Domain models in decay chemistry ecology and public health

A method becomes useful when its variables, units and assumptions match a process. We apply first-order equations and coupled balances across several domains, using illustrative parameters. For the physical models in Sections 2–6, time is forward time $t\ge0$, with initial values specified at $t=0$.

1 Two meanings of a homogeneous first order equation

For a homogeneous linear equation $y'+p(t)y=0$ with continuous coefficient $p$, the positive integrating factor $\mu=\exp(\int p)$ gives $(\mu y)'=0$, so $y=C/\mu$; this is the homogeneous case of the integrating-factor method. Another common use of homogeneous is $y'=F(y/t)$ with continuous $F$ on the ratio range, where simultaneous scaling of $t,y$ leaves the right-hand side unchanged. On an interval with $t\ne0$, putting $y=tv$ gives $tv'=F(v)-v$, which is separable. Constant-ratio branches satisfying $F(v)=v$ must be retained before dividing. These are distinct mechanisms, despite the shared terminology.

2 Geochronology and a closed decay model

Assume an initial parent isotope amount $P_0>0$ and that the parent amount $P$ decays at constant rate $\lambda>0$, produces a stable daughter one-for-one, and the system neither gains nor loses parent or daughter by other routes. Then $P'=-\lambda P$, giving $P=P_0e^{-\lambda t}$. For $t\ge0$ this gives $P(t)>0$, so the daughter-to-parent ratio $D/P$ is defined. Its half-life is $H=\log2/\lambda$.

If daughter amount starts at zero, $D'=\lambda P$ and $P+D=P_0$. Hence

$$\frac DP=e^{\lambda t}-1,\qquad t=\frac1\lambda\log\left(1+\frac DP\right).$$

A ratio $D/P=3$ gives $t=\log4/\lambda=2H$. A nonzero initial daughter amount or exchange with surroundings changes the inference. The differential equation by itself does not establish those geological assumptions, nor does a measured ratio determine an age without a specified rate and model.

3 Consecutive geochemical reactions

For an ideal first-order chain $A\to B\to C$ with rates $\lambda_1,\lambda_2>0$, write amounts as $a,b,c$:

$$a'=-\lambda_1a,\quad b'=\lambda_1a-\lambda_2b,\quad c'=\lambda_2b.$$

Adding shows conservation of $a+b+c$. The first solution is $a=a_0e^{-\lambda_1t}$. Multiplying the middle equation by $e^{\lambda_2t}$ and integrating gives, when $\lambda_1\ne\lambda_2$,

$$b=b_0e^{-\lambda_2t}+\frac{\lambda_1a_0}{\lambda_2-\lambda_1} (e^{-\lambda_1t}-e^{-\lambda_2t}).$$

When the rates coincide, evaluate the integrating-factor integral directly rather than dividing by zero:

$$b=(b_0+\lambda a_0t)e^{-\lambda t}.$$

Finally $c=a_0+b_0+c_0-a-b$. The apparent denominator singularity is a failure of one formula's presentation, not absence of an equal-rate solution. Nonnegative initial amounts remain nonnegative for forward time $t\ge0$ by the integral form of each production-removal equation.

4 Environmental chemistry with periodic input

For a perfectly mixed constant-volume compartment, let $c$ be concentration and assume

$$c'+kc=u_0+u_1\cos\omega t,\qquad k>0,\quad\omega>0.$$

The forcing has concentration-per-time units, and $k$ has inverse-time units. An integrating factor or coefficient comparison gives periodic particular response

$$c_p(t)=\frac{u_0}k+\frac{u_1}{k^2+\omega^2}(k\cos\omega t+\omega\sin\omega t).$$

The full solution is $c=c_p+(c_0-c_p(0))e^{-kt}$. Its oscillation amplitude is $|u_1|/\sqrt{k^2+\omega^2}$: rapid forcing is attenuated relative to a slowly varying input in this model. If $u_0\ge|u_1|$ and $c_0\ge0$, the integral solution is nonnegative because its forcing and exponential factors are nonnegative. A sinusoidal trial with negative forcing need not model a nonnegative chemical input.

5 Ecology and a feasibility threshold

A logistic population with constant harvesting is

$$N'=rN(1-N/K)-H,\qquad r,K>0,\quad H\ge0.$$

Completing the square gives $N'=rK/4-H-(r/K)(N-K/2)^2$. When $H<rK/4$, the equilibria are

$$N_\pm=\frac K2\left(1\pm\sqrt{1-\frac{4H}{rK}}\right).$$

The rate derivative is $r(1-2N/K)$, positive at $N_-$ and negative at $N_+$ when the roots are distinct. The phase line therefore makes the lower equilibrium unstable and the upper locally attracting. At $H=rK/4$, the repeated equilibrium needs a separate one-sided analysis; the distinct-root sign argument no longer applies.

If $H>rK/4$, the rate is negative at every nonnegative population. In fact $N'\le-(H-rK/4)$, so the mathematical trajectory reaches zero no later than $N_0/(H-rK/4)$ if it begins positive. Continuing it into negative populations is algebraically possible but physically meaningless. The model must stop or change at that boundary. A parameter threshold within an assumed model is not a validated ecological harvesting rule.

6 A public-health compartment model as an ODE example

In a closed theoretical susceptible-infectious-recovered model,

$$S'=-\beta SI/N,\quad I'=\beta SI/N-\gamma I,\quad R'=\gamma I,$$

with $\beta,\gamma>0$ and fixed positive total $N=S+I+R>0$. Adding the equations proves conservation. Nonnegative initial compartments remain nonnegative: the first two equations give exponential representations for $S$ and $I$, and $R$ has nonnegative production. Thus total conservation also bounds all three compartments.

During a regime where depletion of $S$ is neglected, replacing it by $S_0$ gives the linear approximation

$$I(t)\approx I_0\exp[(\beta S_0/N-\gamma)t].$$

This explains a threshold in that simplified early-stage equation and shows how a nonlinear coupled system connects to the linear methods studied earlier. It is not a forecast for an actual disease: the approximation, parameter interpretation, contact structure and data validity would all require separate evidence. No treatment or personal health decision follows from these invented calculations.

The repeated threshold and finite-time continuation in the model

At $H=rK/4$, put $z=N-K/2$ and $a=r/K>0$. Then $z'=-az^2$. The equilibrium is $z=0$; for $z_0\ne0$, direct separation and substitution give $z(t)=z_0/(1+az_0t)$ on the interval containing zero where the denominator does not vanish. If $z_0>0$, it decreases to zero for all forward time. If $z_0<0$, it decreases away from zero; the physical solution stops when $N=0$, before the negative-population pole. Thus the threshold equilibrium attracts from above and repels from below. It is not a two-sided locally attracting equilibrium.

For distinct roots the factorization $N'=-(r/K)(N-N_-)(N-N_+)$ gives the sign on every interval. Local uniqueness prevents crossing an equilibrium. Solutions starting sufficiently near $N_+$ are monotone and trapped between their initial value and $N_+$; their limit must be an equilibrium, since a nonzero limiting rate would keep their derivative bounded away from zero and contradict convergence. Thus they tend to $N_+$. Below or above $N_-$ the rate points away, proving instability, with feasibility understood on $N\ge0$.

The continuation step for harvesting is explicit as well. A trajectory staying in a compact population interval has a bounded polynomial rate, say $|N'|\le M$, so $|N(t)-N(s)|\le M|t-s|$. At a finite time endpoint it has a real limit in that interval. Local existence at the limit and uniqueness join it to a continuing solution. Consequently the trapped branches near $N_+$ exist for all forward time before taking their infinite-time limits. If $H>rK/4$, then before hitting zero $0\le N\le N_0$, a compact interval; no earlier finite endpoint can obstruct continuation. Integrating $N'\le-(H-rK/4)$ now proves that zero is reached no later than $N_0/(H-rK/4)$, since otherwise the continued nonnegative solution would become negative by that time.

For completeness of the theoretical compartment example, treat $N>0$ as the fixed initial total parameter. Its polynomial vector field in $(S,I,R)$ is locally Lipschitz on finite boxes, so the proved finite-dimensional theorem gives a local solution. On any interval of existence,

$$S(t)=S_0\exp\left(-\int_0^t\beta I(s)/N\,ds\right),\quad I(t)=I_0\exp\left(\int_0^t[\beta S(s)/N-\gamma]ds\right),$$

and $R=R_0+\int_0^t\gamma I(s)ds$. Hence forward nonnegativity and the conserved total keep the state in the compact simplex $S,I,R\ge0$, $S+I+R=N$. A finite forward endpoint cannot obstruct continuation: the vector field is bounded on a fixed compact neighborhood of that simplex, so the solution is Lipschitz in time and has a limit there at the endpoint. Apply the local theorem at that limit and join the solutions using uniqueness. This proves global forward existence for this specified model; the early constant-susceptible approximation remains a separate assumption.

7 Graded practice

E1 Decay. If remaining parent amount is one-quarter of its initial amount under the closed decay model, express time in half-lives.

E2 Reaction chain. For equal rates $\lambda=1$, $a_0=1$, $b_0=0$, find $b(t)$ and its maximum for $t\ge0$.

E3 Periodic input. For $k=\omega=1$, $u_0=2$, $u_1=1$, find the periodic concentration and amplitude.

E4 Harvesting. For the invented values $r=2$, $K=100$, $H=32$, find equilibria and their local stability.

E5 Linear approximation. With hypothetical $\beta=0.4$, $\gamma=0.2$ and $S_0/N=0.9$ in consistent time units, find the linearized growth rate. What does the number fail to establish?

8 Full solutions

S1. $P/P_0=1/4=2^{-2}$, so $t=2H$. This answer assumes the stated decay law and a known initial fraction; it does not independently verify geological closure.

S2. $b=te^{-t}$. Its derivative is $(1-t)e^{-t}$, positive before 1 and negative after 1, so its global maximum on $t\ge0$ is $e^{-1}$ at $t=1$.

S3. $c_p=2+(\cos t+\sin t)/2$, with amplitude $1/\sqrt2$. Any initial mismatch adds a decaying multiple of $e^{-t}$; it does not change the limiting periodic amplitude.

S4. $rK/4=50$ and the square-root factor is $\sqrt{1-128/200}=0.6$. Thus equilibria are 20 and 80. Rate derivatives are $1.2$ and $-1.2$, so 20 is unstable and 80 locally attracting. These are model results, not an empirical policy recommendation.

S5. The rate is $0.4(0.9)-0.2=0.16$ per time unit. It predicts growth only within the specified linearized model. It neither estimates real parameters nor establishes the validity or duration of the constant-susceptible approximation.

Summary

Balance equations connect a common set of ODE methods to several domains. Conserved totals, integrating factors, equal-rate limits, positivity and feasible boundaries provide independent checks. A valid mathematical solution remains conditional on the process assumptions and should not be mistaken for validated geological, ecological or clinical inference.