A Realistic Host-Vector Transmission Model for Describing Malaria Prevalence Pattern

Abstract

Malaria continues to be a major public health concern all over the world even after effective control policies have been employed, and considerable understanding of the disease biology have been attained, from both the experimental and modelling perspective. Interactions between different general and local processes, such as dependence on age and immunity of the human host, variations of temperature and rainfall in tropical and sub-tropical areas, and continued presence of asymptomatic infections, regulate the host-vector interactions, and are responsible for the continuing disease prevalence pattern.

In this paper, a general mathematical model of malaria transmission is developed considering short and long-term age-dependent immunity of human host and its interaction with pathogen-infected mosquito vector. The model is studied analytically and numerically to understand the role of different parameters related to mosquitoes and humans. To validate the model with a disease prevalence pattern in a particular region, real epidemiological data from the north-eastern part of India was used, and the effect of seasonal variation in mosquito density was modelled based on local climactic data. The model developed based on general features of host-vector interactions, and modified simply incorporating local environmental factors with minimal changes, can successfully explain the disease transmission process in the region. This provides a general approach toward modelling malaria that can be adapted to control future outbreaks of malaria.

Appendices

Appendix A: Immunity Functions

The parameters ρ and η, which depend on the level of immunity of the host, have been incorporated in the model (7a)–(7g) through immunity functions. The details mathematical formulation is described in the following text.

In the proposed model (7a)–(7g), human immunity is considered to act in two ways: (i) reduces the probability (η) of clinical disease and (ii) regulate the rate of removal of parasites (ρ). The immunity, responsible for the first kind of work is called Clinical immunity (C(a,t)) and the immunity, which is responsible for clearance of parasite is called Parasite immunity (P(a,t)).

Clinical Immunity

Every people born with some level of immunity called maternal immunity (C _m ), which decays exponentially with a half life d _m but simultaneously accumulates some immunity (C _f ), which is determined by the force of infection and this also decays exponentially with mean life d _e . Therefore, clinical immunity can be written as

$$C (a, t) = C_{m} (a, t) + C_{f} (a, t) $$

where C _m and C _f can be found by solving the following set of equations:

$$\begin{aligned} \frac{\partial C_{m}}{\partial a} + \frac{\partial C_{m}}{\partial t} =& - \frac{C_{m}}{d_{m}} \\ \frac{\partial C_{f}}{\partial a} + \frac{\partial C_{f}}{\partial t} =& \sigma I_{m} - \frac{C_{f}}{d_{e}} \end{aligned}$$

At the time of birth, maternal immunity (C _m ) is assumed to be half of the total clinical immunity at the age of 25 years (Drakeley et al. 2006) and C _f is zero. The response of the clinical immunity η is assumed to vary by the following way:

$$\eta = \frac{1}{1 + ( \frac{C(a,t)}{H_{s}} )^{2}} $$

Parasite Immunity

This immunity is responsible for clearance of parasite and develops/matures relatively in latter age of life through a delay phase (J _A ). The immunity level in delay phase (J _A ) accumulates through force of infection (σI _m ), matures in parasite immunity (P(a,t)) at a rate (1/d _l ), and decays exponentially with mean life d _A . Parasite immunity can be obtained by solving the following set of equations:

$$\begin{aligned} \frac{\partial J_{A}}{\partial a} + \frac{\partial J_{A}}{\partial t} =& \sigma I_{m} - \frac{J_{A}}{d_{l}} \\ \frac{\partial P}{\partial a} + \frac{\partial P}{\partial t} =& \frac{J_{A}}{d_{l}} - \frac{P}{d_{A}} \end{aligned}$$

At the time of birth, J _A =P=0.

The recovery rate from the disease (ρ) is assumed to be a saturating increasing function of the immunity level P(a,t):

$$\rho = r + \rho_{0} \biggl[ 1 + (w_{A} - 1) \frac{ ( P / H_{A} )^{2}}{1 + ( P / H_{A} )^{2}} \biggr] $$

where ρ ₀ and r are the age dependent and age independent baseline recovery rate, respectively; w _A denotes maximum amplification of baseline recovery rate; H _A is the level of parasite immunity at half saturation.

Appendix B: Existence of Endemic Equilibrium Point

The proof for existence of endemic equilibrium points of the model is presented below.

Proof

The equilibrium point E ^∗ of system (7a)–(7g) can be expressed in terms of $S_{m}^{*}$ as shown in expressions (9a–9f). It can be shown easily that the existence (positivity) and uniqueness of the equilibrium point E ^∗ holds if $S_{m}^{*}$ is positive and ∈[0,1). Condition for $S_{m}^{*}\ \in [0, 1)$ We have the expression for $S_{m}^{*}$ in Eq. (10) as

$$A S_{m}^{*^{2}} + B S_{m}^{*} + C = 0 $$

From the expression for A, B, and C, it is clear that for all set of real parameter values, A<0, B>0, and C<0. Also, the expression for (A+B+C) can be written in terms of basic reproduction number R ₀ as

$$A + B + C = (R_{0} - 1) (\gamma + \mu_{h}) (\rho + \mu_{h}) (R_{S} + \mu_{h})\mu_{m} $$

Now we can observe the following situations, which determine the existence and uniqueness of the equilibrium point E ^∗:

• (i) For R ₀<1, both the roots of $S_{m}^{*}$ will lie between 0 and 1 if the following conditions hold true:

  $$\biggl \vert \frac{C}{A} \biggr \vert < 1\quad \mathrm{and}\quad C (A + B + C) > 0 $$

  But for R ₀<1, the second condition does not hold i.e. $\vert \frac{C}{A} \vert \ > 1$.

Therefore, for R ₀<1, only one equilibrium point exist which is disease free.

• (ii) If R ₀=1, then A+B+C=0.

  In this case roots are $S_{m}^{*} = 1$ and $S_{m}^{*} = \frac{C}{A}$.

  The second root exists if and only if, $\vert \frac{C}{A} \vert \le 1$.

But this condition does not hold true, hence only one equilibrium point $S_{m}^{*} = 1$ exists, which indicates again a disease free state.

• (iii) For R ₀>1, we have C (A+B+C)<0; this implies one and only one root lies between 0 and 1, which is the endemic equilibrium point E ^∗.

Hence, for R ₀>1, there exists one and only one endemic equilibrium point E ^∗, and this proves the lemma.

Hence the Lemma 3.3 can be proved following the above-mentioned conditions.

Appendix C: Coefficients of the Characteristic Equation (18)

The expressions of C ₁, C ₂, C ₃, and C ₄ of the characteristic equation (18) are as follows:

$$\begin{aligned} C_{1} =& \gamma + \rho + R_{S} + 3 \mu_{h} + \mu_{m} \\ C_{2} =& (\gamma + \mu_{h}) (\rho + \mu_{h}) + (R_{S} + \mu_{h}) (\gamma + \rho + 2 \mu_{h}) + \mu_{m} (\gamma + \rho + R_{S} + 3 \mu_{h}) \\ C_{3} =& (\gamma + \mu_{h}) (\rho + \mu_{h}) (R_{S} + \mu_{h}) + \bigl[(\gamma + \mu_{h}) ( \rho + \mu_{h}) \\ &{} + (R_{S} + \mu_{h}) (\gamma + \rho + 2 \mu_{h}) \bigr]\mu_{m} - \alpha \sigma \bigl[\gamma (1 - \eta ) C_{IA} + \gamma \eta C_{IS} \bigr] \\ =& \biggl(1 - \frac{\mu_{m}}{\mu_{h}}R_{0}\biggr) (\gamma + \mu_{h}) (\rho + \mu_{h}) (R_{S} + \mu_{h}) + \bigl[(\gamma + \mu_{h}) (\rho + \mu_{h}) \\ &{} + (R_{S} + \mu_{h}) (\gamma + \rho + 2 \mu_{h})\bigr]\mu_{m} + \frac{\alpha \gamma \sigma}{\mu_{h}}\bigl[ \bigl(1 - \eta^{2}\bigr) C_{IA}R_{S} + \eta C_{IS}\rho \bigr] \\ C_{4} =& (\gamma + \mu_{h}) (\rho + \mu_{h}) (R_{S} + \mu_{h})\mu_{m} - \alpha \gamma \sigma \bigl[\eta C_{IS}(\rho + \mu_{h}) \\ &{} + (1 - \eta ) C_{IA}\bigl\{ (1 + \eta )R_{S} + \mu_{h}\bigr\} \bigr] \\ =& (1 - R_{0}) (\gamma + \mu_{h}) (\rho + \mu_{h}) (R_{S} + \mu_{h})\mu_{m} \end{aligned}$$

Appendix D: Proof of Theorem 3.4

For τ=0, the system (7a)–(7g) is stable around the disease free equilibrium point if all the roots of the characteristic Eq. (18) are negative, and unstable if at least one root is positive.

Three of the seven roots of Eq. (18) can be easily found as −μ _h , −μ _m and −μ _m which are always negative and to know the sign of other four roots we apply Descarte’s rule of sign.

For all real positive parameter values C ₁ and C ₂>0

• (i) If R ₀>1 then C ₄<0⇒ Irrespective to the sign of C ₃, the system (7a)–(7g) is unstable around the disease free equilibrium point.

• (ii) If R ₀<1 then C ₄>0. In this case the stability of the system (7a)–(7g) around the disease free equilibrium point depends on the sign of C ₃. If C ₃>0, the system (7a)–(7g) is stable around the disease free equilibrium point and it is unstable if C ₃<0. The condition for C ₃>0 is as follows:

  $$ \sigma < \frac{(\gamma + \mu_{h})(\rho + \mu_{h})(R_{S} + \mu_{h}) + [(\gamma + \mu_{h})(\rho + \mu_{h}) + (R_{S} + \mu_{h})(\gamma + \rho + 2\mu_{h})]\mu_{m}}{\alpha \gamma [(1 - \eta )C_{IA} + \eta C_{IS}]} $$

• (iii) If R ₀<(μ _h /μ _m ), then C ₃>0 and, as the human mortality rate μ _h is less than mosquito mortality rate μ _m (i.e. μ _h /μ _m <1), therefore, C ₄>0 and, therefore, all the eigenvalues are negative and the system (7a)–(7g) is stable around the disease free equilibrium point.

• (iv) If R ₀=1, then C ₃<0, (as μ _m /μ _h >1). Therefore, all roots of the characteristic equation (18) are not negative. So, for R ₀=1 the system (7a)–(7g) will not be stable.

Hence, the theorem holds true.

Appendix E: Proof of Theorem 3.5

For τ=0, the system (7a)–(7g) is stable around the endemic equilibrium point if all the roots of the characteristic equation (14) are negative, and is unstable if at least one root is positive. Two roots of the characteristic equation (14) are −μ _m , −μ _m <0, and to find the other roots for τ=0 we rewrite the equation in the following form:

$$\begin{aligned} &\lambda^{5} + (A_{1} + A) \lambda^{4} + \bigl(A_{2} + A A_{1} + \alpha \sigma S_{m}^{*}B_{1} \bigr) \lambda^{3} \\ &\quad{} + \bigl\{ A_{3} + A A_{2} + \alpha \sigma S_{m}^{*}(B_{2} + B_{1} \mu_{h})\bigr\} \lambda^{2} \\ &\quad{}+ \bigl\{ A_{4} + A A_{3} + \alpha \sigma S_{m}^{*}(B_{3} + B_{2} \mu_{h})\bigr\} \lambda + \bigl(A A_{4} + \alpha \sigma S_{m}^{*}B_{3} \mu_{h}\bigr) = 0 \end{aligned}$$

(E.1)

where $A = \alpha (C_{IA}I_{hA}^{*} + C_{IS}I_{hS}^{*}) + \mu_{m} > 0$ and the expressions for A ₁, A ₂, A ₃, A ₄ and B ₁, B ₂, B ₃ are given in (17a–17g). For all real parameter values A ₁, A ₂, A ₃, and A ₄>0. By the Routh–Hurwitz criterion (Segel 1980), the polynomial (E.1) has roots with negative real part if all the coefficients of the polynomial are positive, i.e.

• (i) (A ₁+A)>0

• (ii) $(A_{2} + A A_{1} + \alpha \sigma S_{m}^{*}B_{1}) > 0$

• (iii) $\{ A_{3} + A A_{2} + \alpha \sigma S_{m}^{*}(B_{2} + B_{1} \mu_{h})\} > 0$

• (iv) $\{ A_{4} + A A_{3} + \alpha \sigma S_{m}^{*}(B_{3} + B_{2} \mu_{h})\} > 0$

• (v) $(A A_{4} + \alpha \sigma S_{m}^{*}B_{3} \mu_{h}) > 0$

As A ₁ and A both are positive, therefore, (i) is always true. Other conditions hold true if the parameters satisfy the condition (19a) and (19b), where the expressions for X ₁, Y ₁, X ₂, and Y ₂ are as follows:

$$\begin{aligned} X_{1} =& I^{*}_{{m}^{2}}\gamma \sigma^{2}\eta + I^{*}_{m}\gamma \sigma \eta \rho + I^{*}_{m} \gamma \sigma R_{S} + I^{*}_{{m}^{2}}\sigma^{2} \eta^{2}R_{S} + \gamma \rho R_{S} + I^{*}_{m}\sigma \rho R_{S} \\ &{}+ 2I^{*}_{m}\gamma \sigma (1 + \eta )\mu_{h} + 2I^{*2}_{m}\sigma^{2}\eta \mu_{h} + 2 \gamma \rho \mu_{h} + 2I^{*}_{m}\sigma \rho \mu_{h} + 2\rho R_{S}\mu_{h} \\ &{}+ 2I^{*}_{m}\sigma \bigl(1 + \eta^{2} \bigr)R_{S}\mu_{h} + 2\rho R_{S}\mu_{h} + 3\gamma \mu_{h}^{2} + 3I^{*}_{m}\sigma \rho (1 + \eta )\mu_{h}^{2} + 3\rho \mu_{h}^{2} \\ &{}+ 3R_{S}\mu_{h}^{2} + 4\mu_{h}^{3} + \alpha I^{*}_{hS}C_{IS}\bigl[I^{*}_{m} \sigma \bigl\{ \gamma (1 + \eta ) + I^{*}_{m}\sigma \eta \bigr \}\\ &{}+ \bigl\{ \gamma + I^{*}_{m}\sigma \bigl(1 + \eta^{2}\bigr)\bigr\} R_{S} + 3\bigl\{ \gamma + I^{*}_{m}\sigma (1 + \eta ) + R_{S}\bigr\} \mu_{h} + 6\mu_{h}^{2}\\ &{} + \rho \bigl\{ \gamma + I^{*}_{m}\sigma + R_{S} + 3 \mu_{h}\bigr\} \bigr] + \alpha C_{IA}\bigl[I^{*}_{m}\gamma I^{*}_{hA}\sigma \\ &{}+ \sigma \bigl\{ \gamma S^{*}_{m}S_{h}^{*} + \gamma \bigl(I^{*}_{m} + S^{*}_{m} \bigr)I^{*}_{hA} + I^{*}_{m} \bigl(I^{*}_{m} + S^{*}_{m} \bigr)I^{*}_{hA}\sigma \bigr\} \eta \\ &{}+ I^{*}_{hA}\bigl\{ \bigl\{ \gamma + I^{*}_{m}\sigma + \bigl(I^{*}_{m} + S^{*}_{m}\bigr)\sigma \eta^{2}\bigr\} R_{S}\\ &{} + 3\bigl\{ \gamma + I^{*}_{m}\sigma + \bigl(I^{*}_{m} + S^{*}_{m}\bigr)\sigma \eta + R_{S}\bigr\} \mu_{h} \\ &{} + 6\mu_{h}^{2} + \rho \bigl(\gamma + I^{*}_{m}\sigma + R_{S} + 3\mu_{h}\bigr) \bigr\} \bigr] + \bigl[I^{*}_{m}\sigma \bigl\{ \gamma (1 + \eta ) + I^{*}_{m}\sigma \eta \bigr\} \\ &{} + \bigl\{ \gamma + I^{*}_{m}\sigma \bigl(1 + \eta^{2}\bigr)\bigr\} R_{S} + 3\bigl\{ \gamma + I^{*}_{m}\sigma (1 + \eta ) + R_{S}\bigr\} \mu_{h} + 6\mu_{h}^{2} \\ &{} + \rho \bigl\{ \gamma + I^{*}_{m}\sigma + R_{S} + 3\mu_{h}\bigr\} \bigr]{\mu_{m}} \end{aligned}$$

$$\begin{aligned} Y_{1} =& I^{*}_{m}\gamma \sigma (1 + \eta ) + I^{*2}_{m}\sigma^{2}\eta + 3\bigl\{ \gamma + I^{*}_{m}\sigma (1 + \eta )\bigr\} \mu_{h} + 6 \mu_{h}^{2} \\ &{} + \bigl\{ \gamma + I^{*}_{m}\sigma \bigl(1 + \eta^{2}\bigr) + 3\mu_{h}\bigr\} R_{S} + \rho \bigl( \gamma + I^{*}_{m}\sigma + R_{S} + 3 \mu_{h}\bigr) \\ &{} + \bigl\{ \gamma + I^{*}_{m}\sigma (1 + \eta ) + \rho + R_{S} + 4\mu_{h}\bigr\} \bigl(\alpha I^{*}_{hA}C_{IA} + \alpha I^{*}_{hS}C_{IS} + \mu_{m}\bigr) \end{aligned}$$

$$\begin{aligned} X_{2} =& \alpha I^{*}_{hS}C_{IS} \bigl[I^{*}_{m}\sigma \bigl\{ I^{*}_{m} \gamma \sigma \eta + \bigl(\gamma + I^{*}_{m}\sigma \eta^{2}\bigr)R_{S}\bigr\} + 2\bigl\{ I^{*}_{m} \sigma \bigl(\gamma + \gamma \eta + I^{*}_{m}\sigma \eta \bigr) \\ &{} + \bigl(\gamma + I^{*}_{m}\sigma \bigl(1 + \eta^{2}\bigr)\bigr)R_{S}\bigr\} \mu_{h} + 3\bigl\{ \gamma + I^{*}_{m}\sigma (1 + \eta ) + R_{S}\bigr\} \mu_{h}^{2} + 4\mu_{h}^{3} \\ &{} + \rho \bigl\{ I^{*}_{m}\gamma \sigma \eta + \bigl( \gamma + I^{*}_{m}\sigma \bigr)R_{S} + 2\bigl(\gamma + I^{*}_{m}\sigma + R_{S}\bigr)\mu_{h} + 3\mu_{h}^{2}\bigr\} \bigr] \\ &{} + \alpha C_{IA}\bigl[\sigma \bigl\{ I^{*}_{m} \gamma \bigl(I^{*}_{m} + S^{*}_{m} \bigr)I^{*}_{hA}\sigma \eta + \bigl(I^{*}_{m} \gamma I^{*}_{hA} + \bigl(\gamma S^{*}_{m} \bigl(S^{*}_{h} + I^{*}_{hA}\bigr) \\ &{}+ I^{*}_{m}\bigl(I^{*}_{m} + S^{*}_{m}\bigr)I^{*}_{hA}\sigma \bigr) \eta^{2}\bigr)R_{S}\bigr\} + 2\bigl\{ I^{*}_{m} \gamma I^{*}_{hA}\sigma + \sigma \bigl(\gamma S^{*}_{m}S^{*}_{h} + \gamma \bigl(I^{*}_{m} + S^{*}_{m} \bigr)I^{*}_{hA} \\ &{} + I^{*}_{m}\bigl(I^{*}_{m} + S^{*}_{m}\bigr)I^{*}_{hA}\sigma \bigr) \eta + I^{*}_{hA}\bigl(\gamma + S^{*}_{m} \sigma + \bigl(I^{*}_{m} + S^{*}_{m}\bigr) \sigma \eta^{2}\bigr)R_{S}\bigr\} \mu_{h} \\ &{}+ 3I^{*}_{hA}\bigl\{ \gamma + S^{*}_{m} \sigma + \bigl(I^{*}_{m} + S^{*}_{m}\bigr) \sigma \eta + R_{S}\bigr\} \mu_{h}^{2} + 4I^{*}_{hA}\mu_{h}^{3} \\ &{} + I^{*}_{hA}\rho \bigl\{ I^{*}_{m} \gamma \sigma \eta + \bigl(\gamma + I^{*}_{m}\sigma \bigr)R_{S} + 2\bigl(\gamma + I^{*}_{m}\sigma + R_{S}\bigr)\mu_{h} + 3\mu_{h}^{2}\bigr\} \bigr] \\ &{} + \mu_{h}\bigl[\bigl(I^{*}_{m}\gamma + \mu_{h}\bigr)\bigl\{ (\gamma + \mu_{h}) \bigl(I^{*}_{m} \sigma \eta + \mu_{h}\bigr) + R_{S}\bigl(\gamma + I^{*}_{m}\sigma \eta^{2} + \mu_{h}\bigr) \bigr\} \\ &{} + \rho \bigl\{ I^{*}_{m}\gamma \sigma \eta + \bigl( \gamma + I^{*}_{m}\sigma + \mu_{h}\bigr) (R_{S} + \mu_{h}\bigr\} \bigr] + \bigl[I^{*}_{m} \gamma \bigl\{ I^{*}_{m}\gamma \sigma \eta \\ &{} + \bigl(\gamma + I^{*}_{m}\sigma \eta^{2} \bigr)R_{S}\bigr\} + 2\bigl\{ I^{*}_{m}\sigma \bigl( \gamma + \gamma \eta + I^{*}_{m}\sigma \eta \bigr) + \bigl( \gamma + I^{*}_{m}\sigma \bigl(1 + \eta^{2}\bigr) \bigr)R_{S}\bigr\} \mu_{h} \\ &{} + 3\bigl\{ \gamma + I^{*}_{m}\sigma (1 + \eta ) + R_{S}\bigr\} \mu_{h}^{2} + 4\mu_{h}^{3} + \rho \bigl\{ I^{*}_{m}\gamma \sigma \eta + \bigl(\gamma + I^{*}_{m}\sigma \bigr)R_{S} \\ &{} + 2\bigl(\gamma + I^{*}_{m}\sigma + R_{S} \bigr)\mu_{h} + 3\mu_{h}^{2}\bigr\} \bigr] \mu_{m} \end{aligned}$$

$$\begin{aligned} Y_{2} =& \mu_{h}\bigl[\alpha C_{IA}\bigl\{ I^{*}_{m}\gamma \bigl(I^{*}_{m} + S^{*}_{m}\bigr)I^{*}_{hA} \sigma^{2}\eta + \mu_{h}\bigl\{ I^{*}_{m} \gamma I^{*}_{hA}\sigma \\ &{} + \sigma \bigl\{ \gamma S^{*}_{m}S^{*}_{h} + \gamma \bigl(I^{*}_{m} + S^{*}_{m} \bigr)I^{*}_{hA} + I^{*}_{m} \bigl(I^{*}_{m} + S^{*}_{m} \bigr)I^{*}_{hA}\sigma \bigr\} \eta \\ &{}+ I^{*}_{hA}\mu_{h}(\gamma + S^{*}_{m}\sigma + \bigl(I^{*}_{m} + S^{*}_{m}\bigr)\sigma \eta + \mu_{h}\bigr\} \bigr\} \\ &{} + R_{S}\bigl\{ I^{*}_{m}\gamma I^{*}_{hA}\sigma + \sigma \bigl\{ \gamma S^{*}_{m} \bigl(S^{*}_{h} + I^{*}_{hA}\bigr) + I^{*}_{m}\bigl(I^{*}_{m} + S^{*}_{m}\bigr)I^{*}_{hA}\sigma \bigr\} \eta^{2} \\ &{}+ I^{*}_{hA}\mu_{h}\bigl\{ \gamma + S^{*}_{m}\sigma + \bigl(I^{*}_{m} + S^{*}_{m}\bigr)\sigma \eta^{2} + \mu_{h} \bigr\} \bigr\} \\ &{} + I^{*}_{hA}\rho \bigl\{ I^{*}_{m} \gamma \sigma \eta + \bigl(\gamma + I^{*}_{m}\sigma + \mu_{h}\bigr) (R_{S} + \mu_{h})\bigr\} \\ &{} + \bigl\{ \bigl(I^{*}_{m}\sigma + \mu_{h} \bigr)\bigl\{ (\gamma + \mu_{h}) \bigl(I^{*}_{m} \sigma \eta + \mu_{h}\bigr) + R_{S}\bigl(\gamma + I^{*}_{m}\sigma \eta^{2} + \mu_{h}\bigr) \bigr\} \\ &{} + \rho \bigl\{ I^{*}_{m}\gamma \sigma \eta + \bigl( \gamma + I^{*}_{m}\sigma + \mu_{h}\bigr) (R_{S} + \mu_{h})\bigr\} \bigr\} \bigl(\alpha I^{*}_{hS}C_{IS} + \mu_{m}\bigr)\bigr] \end{aligned}$$

Appendix F: Real and Imaginary Parts of the Expression (16)

$$\operatorname{Re} \bigl[\varDelta (i \omega, \tau )\bigr] = \xi_{1} + \alpha \sigma \psi S_{m}^{*}(\xi_{2} \operatorname{Cos} \omega \tau + \xi_{3} \operatorname{Sin} \omega \tau ) = 0 $$

$$\operatorname{Im} \bigl[\varDelta (i \omega, \tau )\bigr] = \xi_{4} + \alpha \sigma \psi S_{m}^{*}(\xi_{3} \operatorname{Cos} \omega \tau - \xi_{2} \operatorname{Sin} \omega \tau ) = 0 $$

where

$$\begin{aligned} \xi_{1} =& (A_{0} + A_{1}) \omega^{4} - (A_{3} + A_{0} A_{2}) \omega^{2} + A_{0} A_{4} \\ \xi_{2} =& B_{3} \mu_{h} - (B_{2} + B_{1} \mu_{h}) \omega^{2} \\ \xi_{3} =& (B_{3} + B_{2} \mu_{h}) - B_{1} \omega^{2} \\ \xi_{4} =& \omega^{4} - (A_{2} + A_{0} A_{1}) \omega^{2} + (A_{4} + A_{0} A_{3}) \end{aligned}$$

with A ₀=α(C _IS I _hS +C _IA I _hA )+μ _m .

Appendix G: Age Distribution of Population in India

To compare the simulated result with the real malaria cases, one should keep in mind the age distribution of population in that locality. Although due to lack of data availability, we have considered the age-wise population distribution, obtained from 2001 census data of India as a standard population distribution in India (Census of India 2001). According to this distribution, the percentage of population at age a is

$$k = \frac{1}{4.92}(12.41 - 0.156a)\quad \mathrm{with}\ \int _{0}^{a_{\max}} k\,da = 100. $$

Here, a _max is the maximum age group of people available in the population.