}^{-1} $	[7,26]
$ \sigma_H $	$ 1;\; (0.1, 10) $	$ \text{day}^{-1} $	[16]
$ \sigma_C $	$ 0.35;\; (0, 1) $	$ \text{day}^{-1} $	Estimated [1,16]
$ \sigma_D $	$ 0.5;\; (0, 1) $	$ \text{day}^{-1} $	Estimated [16]
$ \sigma_{vC} $	$ 0.25;\; (0.1, 1) $	$ \text{day}^{-1} $	Estimated [16]
$ \sigma_{vD}, \sigma_{CT}, \sigma_{DT} $	$ 0.2;\; (0, 1) $	$ \text{day}^{-1} $	Estimated [16]
$ \tau_C $	0.2(0.1429, 0.3333)	$ \text{day}^{-1} $	Estimated [15]
$ \tau_D $	$ 0.25;\; (0.01, 0.3) $	$ \text{day}^{-1} $	[15]
$ \Pi_H $	$ 2.5;\; (1, 5) $	$ \text{day}^{-1} $	[16]
$ \Pi_v $	$ 5000;\; (2500, 6000) $	$ \text{day}^{-1} $	[16]
$ \gamma_1 $	$ 0.23;\; (0, 1) $	Dimensionless	Assumed
$ \gamma_2 $	$ 0.25;\; (0, 1) $	Dimensionless	Assumed
$ \gamma_3 $	$ 0.3;\; (0, 1) $	Dimensionless	Assumed
$ \gamma_4 $	$ 0.23;\; (0, 1) $	Dimensionless	Assumed
$ \delta_C $, $ \delta_D $	$ 1 \times 10^{-3};\; (0.0005, 0.0015) $	$ \text{day}^{-1} $	[16]
$ \delta_{CE} $, $ \delta_{DE} $	$ 1.5 \times 10^{-3};\; (0.00051, 0.0015) $	$ \text{day}^{-1} $	Assumed
$ \delta_{CD} $, $ \delta_{CT} $, $ \delta_{DT} $	$ 1.2 \times 10^{-3};\; (0.0005, 0.002) $	$ \text{day}^{-1} $	Assumed
$ \eta_C $, $ \eta_M $	$ 0.1;\; (0, 1) $	Dimensionless	Estimated [16]
$ \eta_{CD} $, $ \eta_{vC} $, $ \eta_{vD} $	$ 0.1;\; (0, 1) $	Dimensionless	Estimated [16]
$ \eta $, $ \eta_D $	$ 0.12;\; (0, 1) $	Dimensionless	Assumed
$ \xi $	$ 0.01;\; (0.001, 0.021) $	Dimensionless	Estimated [8]
$ \alpha_1 $	$ 0.03;\; (0.15, 0.99) $	Dimensionless	Estimated
$ \alpha_2 $	$ 0.018;\; (0.10, 1) $	Dimensionless	Estimated
$ \omega_1 $	$ 0.015;\; (0, 1) $	Dimensionless	Assumed
$ \omega_2 $	$ 0.013;\; (0, 0.9) $	Dimensionless	Assumed
$ \theta_1 $	$ 0.01;\; (0.005, 0.016) $	Dimensionless	Assumed
$ \theta_2 $, $ \rho_1 $	$ 0.01;\; (0.005, 0.018) $	Dimensionless	Assumed
$ \rho_2 $	$ 0.01;\; (0.0051, 0.01) $	Dimensionless	Assumed

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Table A1. Human reported CHIKV and DENV cases in India [31]

Year	CHIKV	DENV
$ 2010 $	$ 48176 $	$ 28292 $
$ 2011 $	$ 20402 $	$ 18860 $
$ 2012 $	$ 15977 $	$ 50222 $
$ 2013 $	$ 18840 $	$ 75808 $
$ 2014 $	$ 16049 $	$ 40571 $
$ 2015 $	$ 27553 $	$ 99913 $
$ 2016 $	$ 64057 $	$ 129166 $
$ 2017 $	$ 62268 $	$ 157220 $

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Table A2. Human reported CHIKV-DENV co-infection cases in India [13]

Year	No. of cases
$ 2007 $	$ 8 $
$ 2008 $	$ 1 $
$ 2009 $	$ 8 $
$ 2010 $	$ 5 $
$ 2011 $	$ 9 $
$ 2012 $	$ 1 $

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