Modelling cystic echinococcosis and bovine cysticercosis co-infections with optimal control

Abstract

Cystic echinococcosis and bovine cysticercosis are diseases of economic importance especially to rural communities that earn their incomes from livestock. In this paper, a mathematical model for cystic echinococcosis and bovine cysticercosis co-infections is formulated and analyzed to determine parameters that drive the diseases and design the optimal control strategy. The basic reproduction number ${\mathcal {R}}_0$ that governs the dynamics of cystic echinococcosis and bovine cysticercosis is computed by the next-generation matrix method, and the normalized forward sensitivity index is used to derive the sensitivity indices of model parameters. Sensitivity analysis shows that co-infections of cattle with cystic echinococcosis and bovine cysticercosis, open defecation rate by humans who are infected with taeniasis and the rate of slaughtering co-infected cattle play a significant role in the persistence of cystic echinococcosis and bovine cysticercosis. Sobol sensitivity analysis has been carried out to study global sensitivity of model parameters to state variables. The results show that the rate at which exposed dogs progress to infectious class, the rate at which humans who are exposed to taeniasis progress to infectious stage, the rate at which dogs are recruited, cattle per capita natural death rate, cattle recruitment rate and tapeworm eggs’ natural death rate are the most sensitive parameters to exposed dogs, humans who are exposed to taeniasis, infectious dogs, infectious cattle, exposed cattle and tapeworm eggs in the environment respectively. The control strategies, such as cattle indoor keeping, meat inspection, and improved hygiene and sanitation are implemented in combination to determine the possibility of controlling the spread of cystic echinococcosis and bovine cysticercosis. The Pontryagin’s maximum principle is applied to determine diseases’ optimal control strategy. Results show that cattle indoor keeping and meat inspection are more effective in disease control when they are concurrently implemented. Rural communities that earn income from livestock can control cystic echinococcosis and bovine cysticercosis by reducing the number of cattle that are kept in free range system, inspect meat from cattle and improve hygiene and sanitation.

Appendix I: non-negativity and boundedness of solutions

The model system (1) is mathematically and biologically meaningful if its solutions are non-negative and bounded. In this section we investigate non-negativity of model solutions and establish their boundedness.

1.1 Non-negativity of model solutions

Showing that model solutions are positive for all $t\ge 0$, the first equation for susceptible humans in model system (1) can be written as;

$$\begin{aligned} \begin{aligned}&\dfrac{dS_H}{dt}=\Lambda _H-\sigma _H E_VS_H-\Phi _H M S_H-\mu _HS_H,\\&\dfrac{dS_H}{dt}\ge -(\sigma _H E_V+\Phi _H M +\mu _H)S_H. \end{aligned} \end{aligned}$$

Separation of variables leads to;

$$\begin{aligned} \dfrac{dS_H}{S_H}\ge -(\sigma _H E_V+\Phi _H M +\mu _H)dt. \end{aligned}$$

Integration and application of initial condition gives

$$\begin{aligned} S_H(t)\ge S_H(0)e^{-\displaystyle \int _{0}^{t}(\sigma _H E_V(s)+\Phi _H M(s) +\mu _H)ds}\ge 0. \end{aligned}$$

(39)

Applying the same approach for the rest of equations, we obtain;

$$\begin{aligned} \begin{aligned}&E_{HE}(t)\ge E_{HE}(0)e^{-(\gamma _E+\mu _H+\rho _T\Psi _{HE})t}\ge 0,~~\\ {}&E_{HT}(t)\ge E_{HT}(0)e^{-(\mu _E+\tau _E \theta _{HT})t}\ge 0,~~\\ {}&E_{HB}(t)\ge E_{HB}(0)e^{-(\mu _{H}+\gamma _{B})t}\ge 0,~~\\ {}&I_{HE}(t)\ge I_{HE}(0)e^{-(\mu _H+\mu _E)t}\ge 0,~~\\ {}&I_{HT}(t)\ge E_{HT}(0)e^{-\mu _{H}t}\ge 0,~~\\ {}&I_{HB}(t)\ge I_{HB}(0)e^{-(\mu _H+\mu _E)t}\ge 0,~~\\ {}&S_C(t)\ge S_C(0)e^{-\displaystyle \int _{0}^{t}(\sigma _{C}E_V(s)+\mu _C+\delta _1)ds}\ge 0,~~\\ {}&E_{C}(t)\ge E_{C}(0)e^{-(\gamma _{C}+\mu _{C}+\delta _2)t}\ge 0,~~\\ {}&I_C(t)\ge I_C(0)e^{-(\delta _3+\mu _C)t}\ge 0,~~\\ {}&S_D(t)\ge S_D(0)e^{-\displaystyle \int _{0}^{t}(\sigma _{D}M(s)+\mu _D)ds}\ge 0,~~\\ {}&E_{D}(t)\ge E_{D}(0)e^{-(\gamma _{D}+\mu _{D})t}\ge 0,~~\\ {}&I_D(t)\ge I_D(0)e^{-\mu _D t}\ge 0,~~\\ {}&M(t)\ge M(0)e^{-\mu _Mt}\ge 0,\\ {}&E_V(t)\ge E_V(0)e^{-\mu _Et}\ge 0. \end{aligned} \end{aligned}$$

(40)

This indicates that all solutions of the model system (1) are non-negative for all $t\ge 0$.

1.2 Boundedness of model solutions

To establish boundedness of solutions, we consider separately the total populations for humans, cattle and dogs. Considering the total populations for humans, cattle and dogs respectively, we have;

$$\begin{aligned} \begin{aligned}&\dfrac{dN_H}{dt}\le \Lambda _H-\mu _HN_H,\\ {}&\dfrac{dN_C}{dt}\le \Lambda _C-\mu _CN_C,\\&\dfrac{dN_D}{dt}\le \Lambda _D-\mu _DN_D. \end{aligned} \end{aligned}$$

(41)

Solving the first equation in (41) and applying initial conditions, we obtain:

$$\begin{aligned} N_H(t)\le \dfrac{\Lambda _H}{\mu _H}+\left( N_H(0)-\dfrac{\Lambda _H}{\mu _H}\right) e^{-\mu _Ht}. \end{aligned}$$

(42)

Two cases when $N_H(0)>\dfrac{\Lambda _H}{\mu _H}$ and when $N_H(0)<\dfrac{\Lambda _H}{\mu _H}$ are considered to analyze (42). For the two cases we obtain;

$$\begin{aligned} \begin{aligned}&\dfrac{\Lambda _H}{\mu _H}\le N_H(t)\le \dfrac{\Lambda _H}{\mu _H}+\left( N_H(0)-\dfrac{\Lambda _H}{\mu _H}\right) e^{-\mu _Ht}\\ {}&~~~~~~~~~~~~~~~~~~~~~~~~\text {and}\\ {}&\dfrac{\Lambda _H}{\mu _H}+\left( N_H(0)-\dfrac{\Lambda _H}{\mu _H}\right) e^{-\mu _Ht}\le N_H(t)\le \dfrac{\Lambda _H}{\mu _H} \end{aligned} \end{aligned}$$

(43)

respectively. As $t\rightarrow \infty$, equation (43) becomes;

$$\begin{aligned} N_H(t)\le \dfrac{\Lambda _H}{\mu _H}. \end{aligned}$$

(44)

Applying the same procedure for the second and third equations in (41), we have respectively:

$$\begin{aligned} \begin{aligned}&N_C(t)\le \dfrac{\Lambda _C}{\mu _C+\delta _1}~~\text {and}~~ N_D(t)\le \dfrac{\Lambda _D}{\mu _D}. \end{aligned}\nonumber \\ \end{aligned}$$

(45)

Since $N_C(t)\le \dfrac{\Lambda _C}{\mu _C+\delta _1}$ and $N_D(t)\le \dfrac{\Lambda _D}{\mu _D}$ then $I_C\le \dfrac{\Lambda _C}{\mu _C+\delta _1}$ and $I_D\le \dfrac{\Lambda _D}{\mu _D}$ . Without loss of generality, it can be shown that

$$\begin{aligned} M(t)\le \dfrac{\delta _3\Lambda _C}{(\mu _C+\delta _1)(\mu _M+\Phi _H +\sigma _D)}~~\text {and}~~ E_V(t)\le \dfrac{\alpha _D\Lambda _D}{\mu _E\mu _D(\mu _C+\delta _1)}+\dfrac{2\omega _{T}\Lambda _H}{\mu _E\mu _H}. \end{aligned}$$

(46)

The solutions of the model system (1) enter the region

$$\begin{aligned} \begin{array}{lll} \Omega =\left\{ (X_H, Y_C,Z_D, M, E_V)\in {\mathbb {R}}^{15}_{+}: N_H\le \dfrac{\Lambda _H}{\mu _H}, N_C\le \dfrac{\Lambda _C}{\mu _C+\delta _1},\right. \\ \qquad \qquad \left. N_D\le \dfrac{\Lambda _D}{\mu _D} M(t)\le \Gamma , E_V\le \beta \right\} , \end{array} \end{aligned}$$

(47)

where

$$\begin{aligned} \begin{aligned}&X_H=S_H, E_{HE},E_{HT},E_{HT},E_{HB} I_{HE}, I_{HT},I_{HB}; ~ Y_C=S_C, E_C, I_C;Z_D = S_D, E_D, I_D; \\ {}&\Gamma = \dfrac{\delta _3\Lambda _C}{(\mu _C+\delta _1)(\mu _M+\Phi _H +\sigma _D)};\beta =\dfrac{\alpha _D\Lambda _D}{(\mu _C+\delta _1)(\mu _E\mu _D)}+\dfrac{2\omega _{T}\Lambda _H}{\mu _E\mu _H} . \end{aligned} \end{aligned}$$

Thus, the region $\Omega$ is positive invariant and the solutions of the model system (1) that start at the boundary of $\Omega$ enter the region in finite time. Hence, we can consider the flow generated by the model (1) for the analysis. This result is summarized in Theorem 1.

Theorem 5

The solutions of the model system (1) are non-negative and bounded in the region $\Omega$.

Appendix II: global sensitivity indices of model parameters

Variable	Low.ci	High.ci	Sensitivity	Parameter
$S_H$	0.1528	0.2081	$T_i$	$\sigma _H$
$S_H$	0.1461	0.1980	$T_i$	$\mu _C$
$E_{HE}$	0.6418	0.8181	$T_i$	$\gamma _E$
$E_{HT}$	0.5586	0.6973	$T_i$	$\gamma _T$
$E_{HB}$	0.3887	0.4988	$T_i$	$\gamma _T$
$E_{HB}$	0.2538	0.3363	$T_i$	$\sigma _{HT}$
$I_{HE}$	0.1614	0.2118	$T_i$	$\sigma _H$
$I_{HE}$	0.1226	0.1679	$T_i$	$\gamma _E$
$I_{HT}$	0.2809	0.3742	$T_i$	$\gamma _T$
$I_{HT}$	0.1181	0.1584	$T_i$	$\sigma _H$
$I_{HB}$	0.3047	0.3929	$T_i$	$\gamma _T$
$I_{HB}$	0.2766	0.3663	$T_i$	$\sigma _{HT}$
$S_C$	0.2492	0.3354	$T_i$	$\mu _V$
$S_C$	0.2082	0.2796	$T_i$	$\Lambda _D$
$E_C$	0.3821	0.4847	$T_i$	$\Lambda _D$
$E_C$	0.2879	0.3684	$T_i$	$\mu _C$
$E_C$	0.2651	0.3419	$T_i$	$\gamma _C$
$I_C$	0.4739	0.6144	$T_i$	$\mu _C$
$I_C$	0.1455	0.1940	$T_i$	$\delta _3$
$S_D$	0.2293	0.3065	$T_i$	$\mu _C$
$S_D$	0.1960	0.2641	$T_i$	$\Lambda _D$
$S_D$	0.1355	0.1887	$T_i$	$\delta _3$
$S_D$	0.0976	0.1298	$T_i$	$\Phi _H$
$E_D$	0.6100	0.7645	$T_i$	$\gamma _D$
$E_D$	0.2676	0.3442	$T_i$	$\Lambda _D$
$I_D$	0.5384	0.6745	$T_i$	$\Lambda _D$
$I_D$	0.2434	0.3146	$T_i$	$\mu _D$
$I_D$	0.0938	0.1226	$T_i$	$\gamma _D$
M	0.3161	0.4123	$T_i$	$\mu _C$
M	0.1640	0.2181	$T_i$	$\delta _3$
M	0.1360	0.1802	$T_i$	$\Phi _H$
$E_V$	0.3662	0.4715	$T_i$	$\mu _V$
$E_V$	0.2244	0.2885	$T_i$	$\Lambda _D$
$E_V$	0.1714	0.2251	$T_i$	$\alpha _D$