Application of the NSFD method in a Malaria model with nonlinear incidence and recovery rates

Abstract

In this manuscript, we formulate a malaria model with nonlinear incidence and recovery rates to study the impact of the health system’s available resources on disease spread and control. A detailed analysis of the model is provided. It is determined that when the basic reproduction number, ${\mathcal {R}}_0$, is less than unity, the disease may or may not die out due to nonlinear recovery rate. In addition, it is asserted that if the recovery rate is constant, disease eradication is possible for ${\mathcal {R}}_0<1$. The global stability of the unique endemic equilibrium point is established using geometric approach. The model system is also examined for Hopf bifurcation, and it is established that certain conditions on the transmission rate from vector to human lead to the emergence of periodic oscillations in the model system. Keeping in mind the nonlinear nature of the hypothesized model, we develop a non-standard finite difference (NSFD) scheme by discretizing the system. It is proven that conservation law and the positivity of the solutions are maintained by the proposed NSFD method for all finite step sizes. Furthermore, the convergence and error bounds of the developed schemes are also explored. To validate the analytical results, numerical simulations using our computational scheme are presented and these results are compared to two well-known standard numerical techniques, viz. the fourth-order Runge–Kutta (RK4) method and the forward Euler method. It is found that conventional numerical schemes fail to accurately capture the dynamics of the continuous model for certain step sizes, resulting in unstable and negative numerical solutions. In contrast, the developed NSFD scheme successfully retains the fundamental mathematical characteristics of the continuous model. It is worth mentioning that the NSFD schemes also capture the backward bifurcation phenomenon. In addition, numerical simulations show that Hopf bifurcation occurs due to transmission rate.

Appendices

Appendix 1

$$\begin{aligned} {\mathcal {S}}_4=&a_{11}-a_{22}+a_{44}+c_2+\mu _h+2 \mu _v,\\ {\mathcal {S}}_3=&a_{11} c_2-a_{22} c_2+a_{44} c_2-a_{22} \mu _h+a_{44} \mu _h+2 a_{11} \mu _v-2 a_{22} \mu _v+a_{44} \mu _v-a_{11} a_{22}-a_{15} a_{42}+\\ {}&a_{11} a_{44}-a_{22} a_{44}+c_2 \mu _h+2 c_2 \mu _v+2 \mu _h \mu _v+\mu _v^2, \\ {\mathcal {S}}_2=&-a_{22} c_2 \mu _h+a_{44} c_2 \mu _h+2 a_{11} c_2 \mu _v-2 a_{22} c_2 \mu _v+a_{44} c_2 \mu _v-a_{11} a_{22} c_2-a_{15} a_{42} c_2+\\ {}&a_{11} a_{44} c_2-a_{22} a_{44} c_2-a_{11} a_{32} \gamma _h-a_{15} a_{42} \mu _h-a_{22} a_{44} \mu _h-2 a_{22} \mu _h \mu _v+a_{44} \mu _h \mu _v+a_{11} \mu _v^2-a_{22} \mu _v^2-2 a_{11} a_{22} \mu _v-\\ {}&a_{15} a_{42} \mu _v+a_{11} a_{44} \mu _v-a_{22} a_{44} \mu _v-a_{11} a_{22} a_{44}+2 c_2 \mu _h \mu _v+c_2 \mu _v^2+\mu _h \mu _v^2,\\ {\mathcal {S}}_1=&-a_{15} a_{42} c_2 \mu _h-a_{22} a_{44} c_2 \mu _h-2 a_{22} c_2 \mu _h \mu _v+a_{44} c_2 \mu _h \mu _v+a_{11} c_2 \mu _v^2-a_{22} c_2 \mu _v^2-2 a_{11} a_{22} c_2 \mu _v-a_{15} a_{42} c_2 \mu _v+ \\ {}&a_{11} a_{44} c_2 \mu _v-a_{22} a_{44} c_2 \mu _v-a_{11} a_{22} a_{44} c_2-a_{11} a_{32} a_{44} \gamma _h-2 a_{11} a_{32} \gamma _h \mu _v-a_{22} \mu _h \mu _v^2-\\ {}&a_{15} a_{42} \mu _h \mu _v-a_{22} a_{44} \mu _h \mu _v-a_{11} a_{22} \mu _v^2-a_{11} a_{22} a_{44} \mu _v+c_2 \mu _h \mu _v^2,\\ {\mathcal {S}}_0=&-a_{22} c_2 \mu _h \mu _v^2-a_{15} a_{42} c_2 \mu _h \mu _v-a_{22} a_{44} c_2 \mu _h \mu _v-a_{11} a_{22} c_2 \mu _v^2-a_{11} a_{22} a_{44} c_2 \mu _v-a_{11} a_{32} \gamma _h \mu _v^2-a_{11} a_{32} a_{44} \gamma _h \mu _v. \end{aligned}$$

Appendix 2

$$\begin{aligned} &A_2 A_5 A_7-\phi _1 \gamma _h A_4 A_6\\ &\quad =\bigg (1+\phi _1 \left( \frac{\beta _h V_i^{n}}{1+\alpha _v V_i^n}+\mu _h\right) \bigg ) \bigg ((I_h^n+b) +\phi _1 ((\mu _h+\delta _h)(I_h^{n}+b)\\ & \qquad +\mu _0 I_h^{n}+\mu _1 b)\bigg ) \bigg (1+\phi _1(\mu _h+\gamma _h)\bigg )-\phi _1^3 \gamma _h \bigg ( \frac{\beta _h V_i^n}{1+\alpha _v V_i^n}(I_h^n+b)\bigg ) \left( \mu _0+(\mu _1-\mu _0)\frac{b}{I_h^n+b}\right) \\ &\quad =A_2 A_7 \left( I_h^k+b\right) \left( 1+\phi _1(\mu _h+\delta _h)\right) +A_7 \phi _1 \left( 1+\phi _1 \mu _h\right) \left( \mu _0 I_h^n+\mu _1 b\right) \\ &\qquad + \phi _1 ^2 \left( 1+\phi _1 \mu _h\right) \left( \mu _0 I_h^n +\mu _1 b\right) \left( \dfrac{\beta _h V_i^n}{1+\alpha _v V_i^n}\right) >0. \end{aligned}$$