The Impact of Recruitment on the Dynamics of an Immune-Suppressed Within-Human–Host Model of the Plasmodium falciparum Parasite

Abstract

A model is developed and used to study within-human malaria parasite dynamics. The model integrates actors involved in the development–progression of parasitemia, gametocytogenesis and mechanisms for immune response activation. Model analyses under immune suppression reveal different dynamical behaviours for different healthy red blood cell (HRBC) generation functions. Existence of a threshold parameter determines conditions for HRBCs depletion. Oscillatory dynamics reminiscent of malaria parasitemia are obtained. A dependence exists on the type of recruitment function used to generate HRBCs, with complexities observed for a more nonlinear function. An upper bound that delimits the size of feasible parasitized steady-state solution exists for a logistic function but not a constant function. The upper bound is completely characterized and is affected by parameters associated with HRBCs recruitment, parasitized red blood cells generation and the release and time-to-release of free merozoites. A stable density size for mature gametocytes, the bridge to invertebrate hosts, is derived.

Appendix

1.1 Positivity and Positive Invariance Solution

Theorem 7

(Statement of the positivity and positive invariance of solution theorem) Consider system (8)–(14) with initial conditions in (15) and under the conditions given for $\psi (R_h)$ and $H_{i}(E_{i})$ as stated in Sect. 2.2. Then, every solution of the system with initial condition in $\mathbb {R}_+^7$ remains in $\mathbb {R}_+^7$. Additionally, if $\varvec{x}(0)\equiv \varvec{0}$, the solution of system (8)–(14) will remain zero (or positively bounded depending on the form of $\psi (R_{h})$), for all time $t>0$ . That is, $\mathbb {R}_+^7$ is positively invariant and attracting with respect to the system. Furthermore, the system has a forward positive solution in $\mathbb {R}_+^7$ provided that it starts in it.

Proof

We show that the region $\mathbb {R}^7_+$ is positively invariant, that is, whenever $\varvec{x}(0)\in \mathbb {R}^{7}_{+}$, $\varvec{x}\in \mathbb {R}^{7}_{+},\forall t\ge 0$. It suffices to show that there is no solution of the system starting in $\mathbb {R}^7_+$ which is non-positive. Thus, we are required to show that the rate of change of each state variable, that is each $\phi _i, ~1\le i\le 7,$ is non-negative at the origin $\varvec{0}= (0, 0,0,0,0,0,0)$, and on each of the coordinate axis. Notice that at the origin $\varvec{0}$, if $\varvec{x}_0 = \varvec{0}$, then $\varvec{x}'(0) = \Phi (\varvec{0})= \varvec{0}$ if $\psi (R_{h}) = \Lambda -\tilde{\mu }_{h}R_{h}$, or $\varvec{x}'(0) = (\Theta ,0,0,0,0,0,0)$ if $\psi (R_{h})=\frac{\Theta }{R_{h}}$. Thus, if $\varvec{x}(0)=\varvec{0}$, each component of $\varvec{x}$ remains stationary at zero or increases from zero depending on the form of $\psi$. On the other hand, if any one of the components of $\varvec{x}$ is zero, the rate of change of that component with time is non-negative, showing that no trajectory of the system passes out of $\mathbb {R}^{7}_{+}$ through that component’s zero axes. For example, when $R_{p}=0$, $R_{p}'=\frac{\beta _{1} R_{h}M}{1+\xi _0 E_a}\ge 0$, since $R_{h}$ and M are non-negative for all time, showing that no solution of the system passes out of $\mathbb {R}^{7}_{+}$ through the $R_{p}=0$ axis. This implies the vector field of the system is inward pointing on the boundary of $\mathbb {R}^{7}_{+}.$ That is, if $\varvec{x}_0 \in \mathbb {R}^7_+,$ then $\varvec{x} \in \mathbb {R}^7_+, ~\forall t\ge 0.$ Therefore, the region $\mathbb {R}^{7}_{+}$ is positively invariant and attracting.

Next to prove the positivity of the solution, we follow the steps in Ngwa et al. (2016), Page 8. Suppose there exists $t_1 >0$ such that $R_h (t_1) =0, ~~ R_h'(t_1) < 0$ and $R_h, R_p, M, G_e, G_l, E_i, E_a > 0$ for all $0<t < t_1.$ Then

$$\begin{aligned} R'_h (t_1) =\underbrace{R_{h}(t_{1})\psi (R_h(t_1))}_{:=\psi _{0}} -\underbrace{\mu _h R_h(t_1)}_ { :=0}-\underbrace{\frac{\beta _{1} R_h(t_1)M(t_1)}{1+\xi _0 E_a(t_1)}}_{:=0} = \left\{ \begin{array}{lcl} \Theta &{} \text{ if } &{} \psi (R_{h})=\frac{\Theta }{R_{h}}\\ 0 &{} \text{ if } &{}\psi (R_{h}) = \Lambda -\tilde{\mu }_{h}R_{h}.\end{array}\right. \end{aligned}$$

In either case, $R'_h (t_1)\ge 0$, leading to a contradiction to the assumption that $R'_h(t_1) < 0.$ So, no such $t_1$ exists, and hence $R_h \ne 0$. Thus, $R_h (t) > 0, ~~\forall t\ge 0.$ Next suppose that there exists $t_2$ such that $R_p(t_2)=0, R'_p (t_2) <0$ and $R_h, R_p, M, G_e, G_l > 0$ for all $0 <t \le t_2.$ Then, $R'_p (t_2)= \frac{\beta _{1} R_h(t_2)M(t_2)}{1+\xi _0 E_a(t_2)} -\underbrace{(\gamma _{p}+\mu _p)R_p(t_2)}_{:0} = \frac{\beta _{1} R_h(t_2)M(t_2)}{1+\xi _0 E_a(t_2)} >0,$ which is a contradiction to the assumption that $R'_p (t_2) < 0.$ Hence, $R_p(t)> 0, ~~\forall t>0.$ Similarly, one can show that $M(t)> 0,~ G_e (t)> 0, ~G_l (t)>0, E_i(t) >0$ and $E_a (t) >0$ for all $t>0.$ Therefore, any solution of the system with an initial condition in $\mathbb {R}_+^7$ is positive. $\square$

1.2 Boundedness of Solution

Theorem 8

(Statement of the boundedness of solution theorem) Consider system (8)–(14) with initial conditions in (15) and under the conditions for $\psi (R_h)$ and $H_{i}(E_{i})$ as stated in Sect. 2.2. Then, every forward solution of the system in $\mathbb {R}_+^7$, with initial condition in $\mathbb {R}_+^7$, is bounded. Moreover, the system is uniformly dissipative in $\mathbb {R}_+^7.$

Proof

To start the proof of boundedness, we first note the following about boundedness of $f(R_h)$ and $H_i(E_i).$

• 1. For all values of $R_{h}$, we have

  $$\begin{aligned} R_{h}\psi (R_{h}) \le \mathcal {K_R}, \text{ where } \mathcal {K_R}=\left\{ \begin{array}{lll} \Theta &{}\quad \text{ if }&{}\psi (R_{h})=\frac{\Theta }{R_{h}}\\ \frac{\Lambda ^{2}}{4\tilde{\mu }_{h}}&{}\quad \text{ if }&{}\psi (R_{h})=\Lambda -\tilde{\mu }_{h}R_h.\end{array}\right. \end{aligned}$$

  (61)

  We note that the requirement that $\psi$ be monotone non-increasing tacitly comes along with the requirement that $R_{h}\psi (R_{h})$ be continuous from right at the origin. In particular, $\psi (R_{h})$ satisfies conditions (1)–(3) of Sect. 2.2. Other examples, besides the two studied in this manuscript, of recruitment functions $\psi (R_{h})$ found in the biological literature that satisfy conditions (1)–(3) of Sect. 2.2 are $\psi (R_{h})=\Lambda e^{-\tilde{\mu }_{h}R_{h}}$, Ricker recruitment function and $\psi (R_{h})=\frac{\Lambda }{1+ \left( \frac{R_{h}}{L}\right) ^{n}}$, $\Lambda ,L,n,\tilde{\mu }_{h}>0$ which is the Maynard–Smith–Slatkin function. Details on these types of recruitment functions can be found in Brännström and Sumpter (2005) and Ngonghala et al. (2016).

• 2. Similarly, for all values of $E_{i}$ we have

  $$\begin{aligned}&H_{i}(E_{i})\le \mathcal {K}_{i} \text{ where } \mathcal {K}_{i} \nonumber \\&\quad = \left\{ \begin{array}{lcl} \frac{\delta _{i}K_{i}}{4}&{}\quad \text{ if }&{}\quad H(E_{i})=\delta _{i}E_{i}\left( 1-\frac{E_{i}}{K_{i}}\right) \\ \max (A_{1},A_{2},0)&{}\quad \text{ if }&{}\quad H_{i}(E_{i})=\delta _{i}E_{i}\left( 1-\frac{E_{i}}{K_{i}}\right) \left( \frac{E_{i}}{M_{i}}-1\right) \end{array}\right. \end{aligned}$$

  (62)

  where on setting $B = -K_i M_i+K_i^2+M_i^2=\left( M_{i}-\frac{1}{2}K_{i}\right) ^2 + \frac{3}{4}K_{i}^2>0$ we can obtain

  $$\begin{aligned} A_{1}= & {} \frac{\delta _i \left( -\sqrt{B}+K_i-2 M_i\right) \left( -\sqrt{B}+K_i+M_i\right) \left( \sqrt{B}+2 K_i-M_i\right) }{27 K_i M_i}\\ A_{2}= & {} -\frac{\delta _i \left( \sqrt{B}+K_i-2 M_i\right) \left( \sqrt{B}-2 K_i+M_i\right) \left( \sqrt{B}+K_i+M_i\right) }{27 K_i M_i}. \end{aligned}$$

  Thus, the functions $H_{i}$ and $R_{h}\psi$ defined are bounded.

Now to prove the boundedness of the $R_h$ and $R_p,$ let $R (t)= R_{h}(t)+R_{p}(t)$ be the total size of red blood cells within the human at time t, (healthy plus infected red blood cells) with $R (0)= R_{h}(0)+R_{p}(0)=R(0)$. Then, we have from the first two equations of system (8)–(14)

$$\begin{aligned} \frac{\mathrm{d} R}{\mathrm{d} t}= & {} R_{h}\psi (R_{h})-\mu _{h}R_{h}-(\gamma _{p}+\mu _p)R_p-(\rho _{p}+\rho _a E_a)R_pE_i\\\le & {} f(R_{h})-\mu R, \text{ where } \mu = \min (\mu _{h},\gamma _{p}+\mu _p), \end{aligned}$$

where $f(R_{h}) = R_{h}\psi (R_{h})$ with $\psi :[0,\infty )\rightarrow \mathbb {R}_+$ a monotone decreasing continuously differentiable function. So, the function $f:[0,\infty )\rightarrow \mathbb {R}_+$ has a maximum value which is either constant when f is the constant function, or that occurs at the point $R_{h}^{*}\in [0,\infty )$, where $R_{h}^{*}$ satisfies the equation $f'(R_{h}^{*})=\psi (R_{h}^{*})+R_{h}^{*}\psi '(R_{h}^{*})=0$. Set $\mu =\min (\mu _{h},\gamma _{p}+\mu _p)$ and suppose that the maximum value of f is $\mathcal {K}_{R}$, then we have from above,

$$\begin{aligned} \frac{\mathrm{d} R}{\mathrm{d} t} +\mu R\le & {} \mathcal {K}_{R}\Rightarrow R(t)\le \frac{\mathcal {K}_{R}}{\mu }+ A e^{-\mu t}, \end{aligned}$$

where A is an arbitrary constant that can be determined from initial data. Observe that if the initial condition, R(0), is such that $R(0)> \frac{\mathcal {K}_{R}}{\mu }$, then A is always positive and the bound for R(t) is decreasing with time. When $R(0)= \frac{\mathcal {K}_{R}}{\mu }$, then A is non-negative and the bound for R(t) is non-increasing with time. Finally, if $R(0)< \frac{\mathcal {K}_{R}}{\mu }$, A can be a negative number and the bound for R(t) will be an increasing function of t. If at any of the instances we see that

$$\begin{aligned} \lim _{t\rightarrow \infty }{\sup {R(t)}}\le \frac{\mathcal {K}_{R}}{\mu }. \end{aligned}$$

(63)

Thus, $0\le R_{h}(t)+R_{p}(t)\le \frac{\mathcal {K}_{R}}{\mu }, ~\forall t\ge 0.$ So there exist $R_{h}^{\infty }$ and $R_{p}^{\infty }$ with the property that $0\le R_{h}(t)\le R_{h}^{\infty }$ and $0\le R_{p}(t)\le R_{p}^{\infty }$, $\forall t\ge 0$. Hence, $R_h$ and $R_p$ are bounded solutions.

Next we consider the equation for M, namely,

$$\begin{aligned} \frac{\mathrm{d} M}{\mathrm{d}t}= & {} \frac{r \gamma _{p}(1-\sigma ) R_p}{1+\xi _1 E_a (t)} - \mu _m M-\left( \frac{\beta _{2} R_h}{1+\xi _0E_a} + \frac{\beta _{3} R_p}{1+\xi _0E_a} +(\rho _m +\rho _{n} E_{a})E_{i}\right) M, \end{aligned}$$

and observe that when we take into consideration the fact that the quantity $\frac{1}{1+\xi _1 E_a }$ is largest when $E_{a}=0$, we have that

$$\begin{aligned} \frac{\mathrm{d} M}{\mathrm{d} t}\le & {} r \gamma _{p}(1-\sigma ) R_p^{\infty } - \mu _{m}M \Rightarrow M(t)\le \frac{ r \gamma _{p}(1-\sigma ) R_p^{\infty }}{\mu _{m}}+ B e^{-\mu _{m} t}, \end{aligned}$$

where B is an arbitrary constant. As above we arrive at the conclusion that there exist $M^{\infty }$ such that $0\le \sup {M(t)}\le M^{\infty },~\forall t\ge 0$. So M is bounded.

Next to prove the boundedness of $G_e$ and $G_l$, we set $G(t)=G_{e}(t)+G_{l}(t)$ to be the total size of gametocytes within the human and see that

$$\begin{aligned} \frac{\mathrm{d} G}{\mathrm{d} t}= & {} \frac{s \sigma \gamma _{p} R_p}{1+\xi _1 E_a}-\left( \gamma _{l}+ \mu _e\right) G_e -(\rho _g + \rho _{q}E_a)G_{e}E_{i} + \frac{\gamma _{l} G_e}{1+\xi _1 E_a }- \mu _l G_l -\rho _l E_i G_l\\\le & {} s \sigma \gamma _{p} R_p^{\infty }-\min (\mu _{e},\mu _{l})G. \end{aligned}$$

Thus

$$\begin{aligned} \dfrac{\mathrm{d} G}{\mathrm{d}t}+\min (\mu _{e},\mu _{l})G\le r \sigma \gamma _{p,m} R_p^{\infty }\Rightarrow G(t)\le \frac{r \sigma \gamma _{p,m} R_p^{\infty }}{\min (\mu _{e},\mu _{l})}+ C e^{-\min (\mu _{e},\mu _{l}) t}. \end{aligned}$$

Therefore, as before, there exist $G_{l}^{\infty }$ and $G_{e}^{\infty }$ with the property that for $0\le G_{e}(t)\le G_{e}^{\infty }$ and $0\le G_{l}(t)\le G_{l}^{\infty }$, $\forall t\ge 0$. So $G_{e}$ and $G_{l}$ are bounded whenever the preceding variables are bounded.

To establish boundedness of the solutions for the equations of the innate and adaptive immune responses, we proceed as follows. From the last two equations of the general model, system (8)–(14), and using the above results, we get

$$\begin{aligned} \dfrac{\mathrm{d} E_i}{\mathrm{d}t}+\left( \lambda _{1} R_p^{\infty }+ \lambda _{2}M^{\infty } \right) E_i\le & {} \mathcal {K}_{i} + \vartheta _1 R_p^{\infty } + \vartheta _{2}M^{\infty }\\ \dfrac{\mathrm{d} E_a}{\mathrm{d}t}+\left( \mu _a+\theta _{1} R_p^{\infty }+\theta _{2} M^{\infty }) \right) E_a\le & {} \varrho _{1}R_p^{\infty } + \varrho _{2}M^{\infty }, \end{aligned}$$

with the right hand side here being only constants and we can again argue as above to come to the conclusion that each $E_{i}$ will show bounded growth whenever M, $R_{h}$ and $R_{P}$ are bounded. This completes the prove for boundedness. So, if we let

$$\begin{aligned} B^{\infty } = \max \{ R_h^{\infty },R_h^{\infty }, M^{\infty }, G_e^{\infty }, G_l^{\infty }, E_i^{\infty } , E_a^{\infty } \}, \end{aligned}$$

then each of $R_h,~R_p, ~M, ~G_e, ~G_l, ~E_i, ~E_a \le B^{\infty }.$ In the absence of disease, system (8)–(14) reduces to the decoupled equations for the healthy red blood cell population and for the immune cells as follows:

$$\begin{aligned} \frac{\mathrm{d} R_{h}}{\mathrm{d} t}= & {} R_{h}\psi (R_{h})-\mu _{h}R_{h}, ~~\frac{\mathrm{d} E_i}{\mathrm{d}t}= H_i(E_i), ~~ \frac{\mathrm{d} E_a}{\mathrm{d}t}= -\mu _{a}E_{a}. \end{aligned}$$

Note here that in the absence of Allee effect, $H_i(E_i)$ can have similar forms as $F(R_h).$ That is, we can write

$$\begin{aligned} H_i(E_i)= E_i\varphi (E_i)- \mu _i E_i, \end{aligned}$$

where $\varphi :[0, \infty )\rightarrow \mathbb {R}_+$ is a function defined similarly as $\psi$ and satisfies the conditions stated for $\psi .$

So, to prove the boundedness of these functions we observe that the equation for the healthy red blood cell population then satisfies the relation

$$\begin{aligned} \frac{\mathrm{d} R_{h}}{\mathrm{d} t} = R_{h}\psi (R_{h})-\mu _{h}R_{h}\Rightarrow t(R_{h}) = \int \frac{1}{R_{h}(\psi (R_{h})-\mu _{h})}\mathrm{d} R_{h} + C, \end{aligned}$$

where C is an arbitrary constant of integration. For the functional forms of $\psi$ used here, if at time $t=0, R_{h}(t)=R_{0h}$, we have

$$\begin{aligned} t(R_{h}) = \left\{ \begin{array}{lll} \frac{1}{\mu _{h}}\ln \left( \frac{\Theta -\mu _{h}R_{0h}}{\Theta -\mu _{h}R_{h}}\right) &{}\quad \text{ if }&{}\psi (R_{h})= \frac{\Theta }{R_{h}}\\ \frac{1}{\omega }\ln \left( \frac{R_{h}(R_{0h}-K)}{R_{0h}(R_{h}-K)}\right) &{}\quad \text{ if }&{}\psi (R_{h})=\Lambda -\tilde{\mu }_{h}R_{h}\end{array}\right) , \end{aligned}$$

where $\omega = \Lambda -\mu _{h} \text{ and } K=\frac{\omega }{\tilde{\mu }_{h}}$. For both forms of recruitment, it is clear that $t(R_{h})\rightarrow \infty$ when $R_{h}\rightarrow \Theta /\mu _{h}$ or $R_{h}\rightarrow K$, respectively. So the solutions remain bounded. Also, in the absence of disease, the expression for the innate immunity at any time can be written as an exact integral. That is

$$\begin{aligned} \frac{\mathrm{d} E_{i}}{\mathrm{d} t} = H_{i}(E_{i})\Rightarrow t(E_{i}) = \int \frac{1}{H_{i}(E_{i})}\mathrm{d}E_{i} +C, \end{aligned}$$

(64)

where C is a constant whose values can be determined by the initial conditions. So,

$$\begin{aligned} t(E_{i}) = \left\{ \begin{array}{lcl} \frac{K_i \left( \ln \left( E_{i}-M_i\right) -\ln (E_{i})\right) +M_i \left( \ln (E_{i})-\ln \left( E_{i}-K_i\right) \right) }{\delta _i \left( K_i-M_i\right) } +C &{}\quad \text{ if }&{}H_{i}(E_{i})=E_{i} \delta _i \left( 1-\frac{E_{i}}{K_i}\right) \left( \frac{E_{i}}{M_i}-1\right) \\ \frac{\ln (E_{i})-\ln \left( E_{i}-K_i\right) }{\delta _i}+ C&{}\quad \text{ if }&{}H_{i}(E_{i})=E_{i} \delta _i \left( 1-\frac{E_{i}}{K_i}\right) \end{array}\right) , \end{aligned}$$

so that if at time $t=0$, $E_{i}(0) =E_{0i}$, we have the implicit solution

$$\begin{aligned} t(E_{i}) = \left\{ \begin{array}{lcl} \frac{1}{\delta _{i}(K_{i}-M_{i})}\ln \left( \left( \frac{E_{0i}(E_{i}-M_{i})}{E_{i}(E_{0i}-M_{i})}\right) ^{K_{i}}\left( \frac{E_{i}(E_{0i}-K_{i})}{(E_{i}-K_{i})E_{0i}}\right) ^{M_{i}}\right) &{}\text{ if }&{}H_{i}(E_{i})=E_{i} \delta _i \left( 1-\frac{E_{i}}{K_i}\right) \left( \frac{E_{i}}{M_i}-1\right) \\ \frac{1}{\delta _{i}}\ln \left( \frac{E_{i}(E_{0i}-K_{i}}{E_{0i}(E_{i}-K_{i})}\right) &{}\text{ if }&{}H_{i}(E_{i})=E_{i} \delta _i \left( 1-\frac{E_{i}}{K_i}\right) \end{array}\right. \end{aligned}$$

We then see clearly that for the logistic case, $t\rightarrow \infty$ whenever $E_{i}\rightarrow K_{i}$ for any starting value of $E_{0i}>0$. In the case with the Allee effect, if $0<M_{i}<K_{i}$ then $0<E_{0i}<M_{i}$, $t(E)\rightarrow \infty$ as $E_{i}\rightarrow 0+$, while if $E_{0i}>M_{i}$, then again, $t(E)\rightarrow \infty$ as $E_{i}\rightarrow K_{i}$. This shows that in either case, the solutions remain bounded. The inverse function theorem can be applied to obtain the solution $E_{i}(t)$ in some special cases of values of $M_{i}$ and $K_{i}$. We have thus established boundedness of the solutions in all cases in both the presence and absence of the infection. $\square$

1.3 Uniqueness of Solution

Theorem 9

(Statement on the Uniqueness of Solution) The positive and bounded solution for the system (8)–(14) whenever it exists, is unique.

Proof

We show that the function $\Phi$ defined above is globally Lipschitz in $\mathbb {R}_+^7$ and hence the equation $\varvec{x}'(t)=\Phi (\varvec{x}(t)), \varvec{x}(0)=\varvec{x}_{0}$ has a unique solution. It is clear that $\mathbb {R}_+^7$ is a convex set, $\Phi$ is continuously differentiable, since the partial derivatives $\frac{\partial \Phi }{\partial x_i}, ~~i=1,2,\ldots ,7$ exist, and are continuous. We show that these partial derivatives are bounded in $\mathbb {R}_+^7:$ Since $R_h \psi (R_h)$ is monotone decreasing, continuously differentiable function and each state variable $R_h, R_p, M, G_e, G_l, E_i, E_a$ are continuously differentiable, then each component $\Phi _i,~~i=1,2,3\ldots ,7$ of the vector valued function $\Phi$ on right hand side of system (8)–(14) exists and are continuously differentiable because they are rational functions of the state variables. It suffices to show that $\Vert \frac{\partial \Phi }{\partial x_{i}}\Vert _{\infty }$$i=1,2, \cdots , 7$ are bounded where $(x_1,x_{2},x_{3},x_{4},x_{5},x_{6},x_{7})=(R_h, R_p, M, G_e, G_l, E_i, E_a)$. Observe, for example, that

$$\begin{aligned} \left\| \frac{\partial \Phi }{\partial R_{h}}\right\| _{\infty }= & {} \max \left\{ \left| \frac{\partial \phi _{i}}{\partial R_{h}}\right| , i= 1,2,\ldots ,7\right\} \\= & {} \max \left\{ \left| \psi (R_{h})+R_{h}\psi '(R_{h})-\mu _{h}-\frac{\beta _{1}M}{1+\xi _{0}E_{a}}\right| , ~\left| \frac{\beta _{1}M}{1+\xi _{0}E_{a}}\right| ,0,0,0,0,0\right\} \\\le & {} \left| \psi (R_{h})|+R_{h}|\psi '(R_{h})\right| + \left| \mu _{h}+\frac{\beta _{1}M}{1+\xi _{0}E_{a}}\right| \\\le & {} B_{1} \end{aligned}$$

for some $B_{1}$, since M and $E_{a}$ are bounded, and $\psi$ is monotone decreasing so that $|\psi '|$ is monotone increasing and bounded by say K where K is the carrying capacity for $R_{h}$, so $B_{1}$ exists. Similarly, there exists $B_i < \infty$, for $i=2, 3, \cdots , 7$ such that

$$\begin{aligned} \left\| \frac{\partial \Phi }{\partial R_p}\right\| _{\infty }= & {} B_2< \infty , ~\left\| \frac{\partial \Phi }{\partial M}\right\| _{\infty }= B_3< \infty , \left\| \frac{\partial \Phi }{\partial G_e}\right\| _{\infty } = B_4<\infty ,\\ \left\| \frac{\partial \Phi }{\partial G_l}\right\| _{\infty }= & {} B_5<\infty ,~~\left\| \frac{\partial \Phi }{\partial E_i}\right\| _{\infty }= B_6<\infty , ~\left\| \frac{\partial \Phi }{\partial E_{a}}\right\| _{\infty }= B_7 <\infty . \end{aligned}$$

We would have established that the partial derivatives are bounded and hence the function $\Phi (\varvec{x})$ defined by the right hand side of (8)–(14) is Lipschitzian. Now let $\varvec{x}_1, ~\varvec{x}_2$ be two arbitrary points in $\mathbb {R}^7_+.$ Then define,

$$\begin{aligned} \mathbf z (\varvec{x}_1, \varvec{x}_2; \theta ) = \{ \varvec{x}_1+\theta (\varvec{x}_2-\varvec{x}_1),~~0\le \theta \le 1 \} . \end{aligned}$$

Then, $\mathbf z (\varvec{x}_1, \varvec{x}_2; \theta )$ is a line segment joining points $\varvec{x}_1$ and $\varvec{x}_2$ in $\mathbb {R}_+^7$ for $\theta \in [0, 1].$ Furthermore, $\mathbf z (\varvec{x}_1, \varvec{x}_2; \theta )$ is a convex function and since $\mathbb {R}_+^7$ is a convex set, then $\mathbf z (\varvec{x}_1, \varvec{x}_2; \theta ) \in \mathbb {R}_+^7$ for each $\theta \in [0, 1].$

Using the mean value theorem for differentiable functions in $\mathbb {R}_+^n$, one can show that

$$\begin{aligned} \Vert \Phi (\varvec{x}_1)- \Phi (\varvec{x}_2)\Vert _{\infty } =\Vert D_{\Phi }(\varvec{c}; \varvec{x}_1-\varvec{x}_2)\Vert _{\infty }, \end{aligned}$$

where $\mathbf c$ is the mean value point and $D_{\Phi }$ is the directional derivative of $\Phi$ at the point $\mathbf c$ in the direction of the vector $\varvec{x}_1 -\varvec{x}_2.$ Using the expression for the directional derivative, as well as applying the triangle inequality and the Cauchy–Schwartz inequality, we see that

$$\begin{aligned} \Vert D_{\Phi }( \mathbf z ; \varvec{x}_1 - \varvec{x}_2) \Vert _{\infty }= & {} \left\| \sum _{k=1}^7 \nabla \Phi _k (\mathbf z ) \cdot (\varvec{x}_1 - \varvec{x}_2)\mathbf e _k \right\| _{\infty } \\ {}\le & {} \left\| \sum _{k=1}^7 \nabla \Phi _k (\mathbf z ) \right\| _{\infty } \Vert (\varvec{x}_1 - \varvec{x}_2)\Vert _{\infty }\\\le & {} \sum _{k=1}^7 \Vert \nabla \Phi _k (\mathbf z ) \Vert _{\infty } \Vert (\varvec{x}_1 - \varvec{x}_2)\Vert _{\infty } \le \mathcal {L} \Vert (\varvec{x}_1 - \varvec{x}_2)\Vert _{\infty }, \end{aligned}$$

for some constant $\mathcal {L} = 7\max \{B_{1},B_{2},B_{3},B_{4},B_{5},B_{6},B_{7}\}$ where the last inequality comes from the fact that each partial derivative of $\Phi$ is bounded and $\mathbf e _k$ is the $k^{th}$ unit vector in $\mathbb {R}_+^7.$ Hence, there exists a constant $\mathcal {L} >0$ such that

$$\begin{aligned} \Vert \Phi (\varvec{x}_1) -\Phi ( \varvec{x}_2) \Vert _{\infty } \le \mathcal {L} \Vert (\varvec{x}_1 - \varvec{x}_2)\Vert _{\infty }. \end{aligned}$$

Hence, $\Phi$ is Lipschitz continuous and therefore, by the Picard’s existence and uniqueness theorem, the system under study has a unique solution. $\square$